# Attitudinal analysis of vaccination effects to lead endemic phases

### IBMC simulations

In this study, the IBMC system consisted of \(N\) individuals randomly located in a square plane of \({L}^{2}\), corresponding to the population density, \(\rho =\frac{N}{{L}^{2}}\). Individuals represented as structureless points in the model were infected with probability \({p}_{infect}\) when they were within an infectious distance, \({r}_{infect}\), from infected individuals. Infected individuals had incubation time \({t}_{incubation}\), after which they faced two fates: immune after \({t}_{infect-immune}\) or dead with probability \({p}_{fatal}\) during the infected period, \({t}_{infect}\). Immunisation is effective during \({t}_{infect-immune}\). Individuals can also be vaccinated with probability \({p}_{vacc}\) and become immune from \({t}_{vacc-start}\) after vaccination until \({t}_{vacc-start}+{t}_{vacc-immune}\), where \({t}_{vacc-immune}\) is the effective immunisation period. Table 2 lists the parameter values used in this study.

The population density, \(\rho\), of \(0.165 {\mathrm{m}}^{-2}\), which is ten times as high as that of Seoul, South Korea, was used to mimic the human-congested area, and the infectious distance when unmasked was approximated as 1Â m. In addition, probabilities of infection and fatality were set to 1.8% and 0.8% based on the real data from the Centers for Disease Control and Prevention in the Republic of Korea (CDCP)^{44}, respectively, from which the incubation period (7.74 \(\pm\) 4.39Â days) of COVID-19 was also taken. On the other hand, both the infectious period after the incubation period and the time it takes for immunisation after vaccination were approximately set to 14 \(\pm\) 7Â days because the reported values are typically one ton two weeks. In the same spirit, we set immunisation periods from both infection and vaccination to 6 \(\pm\) 1Â months because the recommended vaccination period is reported to be approximately five to six months.

Even though the parameters used in the IBMC simulations are arbitrary, the presence of the recurrent pattern observed in the simulations is rather insensitive to the parameter space. For example, the population ratios of various types of individuals when \(\rho =0.08 {\mathrm{m}}^{-2}\) and \({p}_{vacc}=0\) are shown in Supplementary Fig.Â 1.

Initially, \(N\) healthy individuals were randomly generated in the square plane of \({L}^{2}\), and one individual was selected as infected. Healthy individuals were then randomly vaccinated with a probability \({p}_{vacc}\). Each individual, except for dead individuals, randomly moves with diffusion coefficient \(D=\frac{({\Delta R)}^{2}}{4\tau }\) according to the Einstein relation^{45}, where \(\Delta R\) and \(\tau\) are the step size and time step, respectively. The simulation adopts the periodic boundary condition in both the x- and y-directions to prevent the wall effect at the edges^{45}. The results are the averages and one standard deviation of 10 independent IBMC simulations.

FigureÂ 6 presents the IBMC simulation results. Figures 6 and 7a show the typical time progression of various types of individuals (healthy, sick but not infectious, infectious, immune, dead, and vaccinated) when the probability of vaccination \({(p}_{vacc})\) is 0.4 (the corresponding animation is given in Supplementary Fig.Â 1). Supplementary Fig.Â 1: The animation of the IBMC simulation when \({p}_{vacc}=0.4\) is presented in which green, organ, red, blue, grey, and cyan colours correspond to healthy, sick but not infectious, infectious, immune, dead, and vaccinated individuals, respectively.

The following points are of significance: first, the pandemic occurred recurrently, which implies that it is challenging to control COVID-19. Second, it takes time for the pandemic to bloom. In this case, it took approximately 200Â days for the pandemic to reach its peak. The pandemic had a long incubation period, considering that there was only one infected individual initially. Third, mass infection boosts mass immunisation, reducing the number of infected people. Finally, the peak probability of infection increased as the pandemic recurred, which is shown in Fig. 7b, in which the peak probability of infectious individuals is plotted as a function of the probability of vaccination. The pandemic continues until the peak probability of infectious individuals reaches at least 25%. Moreover, the recurrent trend of the pandemic appears only in the intermediate vaccination probability (\(p_{vacc} = 0.1 – 0.5\)). This indicates that a high mass vaccination rate is required to end the pandemic. It should also be noted that the case for \({p}_{vacc}\ge 0.8\) does not suffer from the pandemic at all.

