Saint Petersburg State University
Department of mathematical game theory and statistical decisions
Disease Propagate Mechanism and Dynamic
Applied Mathematics and Informatics
Master’s Program Game Theory and Operations Research
Dr. Sc.(PhD), Professor
Petrosjan L. A.
Research background and introduction about evolutionary game theory . . . . .
Main contents and section arrangement . . . . . . . . . . . . . . . . . . . . . . .
3 Investigation about cancer therapy
Description of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis of applications about cancer therapies . . . . . . . . . . . . . . . . . . .
Simulation about three basic regimens of cancer therapies . . . . . . . . .
Evolutionary double-bind therapy . . . . . . . . . . . . . . . . . . . . . .
De novo environmental resistance therapy . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Investigation about the impact of population vaccination behaviors
The strategy of children diseases intentions vaccination . . . . . . . . . . . . . .
Description of mathematical models . . . . . . . . . . . . . . . . . . . . .
Nash Equilibrium in the prevention of diseases . . . . . . . . . . . . . . .
Model analysis on Nash Equilibrium . . . . . . . . . . . . . . . . . . . .
Investigation on Vaccination Behavior in Social Networks . . . . . . . . . . . . .
Description of the SIR model and its methods . . . . . . . . . . . . . . .
Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
According to the complexity and uncertainty of situations in the real
world, compared to traditional game theory, evolutionary game theory is more
applicable to practical problems, such as biological problems or economic issues.
Researches on evolutionary game give new ideas to the investigation of complex
With the rapid development of science and technology, nowadays diseases
is still an important problem threatening human survival. Based on the basic framework of evolutionary game theory, in this paper we survey and study
the application of evolutionary game theory in terms of occurrence mechanisms
and propagation mechanisms of disease. We analyze the application respectively from two aspects, which are cancer treatment program and controls of
epidemic spread by population vaccination. Firstly, in cancer treatment, we
mainly analyze evolutionary game results of cancer cells under different treatment regimens and discuss rational application of the ”adaptive therapy” in the
future cancer therapy. Secondly, in the population vaccination behavior, we analyze issues about dominant population strategy in different factors of operated
policy, which was investigated by researchers in different literatures, and discuss the application about evolutionary game theory in the control of epidemic
Key Words: evolutionary game theory, cancer adaptive therapy, vaccination strategy, G-function, strategy dominance, SIR model.
Research background and introduction about evolutionary game
Nowadays, people are being tortured by the unprecedented number of
diseases, but in the future this situation will continue. It also resulting difficult
for us to deal with health issues. Such as the flu crisis, although scientists have
struggle with the flu virus for centuries and influenza vaccine technology has very
mature, the annual flu season can cause a certain degree of social instability.
There is no way to predict that in the future which kind of the specific pathogens
will become a health risk. But according to the current disease has occurred,
one of the important challenges, which need to be solved by scientists, is that
people have able to quickly and efficiently make a control plan and to prevent
Malignant tumor is one of the major diseases seriously, which threat human health. Cancer, cardiovascular diseases and accidents constitute three major causes of death in the world. As a result, the world health organization
(WHO) and health departments in every country’s government all list it as a
priority that it is imperative to conquer cancer . Cancer known as malignant
tumor is harmful to human health, which is caused by a damaged proliferation
mechanism of cancerous cells leading to the cells proliferation at an abnormal
rate like infinite proliferation and damages other parts of tissues in human body
by these abnormal cancerous cells. It is different with the consistent and stable
metabolism of normal cells that cancer cells, deriving a result of a related gene
mutation, are characterized by a damaged metabolic function which causes the
proliferation in infinite circles with a high rate. Therefore, according to such
feature on one hand genes of tumor cells are easily mutable that promotes resistances from anti-cancer drugs, and on the other hand tumor cells easily transfer
to other body organs that damages the other function of human body .
In the earlier time of conventional cancer therapy like chemotherapy, a
main idea of the chemotherapy is an application of a higher dose of single
chemotherapy drug, which depends on the limit of the human body tolerance
that in order to obtain the largest reduction of cancer cells as much as possible. In the earlier stage of the therapeutic process, the chemotherapy indeed
makes a great impact on the population of tumor cells that rapidly kills a large
amount of cancer cells. However the chemotherapy also has much deficiencies.
On one hand, as the chemotherapy proceeding, the treatment with high dose of
anti-cancer drugs can seriously impact on normal cells of the body and at the
same time damage other normal cells of human organs, and on the other hand
even the dose is high enough it is impossible that cancer cells can be destroyed
or eliminated. There indeed exists some surviving cancer cells. Because of
the rapid proliferation and the mutation of cancer cells, these surviving cancer
cells with prominent evolutionary advantages can enlarge the cancer population again by the heredity of resistances. Ultimately it results in the failure of
treatment. Therefore, the most difficult point of the cancer therapy is that in
the infinite proliferation the surviving tumor cells can easily obtain the drug
resistance with the anti-cancer drugs making tumor cells out of control. Then
coming with the impact of tumor cells proliferation which gets out of control,
remaining tumor cells grow rapidly in a very short period of time and ultimately
form a population of resistances.
Based on such reasons, tumor cells are very difficult to get an effective
control from the current therapy. Jessica J. C. and Robert A. Gatenby (2011)
 present a therapy called ”the adaptive therapy” using a simulation model
for cancer treatment concerning the evolutionary game theory. General cancer
treatments have indeed remarkable effects on cancer cells control while treated
on the early stage. Early treatment can make tumor population decline or
stabilize as well. However for the malignant tumor cells have strong evolution
ability, the traditional cancer treatment could quickly produce adaptability,
namely drug resistance. According to the adaption and changes of the tumor in
the whole treatment process, the tumor treatment must be administered basing
on the evolutionary system of dynamic change. The researches apply a tool of
the evolutionary method, namely a fitness generating function (G-function), to
predict the ability of the evolution of resistant cancer cells. This function means
the response respectively to evolutionary adaptation and micro environmental
conditions and response to various therapeutic strategies. With this function,
the population change its characteristics over time.
The other investigation in this research is about vaccination behaviors in
population. Vaccine inoculation is recognized as an outstanding achievement in
biomedical areas and public health areas in the 20th century. Generally as the
disease, the preventing method of vaccines is successfully produced. Smallpox
was completely eradicated in the world in the past century. Due to the implementation of the planned prophylactic vaccination . It is the same for other
diseases, as well as other vaccine can prevent the epidemic and take it under
control. However, in terms of disease control, even with related disease vaccine,
if the vaccination rate is low, the control of disease will not still be realized.
Therefore vaccination is the key point of the study.
Since vaccination has some risks and costs, people would estimate risks
in disease transmission and vaccination. If the risk of vaccination is high or
the possibility of disease infection is small, then people do not want to uptake
vaccinates. Such laissez-faire attitude will lead to the reappear of disease prevalence. Such attitude in turn influences people to reconsider whether consider
Moreover, in the face of vaccination issues, people usually tend to be in
a game dilemma. When some part of people agree to uptake vaccinates, the
possibility of infection for unvaccination people is greatly reduced due to all
neighbours of unvaccination people are vaccinated. Therefore it is not necessary to participate in vaccination. But if anyone in population decide their
behaviours like this, there will not have anyone to uptake vaccinates, in turn
the situation accelerates the spread of diseases, which can cause a new round
of outbreaks of infectious diseases. There is another aspect in the dilemma. In
the case of limited resources, if people fears the presence of the disease, it can
cause a situation that people rush to buy and stock a large number of drugs.
This situation leads to the price gouging by merchants and then causes more
fears and makes the social unrest again. Herd immunity is the most fundamental method to reduce the burden of social health, but on the other hand
it will also encourage more people to take ”free-rider” behavior . Therefore
under the voluntary immunization policy, population vaccination rate will be
much lower than the optimal public health vaccination levels. We analyze different approaches applied game theory methods from new perspective such as
dynamic, adaptive and self-regulating to explore the structural characteristics
of the population and how does the limited rationality of individuals impact an
individual’s immune behavior.
Then the investigations by Chris Bauch and Feng Fu discusses the underlying mechanisms about switching population immunization rates from a low
level of ”Nash equilibrium” to global optimization state in the theoretical level
state. According to this research they applied a method, which is a new immune method to prevent the spreading of diseases based on evolutionary game
theory. It also explores mutual coupling and co-evolution of the spread of infectious diseases and immunity behavior from a new angle. Meanwhile it studies a
physical phenomena and laws, which are reflected in the true complexity of the
system, to help people to understand movement patterns in a complex system
and provides a theoretical reference for measuring policy issues about the prevention and the control of infectious diseases and provides a better way of how
does vaccines distribute in a real social system .
Evolutionary game theory is a new application and a different area with
game theory, which is a combination of analysis of game theory and the dynamic
process of evolution . It is obviously that they are different at the first glance.
The classical game theory often emphasizes one kind of the static equilibrium or
the comparative static equilibrium for the related participants in game . But
in the real world, most of the real game processes for the participants usually
are the dynamic processes, that is to say, such kind of game is a game with a
long process. During the long process, the participants will change and also the
game purpose will change, and these changes will result in the change of the
game system, so as to affect the game. According to these dynamic changes in
game, the evolutionary game theory conforms more to the objective real world
than the game theory. Therefore it is becoming an important method that using
evolutionary game method to study the dynamic development of the practical
problems . We can use such method to solve problems in many diseases
and find an optimal application for this area.
Evolutionary game theory comes from Darwin’s theory of biological evolution . In fact, the earlier idea of evolutionary game could be traced back to
the interpretation of the concept of equilibrium by John Nash. Nash proposed
in his doctoral thesis that the concept of the equilibrium exist two kinds of
explanation: one is the interpretation of the rationalism, another is ”the mass
action interpretation” (Nash, 1950) . The former one of the interpretation
comes from the classic game theory, and another one in fact is the way of the
interpretation from the evolutionary game theory. Nash said that the realization of the equilibrium did not have to assume that all of the participants had
the knowledge of the game structure. The ability to reason a complex process.
He assumed that the participants could accumulate the related experience and
information through various kinds of pure strategies with the comparative advantages in the decision-making (for example, learning a strategy with gains
high). After a period of strategy adjustment, it also could reach to equilibrium. Nash also expounded the basic analysis of the structure of ”mass action”.
Firstly, assuming that each participant took a pure strategy in game, the individuals of the common pure strategy treated as a group, and all the groups
made up a larger population. Next, he assumed that every participate in the
game was randomly selected from the general population. Then, he assumed
that the frequency of strategies with a higher payoff in the population could
increase, on the contrary, the frequency of strategies with a lower payoff in the
population could reduce. Finally, on the basis of these assumptions, a frequency
distribution of the pure strategy equilibrium could be worked out (sometimes
also seemed as the mixed strategy equilibrium). Therefore, although Nash did
not clearly put forward the term of ”the evolutionary game theory”, the thought
of ”mass action” by Nash actually covered the connotation of the evolutionary
game. As the evolutionary game theory are closer to the reality, it is widely
used in biology, sociology, economics and other fields.
