Evolution problem is now a hot topic in the mathematical biology field. This paper investigates the adaptive evolution of pathogen virulence in a susceptible-infected (SI) model under drug treatment. We explore the evolution of a continuous trait, virulence of a pathogen, and consider virulence-dependent cure rate (recovery rate) that dramatically affects the outcome of evolution. With the methods of critical function analysis and adaptive dynamics, we identify the evolutionary conditions for continuously stable strategies, evolutionary repellers, and evolutionary branching points. First, the results show that a high-intensity strength drug treatment can result in evolutionary branching and the evolution of pathogen strains will tend towards a higher virulence with the increase of the strength of the treatment. Second, we use the critical function analysis to investigate the evolution of virulence-related traits and show that evolutionary outcomes strongly depend on the shape of the trade-off between virulence and transmission. Third, after evolutionary branching, the two infective species will evolve to an evolutionarily stable dimorphism at which they can continue to coexist, and no further branching is possible, which is independent of the cure rate function.
Virus leads to the spread of the disease, which attracts the attention and research of many researchers [
The rest of the paper is organized as follows. In Section
The epidemic model itself for influenza and venereal disease is of the simplest kinds, it is assumed that the cured infective hosts recover and become susceptible, and thus infection is not permanent as they have recovery rate. The infective individuals, although possibly subject to a higher mortality and lower fertility than the susceptible, are otherwise active in the population. The model considers a pathogen which increases the mortality of infected hosts by its virulence
In this subsection, in case of a monomorphic infective population with trait
Concerning the effect of virulence
In this model we assume linear density-dependent mortality. Let
To find the invasion fitness for a mutant infected population, we extend the resident population model (
To show its structure, we rewrite matrix
Note that
From (
If
Since mutations are random and very small, the evolutionary model of virulence
A singular strategy
In this model, we choose the cure rate function, transmission rate function, and mortality function of infected hosts like this:
We use “pairwise invasibility plot” (PIP) to study how the strength of treatment affects the outcome of evolution. So we choose three different values of
Pairwise invasibility plots. The resident and mutant strategies are denoted by
Correspondingly, we plot the bifurcation diagram in Figure
Bifurcation diagram for evolutionarily singular strategy
As stated above, we specify the
As such (
The solutions of (
If the trade-off function is concave, we superpose such a trade-off function
From (
If the trade-off function is convex and at the same time less convex than the critical function at the point of tangency, then the singularity is an evolutionary branching point.
The trade-off function is convex, which will not satisfy the ESS condition (
In Figure
Critical function analysis (a)-(b), (a)
In this section we investigate whether further evolutionary branching or evolutionarily stable coexistence of the two resident strains occurs. After branching, population dynamics of two residents with phenotypic traits
The selection gradients
When individual mutations are random and sufficiently small, the evolutionary model of traits
Further evolutionary branching cannot happen in this model whether the cure rate function is concave or convex.
A singular coalition is evolutionarily stable if all its constituent strategies are ESS, that is, if
The condition for absolute convergence stability is given by
Now, the dynamics of evolution in dimorphic pathogen is shown through the coexistence plots and evolutionary branching trees. The dynamics of evolution as predicted by the model are conformed by numerical simulations (Figure
The coexistence plots are depicted, two strains are presented on separate axes and the gray areas define combinations of strains that are mutually invasible and the corresponding evolutionary branching. Shaded areas indicating protected dimorphism are separated by stable (black) and unstable (red) isoclines at which selection gradient vanishes in either
With the effect of increasing
In this paper, we employ the susceptible-infected epidemic model under drug treatment to explore the adaptive evolution of pathogen virulence. Our results show that the strength of treatment noticeably affects the outcome of evolution: continuously stable strategies, evolutionary repellers, and evolutionary branching points. With the methods of critical function analysis and adaptive dynamics, we have shown that evolutionary branching of virulence is possible in a susceptible-infected model with cure rate. Conditions ensuring evolutionary stability and branching for singular strategies and coalitions are derived in both monomorphic and dimorphic environments.
We obtain the biological interpretation for the evolution of pathogen virulence that if there is no treatment or small treatment, then the pathogen virulence can be smaller and evolutionarily stable, while a powerful treatment could accelerate pathogen evolution towards higher virulence.
First, by using the methods of “pairwise invasibility plot” and critical function analysis, we show that if the trade-off function is concave, the evolutionary dynamics of this model can lead to evolutionarily stable, while if the trade-off is convex then the singularity can be an evolutionary branching point. Second, increase in the strength of the drug treatment can increase evidently in the singular strategy value and eventually turns continuously stable strategy into the branching point. This shows that, with increasing in cure rates, pathogen populations will tend towards the evolutionary branching points with higher virulence. Note that this situation is supported by empirical data [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper is supported by the National Natural Science Foundation of China (no. 11371230), Shandong Provincial Natural Science Foundation, China (no. ZR2012AM012), the SDUST Research Fund for Scientific Research Team “Systems Biology and Dynamics,” and a Project of Shandong Province Higher Educational Science and Technology Program of China (Grant no. J13LI05).