After obtaining the output from our IBMC simulations, we transitioned to the game theory model by considering the individual behaviors observed during the simulations. Specifically, we treated the individual’s decision to get vaccinated as a strategic choice in the context of game theory. Each individual’s payoff, in this case, can be understood as the personal benefit derived from either getting vaccinated or refusing vaccination, given the vaccination statuses of other individuals in the population. This is where the concept of an Evolutionarily Stable Strategy (ESS) becomes relevant. An ESS is a strategy that, if adopted by a population in a game, cannot be invaded by any alternative strategy that is initially rare. It is ‘stable’ in the sense that small deviations from it will be reabsorbed into it. We considered an individual’s decision to get vaccinated as an ESS under certain conditions. We approximated the parameters of the game theory model based on the results from the IBMC simulations. The transition from the IBMC simulations to the game theory model involved mapping the individual behaviors and interactions observed in the simulations to the strategic choices and payoffs in the game theory model. This allowed us to capture the complex, individual-level dynamics of infectious disease spread in a theoretically rigorous framework.

### Evolutionary game theory

First, the definitions of the indices, the parameters and the variables used in this research are listed in Table 3.

We proposed a model with â€˜Vaccinated group \(V\)*â€™* and â€˜Unvaccinated group (delay vaccination to get free rides on herd immunity) \(U\)*â€™*. In game theory, the former strategy can be interpreted as a form of cooperation, and the latter as selfish behaviour or betrayal. The individual establishes a strategy based on the perceived risk of the current behaviour^{46,47}. The individual perceived risks of the vaccine and infection are denoted by \({r}_{v}\) and \({r}_{i}\), respectively. The vaccinated groupâ€™s payoff \({\pi }_{V}\) is defined as \({-r}_{v}\) (Eq.Â (2)). An unvaccinated groupâ€™s payoff \({\pi }_{U}\) is defined as \({r}_{i}{\beta }_{U}I\) because it can be assumed that they act according to the infection rate of the unvaccinated group \({\beta }_{U}\) and the proportion of infected people in the group (Eq.Â (3)). We assumed that all individuals received the same information to simplify the model, and this information was accepted completely to perceive the risk^{48}.

$${{{\pi}}}_{{{V}}}=-{{{r}}}_{{{v}}}$$

(2)

$${{{\pi}}}_{{{U}}}=-{{{r}}}_{{{i}}}{{{\beta}}}_{{{U}}}{{I}}$$

(3)

We suppose that the probability that the individual is vaccinated is \(P\). Since the only actions available to the individual are vaccination and no vaccination, the probability of choosing unvaccinated is \(1-P\). If the individual uses a mixed strategy of â€˜choose vaccination with the probability of \(P\) and choose unvaccinated with the probability of the remaining \(1-P\)*â€™*, the average expected payoff of the Individual is described as Eq.Â (4).

$${{E}}={{P}}{{{\pi}}}_{{{V}}}+\left(1-{{P}}\right){{{\pi}}}_{{{U}}}={{P}}\left({{{r}}}_{{{i}}}{{{\beta}}}_{{{U}}}{{I}}-{{{r}}}_{{{v}}}\right)-{{{r}}}_{{{i}}}{{{\beta}}}_{{{U}}}{{I}}$$

(4)

Here, because the individual acts based on the relative perception of vaccines and infection risks, \(r\), it is possible to simplify and represent variables by using \(r={r}_{v}/{r}_{i}\) (Eq.Â (5)).