Besides investigating the evolutionary process, the evolutionary game also
attaches great importance to the analysis of the equilibrium, and usually the
purpose of the investigation of the evolutionary process is going to find the equilibrium. Due to the replication dynamic is nonlinear, it is difficult to work out
the unique solution. Therefore, the purpose of the evolutionary game switches
from solving the equilibrium to take an analysis on the stability of the equilibrium. The basic concept of the theory is Evolutionarily Stable Strategy (ESS).
The evolutionary stable strategy was proposed by Maynard Smith and Price
(1973) . The core idea is that, if an existing strategy is an evolutionary
stable strategy, there must be a positive invasion of the obstacle, which making
sure the existing strategy obtain a higher payoff than the payoff of the mutation
strategy when the frequency of the mutation strategy is lower than this obstacle.
Main contents and section arrangement
This paper mainly includes two sections. The first section describes the
effective treatments for cancer research and analyzes different cancer therapies
and results, which are proposed in the literature. The results show that compared with single drug model namely monotherapy similar to that of tradition
cancer treatments, many drugs for cancer therapy namely multidrug therapy
of cancer cell population density control is more effective, that can keep the
population of cancer cells to a very low level. These methods can help analyze
and research the combined drug therapy by scientists in the future. The second
section analyzes in the complex social network the relationship between individual vaccination behavior and populations vaccination level, and the relationship
between vaccination level and scope of an epidemic. The conclusions of literature provide a basis for specifying the vaccine policy. For different social groups
there should have different vaccination programs. In this way it can guarantee
the effectiveness of the vaccine for the entire population.
Investigation about cancer therapy
According to drawbacks of standard tumor therapies such as chemother-
apy and radiotherapy where cancer cells usually easily obtain drug resistances,
Jessica J. Cunningham, Robert A. Gatenby and Joel S. Brown. investigate a
new cancer therapy named ”adaptive therapy” in ”evolutionary dynamics in
cancer therapy”  by using a certain dose of an anti-cancer drug, which ensures
the drug concentration holding on a reasonable level in an internal environment, which is determined by simulating a develop model of cancer cells through
the application of proliferation model of cancer cells, which is constructed by
a function called G-function, to inhibit the proliferation of cancer cells and to
reduce the tumor size. The ”adaptive therapy” is a sustainable treatment, so as
to control the overall number level of cancer cells keep in a small range. Such
”adaptive therapy” can not only control the development of malignant tumor on
the maximum but also reduce the harm of other normal cells in human body as
low as possible because of anti-cancer drugs, which is controlled in a reasonable
range determined by the model. According to related population parameters
of tumor cells are different in different therapies, the model of tumor cells with
the evolution of the tumor cells adaption and under the micro environment in
proliferation processes of tumor cells is constructed by G-function. On the basis
of such model, authors tested a multidrug therapy, which applied two different
treatments in such program and proved that the multidrug therapy was a more
effective therapy with evolutionary strategies than the monotherapy, which was
based on a single medicine. Analyzing the process, we get a result that an important point of such multidrug therapy is selecting a right time point for using
the second drug, at which tumor cells become vulnerable to the second drug
that not quickly produces resistances to the second drug due. The effectiveness
of the resistance reduction of tumor cells to the second drug depends on the
impact of a drug in the first treatment. A point of the second drug takes advantage of the vulnerability of tumor cells for different types of drugs. Only in
this way resistances of tumor cells can be reduced by the different treatments.
Such process of the multidrug therapy is an optimal approach, which can control
resistance effects of cancer cells by therapies which are produced by the current
process of therapies and continually and effectively inhibit the proliferations of
cancer cells. That would change how we approach medicine.
Jessica J. Cunningham, Robert A. Gatenby, and Joel S. Brown (2011) 
applied G-function to construct a basic model. The exact mathematical model
are useful to simulate the efficiency of cancer therapy and the realization of
cancer therapy. G-function is applied to most biological situations describing
the fitness of a population (here is constructed by Darwinian Theory) . For
a biological population, overall individuals in a population have a same strategy set, which includes different strategies selected by these individuals in the
population. The fitness of each individual is defined by G-function and an evolutionary game theory. Different with the traditional game theory, evolutionary
experiences consider not only one individual but a population and that behaviors of all individuals cannot depend on a rational thought. With evolutionary
game theory we can describe a dynamic process how the density of population
changes over time under some evolutionary behaviors. Therefore the model
could be applied to simulate the population change of tumor cells over time
under some evolutionary resistances.
Description of methods
The two basic ideas of evolutionary game theory are the evolution the-
ory in Darwinian Theory and Lamarckisms biological genetic theory . In
an evolution process, only a group with a higher payoff (reproductive survival
rate) can survive from competitions, and in opposite a lower payoff group can
be eliminated from competitions. This process is namely called ”evolution”.
According to analysis of biological evolution, behaviors of plants and animals
are considered as an instinct of an intuition without thinking. Basing on the
principle of the evolution, for a population behaviors of individuals finally can
tend to Nash equilibrium. The combination of evolution theory and game theory gives a possibility to explain the impact of individuals’ behaviors in the
process of human groups or any biological groups. According to researches of
all behaviors of participants in a group of evolutionary game, the unique condition that regardless of the demand for rational can make it more close to the
biological activity of reality . Nash equilibrium (NS) is a definition of game
theory. It is an optimal strategy called Nash equilibrium if and only if a payoff
of a NS strategy is not less than other strategies in game. To the evolutionary
game theory, a similar concept of Nash equilibrium is evolutionary stable .
The first definition of evolutionarily stable strategy (ESS) was given by
Maynard Smith(1974) : ESS is a strategy such that, if all members of a
population adopt it, then no mutant strategy could invade the population under
the influence of natural selection.
Definition 1. We say that x is an evolutionarily stable strategy (ESS) if for
all y 6= x, there exists some ∈ (0, 1), which may depend on y, such that for all
∈ (0, )
u(x, y + (1 − )x) > u(y, y + (1 − )x).
That is, x is ESS if, after mutation, non-mutants are more successful than
mutants, in that case mutants cannot invade the system and will eventually get
extinct. We refer to as the invasion barrier, the maximum rate of mutants,
against which x is resistant.
G-function can be constructed by three steps. The first step is to select
an appropriate ecological model for current population dynamics. The model
may be taken for a single population or species. It may be a life-history model
with different age and stage classes, or it may be a model of population interactions that includes growth equations for competitors, resources, predators,
etc. The second step is to set strategies and strategy sets associated with the
population, species or communities under consideration. The strategy set may
be continuous or discrete and determined from hypotheses concerning genetic,
developmental, physiological and physical constraints on the set of evolutionarily feasible strategies. The third step is create the G-function by hypothesizing
how the individuals strategy, v, as well as all strategies in the population, u,
influences the values of parameters in the ecological models of population dynamics. As soon as key parameters of a population model become functions of
v, u, x, y, the ecological model becomes a G-function .
Definition 2. (G-function)A function G(v, u, x) is a fitness generating function
(G-function) for the population dynamics if and only if
G(v, u, x)|v=ui = Hi (u, x), i = 1, ..., ns .
Where u and x in G are exactly the same vectors as in Hi . The population
has ns different species. The population size is x = [x1 , ..., xns ]. This is the Gfunction for the simplest problem with scalar strategies .
The fitness generating function G(v, u, x) eq. 2 determines the expected fitness of an individual using a strategy v as a function of its biotic environment that includes the extant strategies found among the different species
within the population u = [u1 , ..., uns ], and the different population sizes xi ,
x = [x1 , ..., xns ]. Because there are ns different strategies, it is reasonable to assume that the fitness of individual i is the sum of the expected payoffs of playing
ui against all strategies in proportion to their numbers in the population.
In the article, authors applied a G-function like this :
K(v) − i x
G(v, u, x) = r
To tumor cells, a real meaning of the fitness relates to the per capita
growth rate. For vectors u, v, they present all phenotypic strategies currently
of each tumor cells. x is the population size of those cells. The cells growth
rate r is the nature rate without limitations like other affections. The carrying
capacity of the tumor cell in population, K, references its ability to survive.
And µ is the cell mortality or the suppression hold in a current treatment.
In reality it is possible to treat a cancer treatment process as an ecological
game process. Therefore tumor cells are treated as a predator and the treatment
is treated as a preying one. It describes a behavior among one tumor cell
and other tumor cells. When two cells select different strategies, they would
produce a different growth suppression between them because of competitions
in resources and space. As to different groups of population, different strategies
of cells will lead to different per capita growth rate.
Therefore the fitness function of one tumor cell is determined by Gfunction with its focal strategy v and other tumor cells’ strategies u ∈ u and
the population size of tumor cells x. For a group which selects the strategy ui
with the population proportion of the current group xi , the evolutionary dynamic including the population dynamics and the strategy dynamics can get
= xi G|v=ui .
xi represents the proportion of those cells which selects the strategy v = ui
in the total population of the evolutionary system xi ∈ x = [x1 , ..., x2 ]; v = ui ∈
u = [u1 , ..., ui ]. The focal cell with its own resistance strategy v = ui is determined by other cells. In eq. 4 and eq. 5,
with time t;
presents the population dynamics
presents the strategy dynamics and shows the resistance trend
of tumor cells with time t. Here S scales the speed of evolutionary change. In
evolutionary dynamics a key factor of adaptation to any therapies is the phenotypic cost of resistance. This gives a model of how does tumor cell adjust
to a population suppressing treatment . It obviously obtains the change of
tumor cells’ adaptation from these functions.
) − µx.
It is applied to measure the population growth rate of tumor cells. Here the
tumor cells population density, x, is a way to measure the change of population
In the eq. 6, tumor cells’ growth rate r is a given number which is known
without limitations like other affections. µ and K are necessary to be measured.
The parameter µ is effected by three aspects of ecological predation. The three
aspects of ecological predation in tumor cells are respectively the encounter rate
of predators (also is chemotherapy dosage attributes), the lethality of predators
without vigilance of preys (also is the resistance of tumor cells to a chemotherapy), and resistances offered by preys (also means resistances of tumor cells).
Those three aspects make up the tumor cell mortality or suppression.
k + be + bp v
Here m is the number of the dose of the anti-cancer medicine in the current
therapy. And k is the mean of the phenotypic resistance without considering
any factors. be represents the resistance from environmental factors. bp is the
mean of the effectiveness of resistances, which act as promotion strategies. v is
the effectiveness of evolutionary strategies. Therefore bp · v indicates the impact
of evolutionary behaviors. Integrating the mortality or the suppression of tumor
cells with all relevant parameters eq. 7, µ indicates the effectiveness of drugs
and also the resistance of tumor cells.
