Abstract:Deterministic approximations to stochastic Susceptible-Infectious-Susceptible models typically predict a stable endemic steady-state when above threshold. This can be hard to relate to the underlying stochastic dynamics, which has no endemic steady-state but can exhibit approximately stable behaviour.Here we relate the approximate models to the stochastic dynamics via the definition of the quasi-stationary distribution (QSD), which captures this approximately stable behaviour. We develop a system of ordinary d… Show more
“…In addition, because turning points Nminfalse(wfalse) become very large under weak selection, fixation probabilities could become less representative of what happens under anti-coordination games in finite populations. Even though pure states are absorbing ones, when mutations are considered the population might spend most of the evolutionary time in particular transient states [33,34], which are called quasi-stationary states in those cases [30–32,47]. In appendix A, we show that average conditional fixation times increase polynomially with Nminfalse(wfalse) as they do with N under neutral fixation, instead of increasing exponentially as is predicted for anti-coordination games under a fixed intensity of selection [34].…”
Section: Resultsmentioning
confidence: 99%
“…Figure 6 a suggests that turning points are inversely proportional to the intensity of selection Nminfalse(wfalse)∼1/w, which means that they would become increasingly large under the weak selection limit. Under this limit, it could be argued that the increase in the population sizes for which fixation probabilities start to increase would lead to a loss of significance of fixation processes due to most of the evolutionary time being spent in transient/mixed states [33,34], also called quasi-stationary states [30–32,47]. Figure 6Fixation process under game with payoff values false[5.5,5,6,3false], corresponding to dove fixation in the HD game.…”
Evolutionary game theory has proved to be a powerful tool to probe the self-organization of collective behaviour by considering frequency-dependent fitness in evolutionary processes. It has shown that the stability of a strategy depends not only on the payoffs received after each encounter but also on the population’s size. Here, we study
2
×
2
games in well-mixed finite populations by analyzing the fixation probabilities of single mutants as functions of population size. We proved that nine of the 24 possible games always lead to monotonically decreasing functions, similarly to fixed fitness scenarios. However, fixation functions showed increasing regions under 12 distinct anti-coordination, coordination and dominance games. Perhaps counter-intuitively, this establishes that single-mutant strategies often benefit from being in larger populations. Fixation functions that increase from a global minimum to a positive asymptotic value are pervasive but may have been easily concealed by the weak selection limit. We obtained sufficient conditions to observe fixation increasing for small populations and three distinct ways this can occur. Finally, we describe fixation functions with the increasing regions bounded by two extremes under intermediate population sizes. We associate their occurrence with transitions from having one global extreme to other shapes.
“…In addition, because turning points Nminfalse(wfalse) become very large under weak selection, fixation probabilities could become less representative of what happens under anti-coordination games in finite populations. Even though pure states are absorbing ones, when mutations are considered the population might spend most of the evolutionary time in particular transient states [33,34], which are called quasi-stationary states in those cases [30–32,47]. In appendix A, we show that average conditional fixation times increase polynomially with Nminfalse(wfalse) as they do with N under neutral fixation, instead of increasing exponentially as is predicted for anti-coordination games under a fixed intensity of selection [34].…”
Section: Resultsmentioning
confidence: 99%
“…Figure 6 a suggests that turning points are inversely proportional to the intensity of selection Nminfalse(wfalse)∼1/w, which means that they would become increasingly large under the weak selection limit. Under this limit, it could be argued that the increase in the population sizes for which fixation probabilities start to increase would lead to a loss of significance of fixation processes due to most of the evolutionary time being spent in transient/mixed states [33,34], also called quasi-stationary states [30–32,47]. Figure 6Fixation process under game with payoff values false[5.5,5,6,3false], corresponding to dove fixation in the HD game.…”
Evolutionary game theory has proved to be a powerful tool to probe the self-organization of collective behaviour by considering frequency-dependent fitness in evolutionary processes. It has shown that the stability of a strategy depends not only on the payoffs received after each encounter but also on the population’s size. Here, we study
2
×
2
games in well-mixed finite populations by analyzing the fixation probabilities of single mutants as functions of population size. We proved that nine of the 24 possible games always lead to monotonically decreasing functions, similarly to fixed fitness scenarios. However, fixation functions showed increasing regions under 12 distinct anti-coordination, coordination and dominance games. Perhaps counter-intuitively, this establishes that single-mutant strategies often benefit from being in larger populations. Fixation functions that increase from a global minimum to a positive asymptotic value are pervasive but may have been easily concealed by the weak selection limit. We obtained sufficient conditions to observe fixation increasing for small populations and three distinct ways this can occur. Finally, we describe fixation functions with the increasing regions bounded by two extremes under intermediate population sizes. We associate their occurrence with transitions from having one global extreme to other shapes.
“…Additionally, because turning points N min (w) become very large under weak selection, fixation probabilities could become less representative of what happens under anti-coordination games in finite populations. Even though pure states are absorbing ones, when mutations are considered, the population might spend most of the evolutionary time in par-ticular transient states [33,34], which can also be called quasi-stationary states [30,31,32,47]. In appendix A, we show that average conditional fixation times increase polynomially with N min (w) as they do with N under neutral fixation, instead of increasing exponentially as is predicted for anti-coordination games under a fixed intensity of selection [34].…”
Section: Fixation Probability Functions With One Minimummentioning
confidence: 99%
“…6a suggests that turning points are inversely proportional to intensity of selection N min (w) ∼ 1/w, which means that they would become very large under the weak selection limit. Under this limit, it could be argued that the increase in the population sizes for which fixation probabilities increase would lead to a loss of significance of fixation processes due to most of the evolutionary time being spent in transient/mixed states [33,34], also called quasi-stationary states [30,31,32,47].…”
Section: A the Effect Of The Weak Selection Limitmentioning
The self-organization of collective behaviour is a topic of interest in numerous research fields, and in this context, evolutionary game theory has proved to be a powerful probing tool. Even though initial evolutionary models with frequency-dependent fitness assumed infinite populations, it has been shown that the stability of a strategy may depend not only on the game's payoff matrix but on the size of a finite population. To perform a systematic analysis of 2x2 games in well-mixed finite populations, we start by proving that 9 of the 24 possible payoff orderings always lead to single mutant fixation probability functions decreasing monotonically with population size as they trivially do under fixed fitness scenarios. However, we observe a diversity of fixation functions with increasing regions under 12 other orderings, which included anti-coordination games (e.g. Hawk-Dove/Snowdrift game), the fixation of dominating strategies (e.g. defectors in the Prisoner's Dilemma), and the fixation of stag hunters under that game (the only exception in coordination games). Fixation functions that increase from a global minimum to a finite asymptotic value are pervasive. These may have been easily concealed by the weak selection limit. We prove under which payoff matrices it is possible to have fixation increasing for the smallest populations and find three different ways this can happen. Finally, we describe two distinct fixation functions having two local extremes and associate them with transitions from ones with one global extreme.
“…And also they state potential control strategies which may eventually curb the epidemic level of the virus whenever there is an outbreak. Also, we have the following authors worked on optimal control strategy on some infectious diseases which are as follows cancer immunology [41] , Covid-19 [40] , [42] , Hepatitis B virus [43] , Cancer therapies [44] , Corruption dynamics [45] , Chronic myelogenous leukemia [46] , Breast cancer [47] , Ebola virus [32] , [48] , stochastic SIS epidemic dynamics [52] and Malaria model [28] , [53] .…”
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