$${{E}}=-{{r}}{{P}}-{{{\beta}}}_{{{U}}}{{I}}(1-{{P}})$$

(5)

We attempted to identify which strategies can be adopted. If the majority of the population adopts strategy \(V\) and an entity adopting another strategy \(U\) always performs lower than that of an entity adopting \(V\), then \(V\) is the best response strategy. If this is true for any \(V\ne U\), V is called the ESS. If \(V\) is the evolutionary stable strategy and everyone is currently playing \(V\), no one should change their strategy. We suppose \(Q\) is the probability that another individual chooses to vaccinate when one individual is agonised about vaccination with the probability of \(P\). Then, whether an individual selects \(V\) or \(U\) depends on the probability \(Q\) that another individual adopts \(V\). If this is expanded to the population \(\varepsilon (0\le \varepsilon \le 1)\), the vaccination rate at the population \(\theta\) is described as Eq.Â (6).

$${{\theta}}={{\varepsilon}}{{P}}+(1-{{\varepsilon}}){{Q}}$$

(6)

The expected payoff to individuals playing \(V\) is given as Eq.Â (7).

$${{{E}}}_{{{V}}}={{E}}({{V}},{{\varepsilon}}{{P}}+(1-{{\varepsilon}}){{Q}})$$

(7)

whereas the expected payoff to individuals playing \(U\) is given as Eq.Â (8).

$${{{E}}}_{{{U}}}={{E}}\left({{U}},{{\varepsilon}}{{P}}+\left(1-{{\varepsilon}}\right){{Q}}\right)$$

(8)

The payoff gains to an individual playing \(\theta\) in such a population are given as Eq.Â (9).

$${{\Delta}}{{E}}={{{E}}}_{{{V}}}-{{{E}}}_{{{U}}}=\left({{{\beta}}}_{{{U}}}{{I}}-{{r}}\right)\left({{V}}-{{U}}\right)$$

(9)

\(\Delta E\) is represented by a scale of attraction from \(U\) to \(V\). EquationÂ (9) shows that the best response \(\varepsilon\) depends on \(r\). Then, depending on the given \(r\), there is a unique strategy \(P={P}^{*}(Q \ne P)\) that satisfies the best response in \(\varepsilon\). That is, the unique strategy \({P}^{*}\) under the condition that \(\Delta E\) is positive means NE, and if the \(V\) strategy is adopted with a probability close to \(P\), it is an ESS^{49}.

### Technical interpretation of game theory

In terms of technical interpretation, we analysed the SIR model. The model is defined as the rate of change in the population proportion in each compartment. Generalising the results of the proposed model when using the SIR vaccination model enables the population compartment to be expressed as shown in Fig.Â 8 and Eqs.Â (10, 11,Â 12) below.

$$\frac{{{d}}{{s}}}{{{d}}{{t}}}={{\mu}}\left(1-{{\theta}}\right)-{{\beta}}{{S}}{{I}}-{{\mu}}{{S}}$$

(10)

$$\frac{{{d}}{{I}}}{{{d}}{{t}}}={{\beta}}{{S}}{{I}}-{{\gamma}}{{I}}-{{\mu}}{{I}}$$

(11)

$$\frac{{{d}}{{R}}}{{{d}}{{t}}}={{\mu}}{{\theta}}+{{\gamma}}{{I}}-{{\mu}}{{R}}$$

(12)

where \(\mu\) is the average birth rate, \(\beta\) is the average infection rate, \(\upgamma\) is the average recovery period, and \(\varepsilon\) is the population. After reaching a dynamic steady state, the vaccine coverage level in the population equals the uptake level. Because we focused on the steady-state solution of the model, our notation \(\theta\) for vaccine uptake is consistent with the payoffs in game theory as Eq.Â (6) in our notation. In the established SIR model, the third equation is redundant because \(S+I+R=1\). Therefore, we can define \(S\) and \(I\) as Eqs.Â (13) and (14)^{50}.