The carrying capacity of cells K is relative to:
K = Kmax exp( 2 ).
The parameter K represents the cost of resistances through the population
carrying capacity. And v is the mean of resistance strategy. And bK represents
the resistance penalty of tumor cells. In a prior process, without any treatments
v = 0, therefore K = Kmax . Over the time of treatment, v will increase and K
will decrease. σK
is the resistance penalty of tumor cells, which is related to the
range of resources. A small value of σK
means lacking resistances and results in
a large penalty.
Analysis of applications about cancer therapies
In this article authors apply the tool of G-function to three basic therapies.
They are monotherapy, multidrug regimens with one response and multidrug
regimens with two responses. The monotherapy is a treatment with a single
drug. The multidrug regimen is a treatment with two process of taking same or
Simulation about three basic regimens of cancer therapies
According to different responses to drugs, authors simulate three regimens
respectively by a single response with a single drug, different responses with one
drugs (different doses with different drugs) and two different responses with two
With same parameters like a same tumor size and same characteristics of
the common group of tumor cells, these parameters set as follows: a constant
size of tumor cells x = 100, a constant carrying capacity Kmax = 100, a constant normal tumor cells growth rate r = 0.1, the resistance penalty of tumor
= 5, a constant effectiveness of the resistance bp = 5, the phenotypic
resistance without considering any factors k = 0.1 and without the resistance
from environmental factors be = 0. In order to get different responses to one
drug, this process settles the single drug with different dosages m1 = 0.1 and
m2 = 0.12 by increasing drug regimens.
In order to test a multidrug therapy, G-function can change the form of
different dose of drugs applied by eq. 6, eq. 7 and eq. 8.
Figure 1: Comparison about evolutionary effects on tumor population.
K(v) − i x
G(v, u.x) = r
− µ1 (v1 ) − µ2 (v2 ).
The trend in fig. 1 shows results of three basic experiments. The vertical
axis indicates tumor population density with three different basic treatments.
The full line represents the population trend with evolutionary abilities. The
dotted line represents the population trend without evolutionary abilities. From
the dotted curve, the population density are obviously drop to zero in a very
short time unit. Differently, processes about the multidrug therapy respectively
with one response and two responses all make the population density curve more
strictly drop than the treatment process with the monotherapy requiring one
single response. Firstly, let us consider cases with evolutionary abilities. In
monotherapy requiring single response with the time point of the curve mostly
close to t = 70, tumor population drops to x = 0. The multidrug therapy
requiring both one response and two responses similarly make the tumor cells
population decreasing to x = 0 in time unit t = 70. The result indicates that the
multidrug therapy has a better effect on the control on the population density
than the monotherapy. Secondly, let us consider cases with evolutionary abilities
as these full lines. These full lines show the results of these treatments processes
with the evolutionary ability in phenotypic resistances. Different from the nonevolution process, the lowest points of population density in these treatments
with evolution are identically far from zero. It means that these evolutionary
abilities advance the survival rate of tumor cells. The population density curve
of the treatment process with the monotherapy with single response has a lowest
point x = 60 at t = 20 and finally reaches to a stable state x = 73 after 130
units of time. The curve of the treatment process with the multidrug therapy
with one response has a lowest point x = 38 at t = 20 and then finally reaches to
a stable state x = 57 after 150 units of time. The curve of the treatments with
multidrug therapy requiring one response has a lowest point x = 23 at t = 25
and then finally reaches to a stable state x = 38 after 150 units of time. These
results indicate that effects of the monotherapy is the worst and effects of the
multidrug therapy requiring two responses are better than the therapy requiring
one response. However all these treatments are not yet able to eradicate the
tumor population density.
According to graphs above, the multidrug therapy requiring two responses
has most high reduction on tumor population density. Under the situation that
tumor cells have evolutionary abilities, the population density will increase over
time. Therefore authors view a further model. Different with the previous
model, the frequency of taking drugs in the model is taking drugs at t = 0 and
then off at t = 50. The cycle of the process is t = 100. The treatment cycle is
used to check how resistances of tumor cells impact the population density of
Top axis: the fractional resistance, vi , to the one or two treatments given
Figure 2: Evolutionary effects on tumor population with time units.
over time; bottom axis: the population density of the tumor. The time axis
means periods of treatment (50 time units) that alternate with periods of no
treatment (50 time units). It means treating for 50 time units and abandoning
for 50 time units.
In the top axis, it shows that the fractional resistance v of tumor cells
to different treatments has a progressive increase over time. The bottom axis
shows the population density of the tumor cells. Initially all these treatments
create obvious reduction in the population density of tumor cells, but the value
of the reduction drops during the first 50 time in one cycle t = 100 according
to the developing resistances of tumor cells. The monotherapy requiring single
response makes the population density drop to x = 58 at the first 50 time units.
Without using drugs in the next relaxation period the population density curve
has a quick recovery closing to x = 99 at the end of the cycle t = 100. The curve
of treatments in the multidrug therapy requiring one response drops to a stable
state x = 43 at t = 20 until the end of the first 50 units of time. In the next
time interval the curve about lacking drugs has a quick increasing from x = 43
to x = 98. Therefore in a 100 time interval the treatment causes results that
the tumor population recover to x = 98. To the multidrug therapy requiring
two responses the population density in a treatment course drops x = 30 and
maintains a stable state during the 50 time interval. In a next 50 time interval
the curve without drugs quickly recovers to x = 85.
From different curves in fig. 2, the trend curve of treatment processes in
the multidrug therapy requiring single response is higher than treatment courses
in the monotherapy. It indicates that the treatment process with a high dose
of drugs is completely ineffective to tumor cells. During each treatment cycle,
the population size is decreasing under the treatment and recovers without
any treatments applied. On the other hand, evolutionary resistance abilities
of tumor cells are directing towards increasing. With enough cycles, tumor
resistances will become completely overall. Therefore there is no effective result
in taking about eradicating tumor cells.
Evolutionary double-bind therapy
In this article , authors putted forward a more strategic therapy. Evolutionary strategy theory shows that predators usually obtain adaptabilities for
certain prey behaviors by the preys during the evolution process. But in most
cases this evolutionary process will hinder the obtaining to the adaptability of
preys by different feeding behaviors with other predators. From another point
of the view, different behaviors of different predators would making the feeding
behavior to be more effective through weakening prey adaptabilities. Concerning the interrelation between different drugs therapies and cancer cells, it is a
kind of a therapy that the implement of one treatment can reduce resistances
of cancer cells to other different treatments. According to this feature, authors
design one therapy that two different treatments alternatingly switch the administration by a periodic way. Under the first stage that applies one treatment
it is realized that cancer cells are killed mostly and at the same time the treatment drives cells to obtain an evolutionary response, making the effect of cancer
cells’ resistances to the second treatment attenuating. It is benefit for the second stage that using the second treatment with a better effect, and at the same
time weakening cancer cells’ resistances to the first treatment, which produces
in the first stage.
According to this scheme, there is new examination with the model by
authors. The main idea of the part is ”predator facilitation”. These processes
in the double bind therapy includes two parts, in which weak cells will be easily
attacked once the second drug is applied. It is said that the evolutionary one
will be targeted by the second therapy. Different with the previous parameter µ,
here authors set µ1 and µ2 presenting the proliferation suppressions by different
treatments which used in each process.
k1 + (1 − v2 )(bp1 v1 )
k2 + (1 − v1 )(bp2 v2 )
Here m1 and m2 are different doses of drugs in each treatment. v1 is tumor
cells’ evolutionary strategy by m1 drug, and v2 is tumor cells’ evolutionary
strategy by m2 drug. bp1 and bp1 represent the effectiveness of resistances with
the first and the second treatment. And G-function could be written as a new
Figure 3: Double bind therapy.
K(v) − i x
G(v, u, x) = r
− µ1 (v1 ) − µ2 (v2 ).
The top axis represents the fractional resistance vi in the double bind
therapy over time; bottom axis represents the population density of the tumor
In fig. 3 the curve shows the fraction of resistance and the population
density by the new function. From curves above, in the double-bind therapy
when using the first treatment the fraction resistance of cancer cells for the first
treatment are increasing from the initial value vt=0 during the first treatment
process. In the meantime the fraction resistance of cancer cells for the second
treatment is decreasing as the first treatment process going. At the right point in
time (here authors set 50 time interval in one treatment process), the multidrug
therapy switches to the second treatment. In the second process the efficacy of
the second treatment becomes more effective, and at the same time phenotypic
resistances of cancer cells to the second treatments are increasing rapidly, but
phenotypic resistances of cancer cells for the first treatment are consequently
reducing over time. But the population density of cancer cells during the treatment has a trend of declining. Even after a period of time with application of
the double-bind therapy, the population density of cancer cells would be a low
population density stable within a certain range and not continually decrease
any more. But cancer cells will continually have resistances in switching two
kinds of evolution reactions, until resources exhaustion.
Authors gave an experiment on a recently research about a new therapy
with p53 cancer vaccine combined with chemotherapy. To a certain extent,
it also supports the validity of the double-bind therapy. The p53 vaccine is
one kind of cancer vaccine. The vaccine could stimulate the body’s immune
response. On one hand, the immune response could suppress tumor cells, and
on the other hand, such response is a degree of resulting cancer cells to become
vulnerable. In this research, it discussed the immunological research and the
clinical effect about the combination therapy with p53 cancer vaccines for lung
cancer patients. According to the immunological research data, p53 cancer
vaccine has no significant clinical response for controlling the population of
Top axis: the fractional resistance vi to the two treatments given over
time; bottom axis: the population density of the tumor.
It is visible to see from the fig. 4 that in the p53 vaccine stage it does
not produced a significant response but promoting the effect of the following
Figure 4: p53 vaccine and subsequent chemotherapy.
chemotherapy. The fraction of resistance in p53 vaccine is increasing after taken.
In the curve of population density, there drops less than 15 during the vaccine
treatment. It indicates that the vaccine has a very small effect to the population
density. The methods for application is choosing a right point in time. The
vaccine process stops at t = 100 and the chemotherapy treatment is applied
at t = 150. The subsequent process of the chemotherapy treatment produces
a significant response as the rate of 62%. Compared with the historical data,
there have a significantly larger improvement of the chemotherapy rather than
the 8% response rate with no p53. From the data of the synchronized blood test
there also gets a support. It presents that the better responses to chemotherapy
certainly present in the patient who has a better immune response for p53
in the treatment process. In other words, the adaption of tumor cells in the
immune therapy drive their evolutionary resistance to become more sensitive to
the chemotherapy drugs.
Figure 5: Denovo environmental resistance.