$$\frac{{{d}}{S}}{{{d}}{{\delta}}}={{f}}\left(1-{{\theta}}\right)-{R}_{0}\left(1+{{f}}\right){{S}}{{I}}-{{f}}{{S}}$$

(13)

$$\frac{{{d}}{{I}}}{{{d}}{{\delta}}}={{R}}_{0}\left(1+{{f}}\right){{S}}{{I}}-\left(1+{{f}}\right){{I}}$$

(14)

where \(\delta =\frac{t}{\gamma }\) is the time and epoch of our model, measured in units of the average infection period, \(f=\frac{\mu }{\gamma }\) is part of the average life span and represents the infection period, and \({\mathcal{R}}_{0}=\frac{\beta }{(\gamma +\mu )}\) is the basic infectious individual production number, a measure of the number of individuals in a susceptible group to an infected person who can spread the virus to^{51}.

The maximum \({\theta }_{crit}\) value of satisfying these ESS conditions is called the invasion barrier for strategy \(V\). If the proportion of strategy \(V\) in the population is less than \({\theta }_{crit}\), \(P\) cannot penetrate the population. According to these payoffs, the individualâ€™s remuneration to select \(P\) at the population level in the early stages of the epidemic can be expressed as Eq.Â (15)^{49}.

$${{{\theta}}}_{{{c}}{{r}}{{i}}{{t}}}=\left\{\begin{array}{c}0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\mathcal{R}}_{0}<1\\ 1-\frac{1}{{\mathcal{R}}_{0}},\,\,\,\,\,\,\,\,{\mathcal{R}}_{0}\ge 1\end{array}\right.$$

(15)

If \(P\ge {\theta }_{crit}\), the epidemiological system converges to the disease-free state \(\left(\widehat{S}, \widehat{I}\right)=(1-P, 0)\), whereas if \(P<{\theta }_{crit}\), it converges to a stable endemic state, as shown in Eqs.Â (16) and (17).

$$\widehat{{{S}}}=1-{{{\theta}}}_{{{c}}{{r}}{{i}}{{t}}}$$

(16)

$$\widehat{{{I}}}=\frac{{{f}}}{1+{{f}}}({{{\theta}}}_{{{c}}{{r}}{{i}}{{t}}}-{{P}})$$

(17)

Because \(S\) and \(I\) are constant in this case, the probability of infection of a person who has not been vaccinated can be expressed by using Eq.Â (18).

$${{{\beta}}}_{{{U}}}{{I}}=\frac{{\mathcal{R}}_{0}(1+{{f}})\widehat{{{S}}}\widehat{{{I}}}}{{\mathcal{R}}_{0}\left(1+{{f}}\right)\widehat{{{S}}}\widehat{{{I}}}+{{f}}\widehat{{{S}}}}=1-\frac{1}{{\mathcal{R}}_{0}(1-{{\theta}})}$$

(18)

Therefore, the condition \(r<{\beta }_{D}I\) for generating a mixed ESS can be written as Eq.Â (19).

$${\mathcal{R}}_{0}\left(1-{{r}}\right)>1$$

(19)

The value of mixed ESS \({P}^{*}\) is obtained by solving \(r ={\beta }_{U}I{P}^{*}\). In situations where vaccination is perceived as an infection risk (\(r>1\)), the individual is unlikely to be vaccinated without the help of the model. According to our game theory analysis, considering the condition of \({\mathcal{R}}_{0}\left(1-P\right)>1\) or ESS, the threshold for the infectious disease control vaccination rate at which the individual can stop immunisation is shown in Eq.Â (20).

$${{{P}}}^{\mathbf{*}}=1-\frac{1}{{\mathcal{R}}_{0}\left(1-{{r}}\right)}$$

(20)

Based on Eq.Â (19), the threshold of the perceived relative risk of vaccination to infection when the individual should stop immunising depends on the basic reproduction number \({{R}}_{0}\).

### Normative interpretation of game theory

By using the SIR model from a demographic perspective, appropriate vaccine thresholds can be derived. However, this model does not consider the variables for strategy transition^{52}. Thus, from a microscopic perspective, we proposed an SIR model that considered the individual’s strategy transition, as shown in Fig.Â 9.

In the proposed model, we considered three infection rates: \({\beta }_{U}\) in the unvaccinated group, \({\beta }_{V}\)Â in the vaccinated group, and \({\beta }_{c}\) in the contact between these two individual groups. We assumed that the probability \({\beta }_{V}\) of infection of the individual who cooperates with the vaccination is less than the infection rate \({\beta }_{U}\) of the individual (unvaccinated group) trying to obtain the benefits of vaccination without effort (\({\beta }_{V}<{\beta }_{U}\)). Vaccinated individuals form neutralising antibodies and are less likely to be infected than nonvaccinated individuals. We also defined \({\beta }_{c}\) as the cross-interaction between the two types of strategies (\(V\leftrightarrow U\)).

We proposed a model incorporating evolutionary game theory and dynamics models by using a parcel approach. The probability that an individual who adoptsÂ \(i\) according to the general evolutionary game dynamics changes strategy to \(j\) is related to its respective remunerations (\({\pi }_{i}\) and \({\pi }_{j}\)). The probability can be expressed using the Fermi rule as Eq.Â (21) ^{53}:

$${{\Theta}}\left({{{\pi}}}_{{{i}}},\boldsymbol{ }{{{\pi}}}_{{{j}}}\right)=\frac{1}{1+{{{e}}}^{-({{{\pi}}}_{{{j}}}-{{{\pi}}}_{{{i}}})/{{k}}}}$$

(21)

This is the probability of enabling strategy modification, and the irrationality of changing these strategies is measured by using the parameter \(k\). In this study, \(k\) = 0.5 was adopted as a constant. Through this, it was possible to determine the ratio of individuals who modified strategies from \(i\) to \(j\), that is, the strategy conversion rate (Eqs.Â (22) and (23)).

$${{\Phi}}_{{{S}}}={{{S}}}_{{{V}}}\left({{{S}}}_{{{U}}}+{{{I}}}_{{{U}}}\right){\Theta }\left({{{\pi}}}_{{{V}}},{{{\pi}}}_{{{U}}}\right)-{{{S}}}_{{{U}}}\left({{{S}}}_{{{V}}}+{{{I}}}_{{{V}}}\right){\Theta }({{{\pi}}}_{{{U}}},{{{\pi}}}_{{{V}}})$$

(22)

$${{\Phi }}_{{{I}}}={{{I}}}_{{{V}}}\left({{{S}}}_{{{U}}}+{{{I}}}_{{{U}}}\right){\Theta }\left({{{\pi}}}_{{{V}}},{{{\pi}}}_{{{U}}}\right)-{{{I}}}_{{{U}}}\left({{{S}}}_{{V}}+{{{I}}}_{{{V}}}\right){\Theta }\left({{{\pi}}}_{{{U}}},{{{\pi}}}_{{{V}}}\right)$$

(23)

In relation to infection dynamics, an infection rate of \({\beta }_{V}<{\beta }_{c}<{\beta }_{U}\)Â exists in general situations. Here, \({\beta }_{c}\) is defined as \({\beta }_{c}=c({\beta }_{V}+{\beta }_{U})/2\). The crossing parameter \(\mathrm{c}\) between the \(V\) and \(U\) is within the range of \(0<c<1\), and in this study, \(c=0.1\) was assumed because of the high possibility of cross-infection owing to vaccine pass release. The differential equations of the SIR vacuum model that consider all the following assumptions are Eqs.Â (24, 25, 26, 27,Â 28)^{52}.