De novo environmental resistance therapy
In the further research, the authors focus on the role of the microenvironment of tumor in the process of driving new resistance. Here microenvironment
mainly refers to the tumor vascular structure and the blood flow. There has two
aspects of the microenvironment. On the one hand, due to the reduced of the
blood flow, it makes the ischemia hypoxia. On the other hand, for drug delivery
blocking it influences the drug effect. This factors involves the parameter be in
the authors model. Keeping the value of parameters except the be in the prior
model, be will be changed from 0 to 1. With using the new coefficient of the
environmental resistance, the µ1 and µ2 in eq. 10 and eq. 11 would be written
in the new formula:
k1 + be1 + (1 − v2 )(bp1 v1 )
k2 + be2 + (1 − v1 )(bp2 v2 )
Top axis: the fractional resistance vi to the two treatments given over
time; bottom axis: the population density of the tumor.
Figure 6: De novo environmental resistance therapy.
The simulation is an application of the previous three basic treatment
which is absent of environmental factor. For such application, comparing with
no environmental resistance of cancer cells (like testicular cancer and lymphoma)
in fig. 1, such results with the impact of environment in fig. 5 are closer to the
majority of the type of tumor. With small doses of the drug treatment, cancer
cells are more likely to have serious drug resistances.
The curve in fig. 6 shows there has 44% reduction in cancer population
density with the de novo environmental therapy. It has 24% increase comparing
results of no environmental resistance. The de novo environmental therapy still
effectively inhibits the formation of cancer cells resistances and at the same time
inhibits the development of cancer cells population, which prevents the tumor
It can be seen from the simulation that this kind of cancer therapy has
a good impact of the issue in the traditional single cancer therapy, which can
quickly increase cancer cells resistances by high doses of anti-cancer drug, not
only inhibits the cancer cell population density but also makes the single drug
disabled. It plays an important role in the development of controlling cancer
cells population density, effectively inhibits the increase of the tumor and regeneration and avoids the emergence of serious drug resistances. There is no doubt
that the ”adaptive therapy” comparing with the traditional chemotherapy is
more powerful and effective, suitable for more patients and can save more lives.
Results from experiments show that even the ”adaptive therapy” makes
significant reductions in the cancer cells population density. But only keeping
the cancer population density to a lower level, it has not fundamentally solved
the problems of eradicating tumor. Moreover, factors influencing environmental
resistances in a real case are more complicated than the model. In fact, this
therapy could make the cancer cells population density keeping down to a level,
which is not ideal, due to the influence of environment resistances. With the
passage of time, resistances of cancer cells during the evolution process will
develop to a serious degree, the therapy leads to cancer cells population density
expanding with no limit, finally making a failure.
From the conclusion of ”adaptive therapy” , it can clearly show that
there has a stationary phase for the tumor population during the simulation
process of the cancer therapy. It is an advantaged way for follow-up treatment. If
making good use of this section, it may receive an unexpected effect. Also it can
be found that the cancer cells population density has a downward trend in the
time interval which has a selection in the first stage of switching two treatments.
Then it may foresee that, if three treatments, or more alternative treatments
could apply to a therapy, the cancer cells population density is possible to
reduce to a lower level and avoids continuing develop towards a high level. An
approach of the ”adaptive therapy” can also be changed flexibly to achieve better
treatment effects according to the current status of the cancer cells population
Investigation about the impact of population vaccination behaviors
Vaccination is recognized as one of the best strategy for the prevention of
diseases in the world, and has developed for more than 200 years, with benefits that vaccinates has already completely eradicated many infectious diseases,
which had a serious threat to human health during past years and had been
disappeared from public view. In a vaccination situation report published by
World Health Organization (WHO) in 2009 , it showed a status report about
the global alliance for vaccines and immunization status. The report pointed
out that immunization, as a core power, played a vital role in disease control in
order to reduce the mortality of new born children under the age of five. More
and more children have accepted vaccinations. From 2005 to 2007, more than
one hundred million children have uptake the vaccine each year. Many lives are
The strategy of children diseases intentions vaccination
Although children vaccinations obtain great achievements, there still have
some children without finishing the degree of immune regulations. After some emergences of bad cases about vaccines, people consider that it must be doubtful
that whether should necessarily take vaccinations. According to lacking popularization of information, a growing number of unsafe factors of vaccines causes
a narrow popularity of vaccinations. However, the reduction of the vaccination level possibly leads to many diseases reappear, which has already become
extinct. Some bad realities have shown the situation in history. Although prob-
lems of vaccines do exist, it is no doubt that the application of vaccines indeed
makes it realize that control and extinct many diseases. Vaccine absolutely
is the most cost-effective way for preventing diseases. In ””vaccination of the
found of games” (2004)  Chris Bauch written, there is a new model, which
combined a game theory framework with epidemic model for predicting vaccination behaviors of individuals and describing how parents decisions of their
children vaccinations, which are influenced by a judgement of risk perceptions
about vaccinations. And their vaccination behaviors will affect the expected
coverage level of vaccination.
In game theory, decision-making behaviors appears in occasions that two
or more individuals interact, the decision of each individual depends on the
prediction of this individual as for other individual behaviors. By selecting the
best decision-making behavior, the aim is to seek profit or utility maximization.
Description of mathematical models
In game theory, the selection of different strategies appears in such occasions that two or more individuals interact, and a decision of each individual
depends on the prediction of this individual as for other individual behaviors.
For selecting the most optimal strategy, the aiming point is to seek profit or
In the prediction for vaccination and vaccine coverage level, the behavior
of each individual in the group has a decisive influence on the result of population. There are two mainly aspects of factors influencing choices of individuals
vaccinations whether it is necessary in population. One aspect is the risk of
the vaccination behavior itself. Another aspect is the infection risk of disease
related the vaccine. In order to make a prediction, it is necessary to build
mathematical models and analyze them. For simplifying such process, authors
assume that all individuals should receive the same information and evaluate
and handle information by the same way.
Firstly for analysis of individuals perceptive risks of vaccines, authors
mainly mentioned perceptive risks, which brought by safety issues of vaccines.
Because of some vaccines are made from deactivated versions of viruses, it also
has a very small possibility of becoming pathogenic for unfortunately technological mistakes or artificial defects . Actually mostly vaccines are fairly safe.
However some opposite events relating to dangers of vaccines occurred in the
past. Even these cases was confirmed false. Public mistakenly thought that such
events revealed many problems of vaccines. It leaded to increase the perceptive
risks of the public and leaded that people missed the best time to vaccinate.
Further these events affected the overall coverage level of vaccinations. According to the vaccinate risk factor, authors applied the parameter rv to define the
risk of vaccine.
Secondly, according to analysis of the infection rate of diseases, authors
mainly mentioned an infection rate, which brings by the disease and besides
under the uptake level of a population impacting individuals’ choices of vaccination behaviors. With the information, on one hand that vaccines could prevent
infecting diseases, individuals should have higher probabilities to choose vaccination, and on the other hand, if the diseases cannot be prevented spreading to
serious consequences by vaccination, individuals should have higher probabilities to reject vaccination. When the vaccination level in population reaches to
a certain level, vaccines will certainly prevent the spread of diseases in a crowd.
According to these disease related factors, authors applied ri to represent the
risk of the infection diseases and πp to represent the probability of the infection
disease in unvaccinated individuals under the vaccination level of p.
In game theory, when decision-making strategies by two or more individuals occur by interactions between, each individual decision depends on the
prediction of behaviors of other individuals. By selecting the best strategy, it
could be realized that the payoff of individuals or groups could get maximize
. For a vaccination strategy, the expected payoff should be calculated as
E(P, p) = P (−rv ) + (1 − P )(−ri πp ).
The first part in the right side of the equal sign is the expected payoff
of the vaccinated individual with the probability P under perceptive risks of
vaccinations and the vaccination level of p. The second part in the right side
of the equal sign is the expected payoff of the unvaccinated individual with the
probability (1 − P ) under risks of diseases and the vaccination level of p. In
this function eq. 15 the morbidity risks from vaccination rv and infection rate
ri could simplified by the relative risk r = rv /ri instead of different types of risk
parameters, then they get a new payoff function,
E(P, p) = −rP − πp (1 − P ).
In the paper, while adding factors of vaccination to the susceptible-infectedresistant model (SIR model), or epidemiological model. The first definition of
SIR model is presented by Kermack and McKendrick (1927) . Authors analyzed the transformation law of the different groups in population under the
action of vaccines. SIR epidemic model is constructed in accordance with the
general transmission mechanism. In the general transformation mechanism, the
population is divided into different parts, and each part could mutually transform step by step. Such model describes the spreading process of infectious disease,
analyses the changing rule of infections, and reveals the development trend of
infectious diseases through the quantitative relation of infectious diseases.
The SIR model is also called ”three-component model”. S is the mean of
susceptible people. I is the mean of infective people. R is the mean of removed
people. The scope of the population of the epidemic model is divided into
three categories: S part (Susceptible) is uninfected individuals but lacking the
immune ability, after contacting with the infected person turned to infection;
I part (Infective) is infected individuals, and it could influence S part; R part
(Removal) refers to isolated individuals, and another part of individuals, which
obtain immunity after illness. Three parts could transform by the way as S →
I → R.
In such model, suppose that individuals with uptaking vaccines will not be
infected any more, therefore: Susceptible part equals unvaccinated individuals
of new born population in unit time minuses healthy people becoming infected in
unit time and then minuses healthy people eventually natural death in unit time.
Infective part equals healthy people becoming infected in unit time minuses
cured patients in unit time and then minuses healthy people eventually natural
death in unit time. Removal part equals vaccinated individuals of new born
population with the permanent immunity in unit time pluses cured patients
in unit time and then minuses healthy people eventually natural death in unit
Such model could be expressed as follows:
= µ(1 − p) − βSI − µS,
= βSI − γI − µI,
= µp + γI − µR,
S + I + R = 1.
Here, µ is the mean birth and death rate of population, µ(1 − p) is unvaccinated individuals, β is the mean transmission rate of diseases, γ is the mean
cure rate of diseases, and p is the vaccine uptake level.
=f (1 − p) − R0 (1 + f )SI − f S,
=R0 (1 + f )SI − (1 + f )I.
The parameters in this function τ = t/γ are time measured in units of
the mean infectious period. f = µ/γ is the infectious period as a fraction of
mean lifetime, and R0 = β/(γ + µ) is the basic reproductive ratio. The basic
reproductive ratio presents that an infected individual causes new infections
during a unit time period.
Herd immunity is a method of controlling the spread of infectious diseases. It means when the level of the vaccination reaches to a range, the overall
population will not infected any more. This method could reduce S0 through
vaccination. That is to say, through the herd immunity it could be realized that
the spread of infectious diseases is stopped by the initial susceptible persons all
So the critical coverage level that eliminating the disease could obtain
from the SIR model.
if R0 ≤ 1,
if R0 > 1.