$$\dot{{{{S}}}_{{{U}}}}=-{{{S}}}_{{{U}}}\left({{{\beta}}}_{{{U}}}{{{I}}}_{{{U}}}+{{{\beta}}}_{{{c}}}{{{I}}}_{{{V}}}\right)+{{{\zeta}}{{\Phi}}}_{{{S}}}$$

(24)

$$\dot{{{{S}}}_{{{V}}}}=-{{{S}}}_{{{V}}}\left({{{\beta}}}_{{{c}}}{{{I}}}_{{{U}}}+{{{\beta}}}_{{{V}}}{{{I}}}_{{{V}}}\right)-{{{\zeta}}{{\Phi}}}_{{{S}}}$$

(25)

$$\dot{{{{I}}}_{{{U}}}}={{{S}}}_{{{U}}}\left({{{\beta}}}_{{{U}}}{{{I}}}_{{{U}}}+{{{\beta}}}_{{{c}}}{{{I}}}_{{{V}}}\right)-{{\gamma}}{{{I}}}_{{{U}}}+{{{\zeta}}{{\Phi}}}_{{{I}}}$$

(26)

$$\dot{{{{I}}}_{{{V}}}}={{{S}}}_{{{V}}}\left({{{\beta}}}_{{{c}}}{{{I}}}_{{{U}}}+{{{\beta}}}_{{{V}}}{{{I}}}_{{{V}}}\right)-{{\gamma}}{{{I}}}_{{{V}}}-{{{\zeta}}{{\Phi}}}_{{{I}}}$$

(27)

$$\dot{{{R}}}={{\gamma}}({{{I}}}_{{{U}}}-{{{I}}}_{{{V}}})$$

(28)

In the normative interpretation of game theory, dynamic scenarios based on behaviour can be analysed by using the game-theory-based SIR dynamics model. In this study, parameters \({r}_{V}=1, {r}_{i}=10, \zeta =1, \gamma =1.25, {\beta }_{V}=1, {\beta }_{U}=10, k=0.5,c=0.1\) were adopted as constant variables, and the variations in the infection risk \({r}_{i}\) and infection rate \({\beta }_{U}\) of the unvaccinated group were analysed^{52}. In this case, \({I}_{0}=0.01\) and \({S}_{0}=1-{I}_{0}\) were set between the \(V\) and \(U\) strategies because the number of infected individuals was minimal at the beginning of the infection. Simultaneously, we described an ideal situation in which a person adopting the V strategy was vaccinated with the probability of \({P}^{*}\).

Each individual can receive an optimal reward if the least selected strategy was selected on average. This payoff matrix can be considered a noncooperative game, and the best strategy in a noncooperative situation is to reverse what the other party does^{54}. This is like the problems presented in vaccination. Individuals should be vaccinated; however, if a majority of the population is vaccinated, an individualâ€™s motivation to not get vaccinated increases. The noncooperative factor is related to the risk of a relative vaccine against infection risks. Moreover, the risk of infection \({r}_{i}\) fluctuates based on the infected individual and is a central medium for attaining a continuous infection peak.

When the risk of infection is low, the individual feels that the risk of the vaccine is greater than that of the infection. Thus, the individual withdraws the \(V\) strategy and adopts the \(U\) strategy within a short period, as shown in Fig.Â 10. In this case, many people become infected within the same time, leading to shortages of beds and medical personnel and more damage. However, when the risk of infection is high, the perception of risk of infection increases as the peak of infection persists. Consequently, the payoff for vaccination and the number of individuals cooperating with it increase. The proportion of simultaneous infections decreases, increasing the possibility of preventing a pandemic. In other words, it is possible to have time to suggest pre-emptive measures to adjust the peak size of simultaneously infected people when infectious diseases spread. Therefore, to prevent simultaneous infections through vaccination, we should not only emphasise the importance of vaccines but also the altruistic attitude to form herd immunity, giving individuals a reason to get vaccinated to increase the payoff of relative vaccines.

Additionally, \({r}_{i}\) influences the variance in confirmed cases. When \({r}_{i}\) is low, the number of infected people increases exponentially over a short period. However, as \({r}_{i}\) increases, the distribution of confirmed cases is dispersed, and multiple oscillation terms exist. However, there may be insufficient confirmed cases to paralyze the medical system. We changed the following values to clarify the infection risk perception for the size and duration of infection peaks, which greatly rely on \({r}_{i}\).

Therefore, the size of the infection peak is critical when investigating the epidemiology of an epidemic. The analysis results for the variations in \({I}_{max}\) with \({r}_{i}\) and \({\beta }_{U}\) are shown below. That is, the size of \({I}_{max}\) increases as \({\beta }_{U}\) increases but decreases as \({r}_{i}\) increases. Therefore, the higher the risk of infection is, the higher the individual’s benefit from the vaccine, indicating that the individual exhibits a more cooperative attitude towards the vaccine; hence, herd immunity can be rapidly achieved.

The probability \({\beta }_{U}I\) that an individual who chooses the \(U\) strategy is infected should decrease as \(\theta\) increases until \(\theta\) reaches \({\theta }_{crit}\). Currently, all parameters are greater than 0. Thus, the following maximum expected reward values are obtained by using Eq.Â 4 when \(P=1\) (always vacuum), the maximum expected payoff can be obtained if \({\beta }_{U}I>r\), and the maximum expected payoff can be obtained if \({\beta }_{U}I<r\) when \(P=0\) (always unvacuumed). Therefore, we can define the strategy change point \(I^{\prime}\) as Eq.Â (29).

$${{{I}}}^{{^{\prime}}}=\frac{{{{r}}}_{{{v}}}}{{{{r}}}_{{{i}}}{{{\beta}}}_{{{U}}}}$$

(29)

The vaccinated group ratio can be expressed as \(V=({S}_{V}+{I}_{V})/(S+I)\). Since only two strategies exist in the dynamics model, it can be expressed as \(U=1-V\). The rate of change of strategy varies with strategy flux terms \({\Phi }_{S}\) and \({\Phi }_{I}\); that is, the rate of change of strategy *V* Ì‡ can be expressed as Eq.Â (30).

$$\dot{{{V}}}={ }{-{{\Phi}}}_{{{S}}}-{{{\Phi}}}_{{{I}}}=\left({{{S}}}_{{{U}}}+{{{I}}}_{{{U}}}\right)\left({{{S}}}_{{{V}}}+{{{I}}}_{{{V}}}\right){{\Theta}}\left({{{\pi}}}_{{{U}}},{{{\pi}}}_{{{V}}}\right)-{ }\left({{{S}}}_{{{V}}}+{{{I}}}_{{{V}}}\right)\left({{{S}}}_{{{U}}}+{{{I}}}_{{{U}}}\right){{\Theta}}({{{\pi}}}_{{{V}}},{{{\pi}}}_{{{U}}})$$

(30)

If we rearrange \(\dot{V}\) according to \(V(S+I)=({S}_{V}+{I}_{V})\), \(U(S+I)=({S}_{U}+{I}_{U})\), and \(S+I+R=1\), the expression is Eq.Â (31).

$$\dot{{{V}}}={\left(1-{{R}}\right)}^{2}{{V}}{{U}}[{{\Theta}}\left({{{\pi}}}_{{{U}}},{{{\pi}}}_{{{V}}}\right)-{{\Theta}}\left({{{\pi}}}_{{{V}}},{{{\pi}}}_{{{U}}}\right)]$$

(31)

Here, \({\left(1-R\right)}^{2}\) controls the rate of change of the strategy because it relates to the total available population, which can vary its strategy. However, the most critical factor is the remainder of the equation. This is the general mean-field form of the master equation for the evolution of cooperation in two-strategy games. The proposed model is consistent with and returns to an evolutionary game considering only the strategy density.