When p > pcrit , S = 1 − p, I = 0, this situation means that with the
vaccine uptake level p reaching to the threshold. Under this threshold level
overall population will be immunity and population outbreaks will not happen.
Thus nobody else gets sick in this situation. When p < pcrit , it means that
the vaccine uptake level p is smaller than the level of disease eradication, then
some people must be infected. Thus the vaccine uptake level p should be stable
coverage to the threshold. The SIR system could converge to a stable endemic
state as follows:
Ŝ =1 − pcrit ,
(pcrit − p).
Here is a problem of the convergent stability. Before explaining the convergent stability, let us introduce the application of Nash Equilibrium applied
in the article.
Nash Equilibrium in the prevention of diseases
Definition 3. (Nash Equilibrium): The Nash equilibrium is a collective strategy
in a game involving two or more players, where no player has anything to gain
by changing only their own strategy. Formally, a set of two strategies played by
players A and B (pA , pB ) is a Nash equilibrium if
EA (pA , pB ) ≥EA (qA , pB ), f or all qA ,
EB (pA , pB ) ≥EB (qA , pB ), f or all qB ;
Where qA and qB describe any other strategy that can be played by players
A and B respectively, and EA and EB are the players payoffs. The pA and pB
to mean a particular strategy like cooperate or defect in the prisoners dilemma
(these are called pure strategies). But games can also be played with a random
element, where pA and pB are the probability distributions for all the pure
strategies a player can play (these are called mixed strategies). In the case
of mixed strategies, EA and EB is the expected payoff obtained from random
encounters between mixed strategy pA and pB .
As a stable and dominant state, the Nash equilibrium is able to give a
judgment strategy to determine the optimal strategy for the group. For the real
case, the Nash equilibrium is the strategy with a better payoff. If the majority
of individuals selection is strategy P , individuals with strategy Q always obtain
the lower payoff comparing with the mostly individuals choosing P . Then the
strategy P is called a Nash equilibrium. If P is a Nash equilibrium and currently
each individuals choose P , there is no one will change own strategy. In the
population strategy P has be chosen by proportion ratio of the population for
epsilon, and strategy Q with ratio of (1 − ε). Then the ratio of entire population
p = εP + (1 − ε)Q,
With the payoff gain we could measure the incentive of individuals which
switching strategy in Q and P ,
∆E = EP − EQ = (πεP +(1−ε)Q − r)(P − Q),
When ∆E > 0, the strategy P is the dominant of those strategies and will
be chosen. Where Q is the dominant strategy.
Payoff functions can be constructed respectively for P individuals and Q
EP (P, Q, ε) =E P, εP + (1 − ε)Q ,
EQ (P, Q, ε) =E Q, εP + (1 − ε)Q .
Here authors gave a definition of the convergent stability.
Definition 4. (Convergent Stability): If the majority of individuals same selection is strategy Q and some individuals choose the strategy closer to P than
Q, the payoff of the strategy closer P obtains the higher payoff than those
who individuals with strategy Q (from P strategy to lower revenues). Under
this situation if any Q 6= P is true, then P is known as the convergence and
Then the probability of an unvaccinated individual becoming infected πp
could be expressed with the vaccine uptake level of p:
R0 (1 + f )Ŝ Iˆ
R0 (1 − p)
R0 (1 + f )Ŝ Iˆ + f Ŝ
Here Ŝ and Iˆ represent estimated values of S part and I part. For the real
case, when the perception risk of vaccinations is larger than the infection rate
of disease r > π0 , the result of CSNE is rejecting vaccinations P ∗ = 0; when the
perception risk of vaccinations is less than the infection rate of disease r < π0 ,
the result of CSNE is uptaking vaccinations with the probability 0 < P ∗ < 1.
P ∗ is the unique solution of the equation πP ∗ = r, which can be calculated by
such function as follows:
P∗ = 1 −
R0 (1 − p)
Model analysis on Nash Equilibrium
Figure 7: Vaccine coverage p∗ at the CSNE versus relative risk r.
In the fig. 7, it shows that different disease with different values of the
basic reproduction R0 have different values of the vaccination threshold pcrit .
For R0 = 2, pcrit = π0 = 0.5, for R0 = 5, pcrit = π0 = 0.8, for R0 = 10,
pcrit = π0 = 0.9 and for R0 = 20, pcrit = π0 = 0.95. For any relative risks as
r > 0, the expected vaccination level P ∗ is definitely lower than the threshold
value. Under the same vaccine relative risk, the disease with the higher R0
have a high coverage level of vaccination. When the vaccine relative risk equals
to 0, r = 0, there has the largest vaccination rate; as the vaccine relative risk
increasing, vaccination level decreases; when the relative risk is big enough as
the value is greater than 1 or greater than the π0 , r 1, vaccination level
equals to 0, namely individuals will not vaccinate. For childhood diseases R0
(5 < R0 < 20), we can get that the value of the vaccination threshold, which is
closing to 1, shows in generally the children disease vaccination uptake level is
high. Considering actual reasons according to immune systems of children, it is
not perfect and does not have enough abilities to resist diseases. It is necessary
to take vaccinations to newborn children.
Let us considering the influence of the relative risk on changing vaccination
strategies. During the time of the vaccine scare, the relative risk will increase.
It will affect the vaccination level of the population. According to the previous
analysis, if r < π0 , individuals consider vaccinations with the probability of P ; if
r > π0 , individuals do not vaccinated. With different new CSNE P 0 and P , P 0 is
associated with the perceived relative risk r0 and P is the CSNE associated with
the relative risk r. According to values of the payoff gain always are positive,
the value of the payoff gain ∆E measures the incentive to switch from the value
of CSNE as P to the value of CSNE as P .
When the vaccine safety incident happening, the perceived relative risk r0
must increase in fig. 8. The different lines show us the relationships between the
vaccine risk r0 and the ∆E, and between the vaccine risk r0 and ∆P with different
values of R0 , correspondingly different values of threshold value of vaccination
Figure 8: Analysis of vaccine scares.
level pcrit = π0 , which was got early. As the vaccine risk r0 increasing, ∆E
exponentially increases and ∆P exponentially decreases until r0 = π0 . For R0 =
2, when r0 ∈ (0, 0.5) with the constraint by π0 = 0.5, ∆E increases exponentially
and ∆P decreases exponentially, and then when r0 increases exceed 0.5, ∆E
increases as a slower trend and ∆P decreases as a slower trend. For R0 = 5, the
turning point gotten when r0 = 0.8 by π0 = 0.8, after that ∆E increases as a
slower trend, but faster than the situation, when R0 = 2, and the turning point
of ∆E(R0 = 5) = 0.52 higher than ∆E(R0 = 2) = 0.52, the turning point of
∆P (R0 = 5) = 0.69 higher than ∆P (R0 = 2) = 0.45. And for R0 = 20, when
r0 = 0.95, ∆E increases exponentially. With the increase of r0 > 0.95, ∆E with
R0 = 20 increases as a slower trend than the previous process but better than the
disease with R0 = 2. For different R0 , ∆P decreases exponentially with r0 < π0 .
When r0 increase to 0.5, ∆P decreases exponentially. When r0 ≥ 0.5, ∆P = P .
It means that when the relative risk is greater than π0 , individuals choose not
to vaccination and the population proportion is as a trend of declining.
Figure 9: Public education programs impact counteract vaccine scares.
Vaccine safety issues does make much confusions to a lot of people about
the vaccination. However there is no denying that vaccination is still by far the
most effective method of preventing disease outbreaks. Although vaccine safety
events have the bad influence on the public’s perception of vaccination, people’s
cognition will get change by the popularity of the public education project. The
curve in fig. 9 presents relative risk gradually reduce with the popularity of
public education programs after the vaccine panic. That is the process occurs
from r > π0 to r0 , with the means less than π0 .
During the vaccine safety incidents, perceptive relative risk increasing is
greater than the disease infection rate r > π0 . With the popularity of public
education programs, relative risk gradually reduces, r0 < π0 .
It is obviously shown that different curves have same trends. For one
line, with the increase of risk of vaccine r0 , ∆E exponentially decreases and
∆P exponentially decreases. Until r0 increasing to π0 , (different diseases have
different infection rate, corresponding to different π0 ), ∆E reduces to 0, ∆P also
reduces to 0. On the contrary with r0 drops from π0 gradually, within the scope of
the greater than π0 , due to r and r0 all are larger than π0 , so individual dose not
choose to vaccination, ∆P = 0; With the public education reducing the relative
risk, r0 gradually drops under π0 in the process, the population vaccination rate
increases gradually, that shows as an exponential trend. Comparing with the
process of vaccination panic, under the help of the public education programs,
the increased degree of the vaccination level is much slower than the reductive
degree when vaccine panic. When relative risk is eliminated eventually, the
overall level of vaccination keeps as a low state than the previous state.
Investigation on Vaccination Behavior in Social Networks
Vaccine inoculation is recognized as an outstanding achievement in biomed-
ical areas and public health areas in the 20th century. Vaccination is a successful
production preventing for preventing some epidemics spreading among population. Many infectious diseases, such as smallpox, have already been effectively
controlled. From the 16th century to the 18th century, the number of people
died each year from smallpox was about 50 million people in Europe and about
80 million people in Asia. Roman Empire decayed in the 2nd century and the
3rd century legend because of the ravages of smallpox, which cannot be curbed.
But now smallpox has completely been eradicated in the world . These successful cases depend on not only the effective vaccine but also the implement of
the planned prophylactic vaccination. Those cases has proved that the specific
vaccines can prevent some epidemics from population and take some epidemics
spreading under control. But in terms of some other disease, which is not easily
controlled, even with a specific related disease vaccine, if the vaccination rate
maintains a low level, a well prevention of the epidemic spread can still not be
realized. In the real case of immunization programs to hepatitis B, although the
hepatitis B (HB) vaccine has already have a quite mature development, which
means that many hepatitis B vaccine can efficiently prevent hepatitis B, the
actual incidence of HB remains high for a time due to gaps in coverage persist.
According to the data of survey in 2015, there are 120 million people infected
with HB in China, Where chronic hepatitis B patients about 30 million cases.
About 35 million people die each year from diseases associated with chronic
hepatitis B. Therefore an efficient measure, which makes an effective strategy
or a policy involving improvements of the population vaccination rate is the key
to prevention works.
In the article ”imitation dynamics of vaccination behavior on social networks” written by Feng Fu, Daniel I. Rosenbloom, Long Wang and Martin A.
Nowak (2011) authors consider an issue about research the social vaccination
behavior in definite population groups . Decisions of individuals behaviors
about whether to take up vaccinations decide the population vaccination rate.
Scholars in their investigation propose a model that not only would measure the
probability of vaccination take-up within a definite social network but it takes
into account the phenomenon of the anecdotal information implicit influence in
making vaccination choices as well.
It is necessary to note that an individuals vaccination and, thus, immunity
status contributes to the whole populations immunity. The more individuals
decide to take up a voluntary vaccination, the higher is the chance to escape
an infection risk to those who lay their hopes on the consciousness of the other
society members and preferred not to bear any vaccination costs by themselves.
In our investigation papers we call them ”free-riders”. In fact we should admit
that ”free-riding” is the most favorable scenario this kind of behavior bears no
costs at all. However, irrational but successful example of those ”free-riders”
gives a green light to their social neighbors and in the next epidemic round when
much less members would take up a vaccine, herd immunity would evidently
plummet and suffer under the epidemic outburst. Thus, this work highlights
the effect of the imitation behavior and its consequences on the whole nation
On one hand, it is obvious that pre-emptive vaccination can save us from
the great variety of possible diseases and their adverse influence on our health.
On the other hand, each of us faces definite rational and irrational arguments
why he or she may not take up a vaccination. Rational reasons for the vaccine
rejection underlie in its cost that involves in its turn financial costs, temporal
costs and probable adverse costs caused by a vaccine on the individuals health
status. Irrational motives include probable anecdotal information, public fears,
personal examples of the close friends and relatives. If we assume that individuals act rationally and with perfect knowledge of their infection risks, then
public vaccination decisions would obviously converge to a Nash equilibrium.
Nevertheless, irrational variable in peoples behavior leads to inadequate level of
the vaccination coverage in the community. Let us take down to the proposed
model and its methods.
Description of the SIR model and its methods
Before the main method, we make the introduction of the social network.
In recent years, as the rise and development of the complex network research, it
makes the people have a clearer understanding of the structural evolution and
the complexity of various networks in reality. Especially in 1998, Watts and his
supervisor Strogatz published an article in Nature about the small world network
model . The researches of complex networks quickly attracted attentions of
many researchers in different fields, such as the physics and the biology, complex
network theory has been full of exploration and development.
Complex network theory provides the convenient frameworks for describing the relationship of games among individual. The nodes in the network
represent individuals in game. The sides represent the relationships between
individual and his neighbors in the game. In this way, we can use topological
relations of the complex network to study the complex game. The well-mixed
assumption of the game theory can be viewed as the total connected graph. The
game with two dimensional lattice or one dimensional ring can be converted to
the regular network game. However, in the real world networks are heterogeneous as the most number of neighbors are different. Therefore, it is significant
to research the impact on the heterogeneity of the network of contacts to the
dynamics game .
To study the voluntary vaccination dilemma the authors take as a basis
a simple agent-based model in the spirit of evolutionary game dynamics. This
model in its turn is combined with the susceptible-infected-resistant (SIR) model, or epidemiological model. The first definition of SIR model is presented by
Kermack and McKendrick in 1927 . In this model the population is divided
into three categories: susceptible, infected and resistant ones. Susceptible population includes uninfected individuals but lacking the immune ability to diseases,
who will easily turn to be infection after contacting with the infected persons;
Infective population includes infected individuals, and they could influence from
the susceptible part; Removal population includes to healthy individuals, which
are recovered from infected individuals. For well-mixed populations, the time
evolution states can be expressed by following functions :
= − rN SI,
=rN SI − gI,
S + I + R = 1.
Where r means the disease transmission rate, and g means the rate of
recovery from infection. N is the total population size.
Denote rN/g by R0 , commonly called the basic reproduction radio. Dividing eq. 27 by eq. 28, there obtains a function that
= −R0 S.
Using the initial condition S(0) 1 and R(0) = 0, the final state I(∞) = 0
and S(∞) = 1 − R(∞), and R(∞) is the final fraction of individuals who had
been infected during the epidemic outbreak, we obtain
R(∞) = (1 − e(−R0 R(∞)) ).
If we consider preemptive vaccination by supposing that a proportion x
of the population initially vaccinated, eq. 29 can be rewritten as
R(∞) = (1 − x)(1 − e(−R0 R(∞)) ).
Figure 10: Two-stage game.
The experiment is held within one epidemic round as vaccines are mostly
effective for only one season due to the capability of the pathogens to evolve
mutation and adapt to the vaccine. Authors model the vaccination dynamics as
a two-stage game in fig. 10. For simplicity, we assume that any of the vaccine
applied would be highly efficient and protect the individual from the probable
infection risk. The first stage represents a public vaccination campaign that
ordinarily takes place before any epidemic occurs. The necessary admissions:
X is a fraction of vaccinated individuals; the cost of vaccination in general
should be substantially lower than the infection cost. This is a period when
each individual thinks carefully about his personal pros and cons whether to
take up a vaccine or deny and makes his final decision in this epidemic round.
If a decision is made to vaccinate, a definite cost of the vaccination V is taken
by the vaccinated individual.
The second stage depicts the process when unvaccinated individuals can be
subdivided into two categories. One of them is that those who face the infection
risk and bear infection cost I (that includes healthcare expenses, lost productivity and even probability of lethal outcome), and those lucky ones unvaccinated
called free-riders, but nevertheless, not infected. In other words, free-riding
is the phenomenon when unvaccinated individuals remain healthy and benefit
from the vaccination consciousness of the others. In our model, the epidemic
initially infects individuals I0 , and then spreads according to the SIR dynamics model with per day transmission rate r and recovery rate g. Conclusion of
these two-stage game is the following: the highest benefits get those unvaccinated participants who managed not to get infected within the epidemic round.
Thus, each individual wants to escape both vaccination and infection. Besides,
successful examples of unvaccinated individuals and personal fears linked with
some isolated cases influence much on him. However, group interest lays just
on the opposite side, that the more individuals would get vaccinated, the more
solid immunity to the society would be provided.
The assumption of rationality is relaxed in this model. From the evolutionary perspective individuals revise their vaccination strategy each season
basing on the current payoff. Besides, we assume that individual i randomly
chooses individual j from their common network as a role model. Clearly, the
strategy with the higher benefits will be likely to be copied in the next season.
Thus, here the author introduces the variable β standing for the strength of
selection. For small value of β (weak selection) individuals are less responsive
to payoff differences. In other words, individual may choose a strategy of a less
successful role model, and interact more rationally not only retrospectively but
with some definite future expectations. Large β stands for the behavior model,
when individuals keep to the strategy with the higher observed payoff, even if
the payoff difference is relatively negligible. An individual i randomly chooses a
neighbor j. We suppose that the probability that individual i adopts individual
js strategy is given by the Fermi function
f (Pj − Pi ) =
1 + exp[−β(Pj − Pi )]
It means that individuals choose a strategy dependent on the neighbors
behaviors and is decided by the difference of payoff. And in this function denotes
the strength of selection where 0 < β.
Results and Conclusions
In the Fig. 11 and Fig. 12, it shows vaccination dynamics and epidemic
dynamics in well-mixed populations. Lines of open and filled squares and open
inverted and filled inverted triangles mean simulation results. For open squares
and open inverted trangles, β = 1; For filled squares and filled inverted trangles,
β = 10. Lines of dotted line and solid line show theory results. For dotted
line, β = 1; for solid line, β = 10. The horizontal axis is the relative cost of
Figure 11: Vaccination level dynamics in well-mixed populations.
Figure 12: Epidemic dynamics in well-mixed populations.
For the curve in well-mixed population, with β = 10, when increasing
the cost c the vaccination level have a drop from the initial vaccination level
0.70. The curve has a stable 0.38 with the relative cost of vaccination v = 0.20.
Then the curve from v = 0.75 to v = 1.0 drops to vaccination level 0. For the
same value of β, the final epidemic size has a increasing trend from 0 under
the vaccination level 0.70. The epidemic curve increases to a stable state 0.38
under the vaccination level 0.38 and v = 0.20. Then with the increasing cost the
epidemic size closing reaches to 0.90. The two graphs above have a same stable
range with the relation cost of vaccination 0.20 < v < 0.80. When β = 1, the
value of vaccination level and epidemic size have a stable trend like a straight
line decreasing from the initial point to final vaccination level 0 and increasing
to final epidemic size 0.90 with the increasing relative cost of vaccination from
0 to 1.0.
In lattice populations, Fig. 13 and Fig. 14 show vaccination dynamics
and epidemic size dynamics. Open squares with the solid line and open inverted
triangles with the solid line have β = 1; filled squares with the solid line and
filled inverted triangles with the solid line have β = 10. The horizontal axis is
the relative cost of vaccination c.
For the curves in lattice population, when β = 10, as increasing cost the
vaccination level straightly drops from the initial level 1.0 to a stable 0.1 under
the relative cost of vaccination v = 0.27. Then have a long stable state until
v = 0.75. After adding the relative cost the vaccination level drops to 0 with
the relation cost of vaccination v = 0.87. For the same value of β = 10, the
final epidemic size have an increasing trend from 0 to 0.60 respectively with the
vaccination cost v = 0 to v = 0.27. Similar with the vaccination level, final
Figure 13: Vaccination level dynamics in lattice populations.
Figure 14: Epidemic dynamics in lattice populations.
epidemic size have the same cost value v = 0.87 increasing to 0.90. With β = 1,
the value of vaccination level straightly decreases and until vaccination cost
v = 0.60 decreases to 0. For β = 1, the final epidemic size have an increasing
trend from 0 to 0.90 with the interval of vaccination cost from v = 0 to v = 0.60.
And with increasing the relative cost of vaccination, the final epidemic size stays
0.9 under the vaccination level 0. In lattice population, simulations and theory
For the Fig. 15 and Fig. 16, they show trends of vaccination dynamics and
final epidemics size dynamics in random network populations. Open squares
with solid line and open inverted triangles with solid line have β = 1; filled
squares with solid line and filled inverted triangles with solid line have β = 10.
The horizontal axis is the relative cost of vaccination c.
Figure 15: Vaccination level dynamics in random network populations.
Figure 16: Epidemic dynamics in random network populations.
For these curves in random network populations, the simulations and theory are same. With β = 10 as the increasing cost v the vaccination level
straightly drops from the initial value 1.0. Then keeping to a stable 0.40 when
an interval of the relative vaccination cost 0.34 < v < 0.80. Finally it drops
to 0 with increasing the relation cost of vaccination from 0.8. For β = 10, the
final epidemic size have an increasing trend from 0 when the vaccination cost
v = 0.04. Then the vaccination level keeps to a stable state 0.30 when the vaccination cost v = 0.35. The two graphs between vaccination level and epidemic
size have a similar stable range with the interval of the relation cost of vaccination 0.35 < v < 0.60. When β = 1, the value of vaccination level do not move
before v = 0.20. Increasing the vaccination cost from v = 0.20 the vaccination
level straightly drops to 0 until the relative cost of vaccination v = 0.85. With
the same value of β = 1, the final epidemic size also does not have any changes
in the interval of the relative cost of vaccination from 0 to 0.20. Then the final
epidemic size have an increasing trend from 0 to 0.9 under the vaccination cost
from v = 0.20 to v = 0.85. As increasing the relative cost of vaccination until
v = 1.0, the vaccination level maintains 0.90 under the vaccination level 0.
Comparing the graphs, we see slightly changing value of c enlarge with the
β(strength of selection). In the random network populations the trends could be
controlled well enough when the cost c is located in a small range. But with the
increasing cost c the trend gets out of control. Basing on the initial data used in
the investigation, we can get following curious conclusions: in the vaccination
game, if all of ones neighbors adopt one strategy, it occurs to be beneficial to use
the opposite strategy. By this reason vaccinated and unvaccinated individuals
change their roles throughout the experiment.
For small value of β (weak selection) we see that the function is roughly
linear, the higher is the vaccination cost, the less individuals would take up
a vaccination. Therefore, as vaccination level falls with the growing cost, the
final size of the epidemic grows. There is a point on the graph indicating a
relatively high level of vaccination cost, above which no one chooses vaccination
and the epidemic reaches its maximum size. As for the large β (strong selection)
results we see that individuals will rather attempt to ”free-ride” than adopt a
rational decision. And this kind of irrational and imitating behavior reflects on
the graph correspondingly. It is notable that if we restrict interaction between
an individual and his neighborhoods, we can partly eliminate the ”free-riding”
syndrome, but instead we get a higher sensitivity to the parameter of the cost of
vaccination, c. Thus, restricted spatial interactions can provide us with results
close to Nash equilibrium optimal to the social wealth. On both our graphs
we can allocate the point defining the critical vaccination cost below which the
epidemic is prevented.
The next notable conclusion is that higher vaccination coverage is typically
required to achieve herd immunity in populations with greater degree of heterogeneity. Particularly, degree of population heterogeneity defines many aspects of
social behavior. Thus, we can assign so called social ”hubs”, or important social
centers of great influence on the neighbors. These are, for instance, physicians,
teachers and many others occupied in the social sphere. Such ”hubs” are likely
to get vaccinated because of the wide range of interaction and therefore they
are subject to greater risk of getting infected. In their turn, ”hubs” can spread
disease to a great number of peers if infected. And vice versa, hubs government
policy vaccination plays a dramatically significant role in disease prevention.
Thus, the considered above investigation shows how anecdotal information
and public fears affect our social behavior and form strong selection imitating
model. In the whole society scale, such irrational behavior responds for the
vaccination coverage fall below Nash equilibrium. It is a question if a voluntary vaccination really makes sense in our liberal society on condition that the
majority prefers the strategy of ”free-riding”, contradicting the whole society’s
interests. One can compare the voluntary vaccination to public goods studies
as herd immunity provides a communal benefit. However, here we should relax
the assumption of rational behavior of individuals and mind that public fears
can significantly influence on vaccine take-up and public health.
In ”World Cancer Report 2014” published by WHO , there were 14.09
million of malignant tumor incidences in the whole globe in 2012, including
7.43 million male patients and 6.66 million female patients; and there were 8.2
million people died from different kinds of cancers, including 4.65 million males
and 3.55 million females. The most common cancer was lung cancer, accounting
for 13% of the total. The second was breast cancer, accounting for about 11.9%.
Then there are colorectal cancer, prostate cancer, stomach cancer, liver cancer
and so on. In cancer deaths patients, lung cancer occupied the first place,
accounting for about 19.4%. Next is liver cancer, accounting for about 9.4%.
Then followed by gastric cancer, there are colorectal cancer, the breast cancer
and so on. In China, according to the national cancer center about the statistical
report of the latest status of cancer, there pointed out that the cancer rate in
China was 0.55%. According to the data in 2011, it showed that the number
of cancer patients was 3.37 million, including 1.91 million men and 1.45 million
women. Among the top 10 of malignant tumors, the top three about male
are: lung cancer, stomach cancer and liver cancer, while for females they were
breast cancer, lung cancer and colorectal cancer. With the rapid development
of science and technology, cancer is a difficult risk for people because of the lack
of an effective method making people puzzling. It seems be sentenced to death
once having a cancer .
Thankfully, the cancer treatment in the last decades has made some new
progresses. For the treatment of tumors in addition to the surgical removal of the
entity lesions, the chemotherapy, the radiation therapy and the targeted drugs
are widely used. To some extent, these methods can control the development of
cancer. However there exists many subsequent problems in the process of the
treatment. To the chemotherapy or the radiation therapy, due to lacking the
specificity, drugs not only kill tumor cells but also kill normal cells, thus have a
large toxicity and damage the immune system, seriously affecting the patient’s
quality of recovery. Then researchers put forward the targeted drugs with strong
specificity for cancer. Targeted drugs can targetedly kill tumor cells with certain
genetic mutations without hurting normal cells. But it also has limitations due
to only specific mutation patients can benefit from such targeted drugs. More
importantly, however, cancer cells can often evolve a variety of mechanisms to
combat these drugs, namely we often say the resistant. According to these
factors, the cancer development of some patients has been effectively controlled
at the beginning of the treatment, but soon after be relapsed again.
In recent years, researchers have proposed new treatments based on the
characteristics of the body’s own immune, which was called immunotherapy.
It appears to be effective to solve the shortage of the previous treatment. Immunotherapy is used by activating the body’s own immune system to kill tumors.
This method is considered can effectively kill tumor cells, inhibit tumor evolution, and effects are relatively mild and controllable, and the recurrence rate is
In a wide variety of cancer clinical trials it shows a surprising result and
brings hope to the vast number of cancer patients. For the normal immune
system, the immune system in the human body always plays in charge of recognizing and eliminating the bacteria, viruses and other pathogens for protecting
the body from damage, ensuring the normal operation of the body.
Normal immune system has an effective mechanism for distinguishing be-
tween the pathogens and the molecular of the body’s own. It can selectively
eliminate these pathogenic factors without hurting the bodys own molecular.
In the process of recognition, there is a certain type of signaling molecules to
prevent the immune system attacking the body’s own molecules, called immune
checkpoint. Some of them can activate the immune response, and some others
can suppress the immune response. If immune check points have problems, the
immune system will be unable to distinguish between the external factor and
the internal factor, then lead to the immune system attacking the normal tissues
and cells. When the human body cell gene have the mutation because of various
factors, the accumulation of these mutations finally cause cells out of control,
also known as cell cancer.
At this time due to the cancerous cells have some characteristics molecules
different from normal cells, such as cancer gene mutation, the immune system
will start to recognize and eliminate cancerous cells. However, according to the
mutation rate with extremely high, in the face of the cleaning function of the
immune system, the cancer cells will not give in easily, and will change their own
recognition characteristics, evade the identification of the immune system. This
mechanism is called immunosuppressive or immune escape. Once the cancer
cells evade from attacking of the immune system, the cancer cells will crazily
breeding and occupy other tissues and organs, and out of control. Therefore in
the face of the limit resulted from chemotherapy drugs, the combination with
the person’s own immune system, known as the application of immunotherapy,
has brought hope to cancer patients. Currently in clinical the immunotherapy
mainly includes two categories: cell therapy and intervention therapy.
However, like any kinds of therapies, Immunotherapy is not perfect. Im-
munotherapy fights cancer by activating the body’s own immune system. In
clinic, if the control of the activated immune system is not very well as too
active, it may harm to normal cells, causing immune related side effects, and
more seriously causing organ failure even death. Therefore, groping for the
right dose of drug and using the right medication to control these side effects
are particularly important.
According to the models applied the G-function created by Jessica J. Cunningham, Robert A. Gatenby and Joel S. Brown creates (2011) , it provides
an important tool for the cancer treatment. The model can get the resistance
trend of cancer cells and population density trend of cancer cells under the drug
dose and environmental parameters. According to the corresponding analysis
of different doses of the trends, and the analysis of the resistance development
trend, there can find the right delivery time and optimal dose of drug. To a
certain extent, it can simulate in theory, get the quantitative analysis of direct effect of drugs on cancer cells, and provide a method to evaluate cancer
Now the effectiveness of the immunotherapy of lung cancer has been tentatively identified. The research of the immunotherapy of non-small cell lung
cancer showed that the efficacy of monoclonal antibodies for advanced NSCLC
has a clear curative effect, such as can solve the problem of allergies and adverse reactions. The immunotherapy of the monoclonal antibody is expected to
become the standard supplementary treatment of the lung cancer and to enrich
the content of comprehensive treatment of the lung cancer. The method, which
is presented by Jessica J. Cunningham, Robert A. Gatenby and Joel S. Brown
(2011) , can be used to simulate the effect of monoclonal antibodies for the
treatment of the lung cancer, and simulate the trend of cancer cells resistance for
analyzing the effective treatment and providing an important basis for clinical
In another aspect, there is an application about combined therapy. According to the multi-drug therapy presented by monitoring the changing trend of
cell resistances, Jessica J. Cunningham, Robert A. Gatenby and Joel S. Brown
(2011)  put forward the multi-drug therapy combined immunotherapy and
the traditional radiation or the chemotherapy or the targeted therapy for the
study of controlling cancer. Aiming at the resistance due to the immune escape
in immunotherapy, the model of multi-drug therapy can develop the targeted
combined cancer therapy of treatments, such as combining cancer immune suppression mechanism for improving the success rate of the treatment. In the
combination of immunotherapy and traditional radiation or the combination of
immunotherapy and chemotherapy scheme, firstly during the first phase of the
immunotherapy it stimulates the immune system, which relying on the immune
function reducing cancer cells and at the same time, to some extent weakening
the resistance of cancer cells making cancer cells becoming vulnerable. It benefit
the second phase of chemotherapy drugs.
The effects of drugs under such conditions greatly enhance and eliminate
most of the cancer cells. This method do not eradicate cancer, but according
to the results of the research, comparing with the general single therapy, the
combined multi-drug therapy can control the population of cancer cells at a
lower level. With the increase of the further research and clinical experience,
there will be more mature and perfect immune therapy to help people cope with
various kinds of cancer and benefit the patients.
China as such a large social group with a big country with 1.3 billion people, nearly closing to a quarter of the total population in the world when facing
the outbreak of the influenza and other infectious disease often feels helpless.
Due to the large amount of population, the high rate of the population mobility and the complex and diverse social structure, once happening the epidemic
outbreak the rate of the outbreak will be difficult to be effectively controlled in
China. In the 2003 SARS outbreak case, there were 5, 327 of cumulative cases
in China, and the death of 349 people; in the whole world there were 8, 069 people infected and 775 people died from SARS. In the 2009 flu pandemic, there
were 120, 498 people infected and 648 deaths in China from the H1N1 flu and
the global number of infected person is 1, 353, 141 and death person is 15, 934.
Statistic data shows that China has made ”outstanding contributions” to the
world epidemic outbreak situation. Therefore the epidemic control in China
plays an important role in the global epidemic outbreak.
The development of an effective and timely outbreak control program
has to be solved as a major and important issue. Influenza is an acute febrile
respiratory disease caused by influenza viruses. In history there were many cases
of the flu outbreak all over the world. The influenza is caused by various kinds
of influenza viruses.
In some given years of the outbreak, some species of the influenza virus
may become extinct, but some other species of the influenza virus are generated
over time or become mixed virus with the mixed generation of other types of influenza viruses. Since the influenza viral antigens mutate frequently, they could
easily lead to repeated infections and high incidence of populations. Therefore
these flu viruses can cause the influenza pandemic. Generally, to the influenza
season of a year in North and South, the variations of influenza viruses cause
about 50 million deaths in the world. In these variations, some newly created
viruses of variations are milder treated as a general epidemic. Besides, some
other types of the newly created viruses are more serious viruses that can cause
a significant influenza pandemic. For the infected person, some complications
of the influenza seriously threat the human health, such as pneumonia, otitis
media, rhinitis, myositis, Leiyi Shi syndrome and other serious complications.
So comparing with the common cold, the influenza has more severe symptoms, more contagious ability and are useless to the antibiotic therapy. Although influenza is difficult to be cured, it can be prevented by vaccinations.
The influenza vaccination is a most effective method to prevent the flu outbreak.
Therefore, there is a growing emphasis on the study of the influenza vaccine.
Influenza vaccine is used for the prevention of the influenza, and applies to any
healthy people with the possible opportunity to be infected by influenza viruses. In the policy of vaccination, people should better take one dose before the
epidemic season annually and the immunity sustainable will maintain one year.
As one of the main measures for the prevention and control of influenza, the influenza vaccine can reduce the chance of infection or alleviate the
symptoms of influenza. The study of influenza vaccine has received an effective
progress. We have already developed results about the flu vaccine for seasonal
flu outbreaks. WHO has estimated that in May 2009, according to assumptions
about the best scenarios, the global annual production capacity of the pandemic
influenza vaccinate is about 5 billion. But thereafter, according to more real
information obtained by the real vaccine production and the appropriate dosage
of vaccine area, now the estimate of WHO makes that the annual production
capacity of the pandemic influenza vaccine in the global is about 3 billion. This
figure is less than previously estimate. These vaccine is still not enough to
supply and cover 6.8 billion of the world’s population. Almost everyone in the
world has a susceptibility to the regular development of the highly infectious
virus. Now the vaccine supply can be protected the number doubled.
As the global capacity for influenza vaccines is limited and cannot meet
the demand problem and the flu vaccine is an important preventive strategy, the
effective vaccination strategy is an effectively method to improve immunization
rates in the general population under such limited condition of vaccines. For a
country with low population density like Russian, epidemic is not easy to spread
and the herd immunity is easy to realize. Different with these low population
density countries, in China even for a general small city, the social network of
the small region is complex as closing 0.5 million people. The social structure
in the small region is similar considered to a scale-free network . In China for
a social group, according to the more complex social structure, people become
vulnerable to be attacked by viruses. Therefore epidemic is easily transmitted
among the social group. It is necessary to develop an efficiency strategy of
In the book of ”the complex network theory and application”, it introduces
five immunization strategies: the random immunization, the immunization target, the acquaintances immunity, the cyclic immune and the contact tracing.
Feng Fu, Daniel I. Rosenbloom, Long Wang, and Martin A. Nowak investigates
the vaccination behavior on social networks under the condition of the vaccination cost . They provided a useful method for researching the effect of
vaccination behavior in different social networks. The random immunity refers
to randomly select some people immunize. Under the random immunization
strategy, in order to achieve the herd immunity, all population need to take up
vaccines. It is realistic to a small group rather than a large size group with the
large amount of population. The target immunity refers to a selection of nodes
with a large degree to be immunized. After immunizing these nodes, it means
that relatively large number of people around these nodes reduce the way to
exposure to the virus and then prevent the virus infection. It is an efficient way
to a social network with a low population mobility like a closed small village or
an area with a low population density. The acquaintance immunization strategy refers to randomly find a node and a neighbor of the node to immunization.
Acquaintance immunization strategy is the most effective of local immunization. The ring vaccination refers to isolate or immunity all neighbors of infected
There is a real application, which is Ebola vaccine for clinical trials, adopting the ring vaccination strategy. By isolating all possible individuals contacting
with the Ebola virus infectors and vaccinating the isolators, there formed a ”ring
isolation zone” surrounding Ebola virus infectors to prevent further spread of
the virus. The clinical trial results showed that this strategy ensures a very
significant effect of control of the vaccine for Ebola virus . The ring strategy
of Ebola vaccine got tested in the real Ebola case. This strategy has also been
used in the smallpox virus prevention. The contact tracing refers to trace those
who have contact with contagious individuals, then vaccinate with a certain
probability. Efficient contact tracing and testing made it possible to conclude,
with authority, that the outbreak was over in record time. In the outbreak of
SARS, the date has shown that SARS has been controlled using conventional
measures such as rapid detection, infection control, isolation, quarantine and
contact tracing . The imitation dynamics of vaccination behavior is a good
way to simulate the epidemic dynamic before the epidemic outbreak and specifies a valid immunization strategy provided to prevent the epidemic spread.
According to ”vaccination and the theory of games” written by Chris T.
Bauch and David J. D. Earn and ”imitation dynamics of vaccination behavior
on social networks” written by Feng Fu, Daniel I. Rosenbloom, Long Wang,
and Martin A. Nowak, there are two integrated model combining the epidemic
model, as called SIR model separately related to the vaccination cost and the
vaccination risk . They are helpful to the development of the vaccination
policy according to predict the trends in the epidemic under different conditions.
We analyze different approaches applied game theory methods from new
perspective such as dynamic, adaptive and self-regulating to explore the structural characteristics of the population and how does the limited rationality of
individuals impact an individual’s immune behavior. Then from the theoretical level state the study discusses the underlying mechanisms about switching
population immunization rates from a low level of ”Nash equilibrium” to global
optimization state of. Through this research we try to estimate a new immune
method to solve immune problems based on evolutionary game theory, explore
mutual coupling and co-evolution of the spread of infectious diseases and immunity behavior from a new angle, study physical phenomena and laws, which are
reflected in the true complexity of the system, to help to understand movement
patterns in complex systems and provide a theoretical reference for measuring
policy issues about the prevention and control of infectious diseases and the way
of vaccine distribution for real social systems.
 Azhar I. Studies in the theory of quantum games, 2004. P. 21-24.
 Bernard S., Christopher P. W. // World Cancer Report 2014, 2014.
 Chris T. B., David J. D. Earn . Vaccination and the theory of games article
// Proc Natl Acad Sci USA, 2004. Vol. 101. No 36. P. 1339113394.
 Fu F., Daniel I. R., Wang L., Martin A. N. Imitation dynamics of vaccination behavior on social networks // Proceedings of the Royal Society of
London. Series B. J., 2011. Vol. 278. No 1702. P. 42-49.
 Gjorjieva J., Kelly S., Jessica S., Gerardo C., Fabio S. The role of vaccination in the control of SARS // Math Biosci Eng. J., 2005. Vol. 2. No 4. P.
 Jessica J. C., Robert A. G., Joel S. B. (2011). Evolutionary dynamics in
cancer therapy. Mol. Pharmaceutics .J. Vol. 8, No 6. P. 20942100.
 Kermack W. O., McKendrick A. G. A contribution to the mathematical
theory of epidemics // Proc. R. Soc. Lond, 1927. Vol. 115, No 772. P.
 Liu Y. J., Wang Z. Y. Evolutionary game and propagation mechanism of
public opinion in emergency // Chinese Journal of Management Science.
J., 2014. Vol. 22, No 11. P. 87-96.
 Maynard S. J. Evolution and the Theory of Games. Cambridge university
press, 1982. P. 1-5.
 Maynard S. J., Price G.R. The logic of animal conflict // Nature. J., 1973.
Vol. 246. P. 15-18.
 Nash F. J. Non-cooperative games. Mathematics Department, Princeton
// Ph.D. Thesis. University, 1950. Vol. 1950b. P. 21-22.
 Ni L., Rongshou Z., Siwei Z. An analysis of incidence and mortality of
cervical cancer in China, 2003-2007 // China Cancer. J., 2012. Vol. 21, No
11. P. 801-804.
 Pastor S., Vespignani R. A. Immunization of complex networks // Phys.
Rev. E, 2002. Vol. 65. P. 036104.
 Peter D. T., Leo B. J. Evolutionarily stable strategy and game dynamics
// Mathematical Biosciences. J., 1978. Vol. 40. P. 145-156.
 Petrosyan L. A. Dynamic games and appilication, 2009.
 Ral I., Karl E. L. On the mathematical interpretation of epidemics by Kermack and McKendrick // Gen.Math. Notes, 2013. Vol. 19, No 2. P. 83-87.
 Rong Z. Y. The spread of the epidemic research on complex networks. East
China Normal University, 2013. P. 2-7.
 Vincent T. L., Brown J. S. Evolutionary Game Theory. Natural Selection,
and Darwinian Dynamics, 2005. P. 60-110.
 World health organization global eradication commission. The disease status report, 1980. P. 2-7.
 Watts D. J., Strogatz S. H. Collective dynamics of ”small-world” networks
// Nature. J., 1998. Vol. 393, No. 6684. P. 440-2.
 Weibull W. J. Evolutionary game theory, 1997. P. 56-78.
 World Health Organization. Global Cancer Report, 2012. P. 3-5.
 World Health Organization, UNICEF and the World Bank. The global
basic status of vaccines and immunization, 2009. P. 2-4.
 Yang L., Li J., Gao G. F., Liu W. Overview of Ebola virus vaccine //
Chinese Journal of Biotechnology, 2015. Vol. 31, No. 1. P. 1-23.
 Zhang L. Q. Evolutionarily stable equilibrium and nash equilibrium - and
talk about the development of evolutionary game theory // Economic Science. J., 2001. Vol. 3. P. 103-111.
 Zhang L. Q., Fu C. W.. Rationality and bounded rationality-on the relationship of classic game theory and evolutionary game theory // World
Economy. J., 2004. Vol. 8. P. 74-78.
 Zhao T., Wang B. H., Huang Z. P., Mingsheng S. Social network analysis
and its application in the prevention and control of propagation for public
opinion and the epidemic // Journal of Systems Engineering. J., 2010. Vol.
6. P. 742-754.