Optimal Virulence, Diffusion and Tradeoffs

Abstract

In this work we propose a variant of a classical SIR epidemiological model where pathogens are characterized by a (phenotypic) mutant trait x. Imposing that the trait x mutates according to a random walk process and that it directly influences the epidemiological components of the pathogen, we studied its evolutionary development by interpreting the tenet of maximizing the basic reproductive number of the pathogen as an optimal control problem. Pontryagin’s maximum principle was used to identify the possible optimal evolutionary strategies of the pathogen. Qualitatively, three types of optimal evolutionary routes were identified and interpreted in the context of virulence evolution. Each optimal solution imposes a different tradeoff relation among the epidemiological parameters. The results predict (mostly) two kinds of infections: short-lasting mild infections and long-lasting acute infections.

Appendix A: The Diffusion Term

We assume that the pathogen is characterized by some (phenotypic) trait $x\in [0,T]$ that can mutate. The mutation process is modeled according to a random walk with steps of length $\delta x > 0$.

Fig. 9

Mutation of trait x modeled according to a random walk

Let v(t, x) denote the density of infectious hosts with pathogen of trait x at time t. We want to describe the time evolution of v(t, x). Let $p\in [0,1]$ be the probability that an infective host transmits a mutant pathogen. If h denote an infinitesimal time, then (see Fig. 9)

$$\begin{aligned} v(t + h,x)= & {} v(t,x) \quad \text {(infected with trait} x \text {at the time} t)\\&+ \frac{p}{2}h\beta (x-\delta x)v(t,x-\delta x)S(t) \quad \text {(infected from trait} x-\delta x) \\&+ (1-p)h\beta (x)v(t,x)S(t) \quad \text {(infected from trait} x) \\&+ \frac{p}{2}h\beta (x+\delta x)v(t,x+\delta x)S(t) \quad \text {(infected from trait} x+\delta x). \\ \end{aligned}$$

Rearranging the terms

$$\begin{aligned} \begin{aligned}&v(t+h,x) = v(t,x) + S(t)\beta (x)v(t,x)h\\&\quad \quad \quad \quad \quad \quad +\,\frac{hp}{2}S(t)[\beta (x-\delta x)v(t,x-\delta x) - 2\beta (x)v(t,x) + \beta (x+\delta x)v(t,x+\delta x)]. \end{aligned} \end{aligned}$$

(17)

Assuming that $\beta (x)$ and v(t, x) are analytic functions, the Taylor’s series expansion for the product $\beta (x)v(t,x)$ implies, for $\delta x$ small, that Eq. (17) can be approximated by

$$\begin{aligned} v(t+h,x) = v(t,x) + hS(t)\beta (x)v(t,x) + \dfrac{hp(\delta x)^2}{2}S(t)\dfrac{\partial ^2}{\partial x^2}\big ( \beta (x)v(t,x)\big ). \end{aligned}$$

In the limit ($h \rightarrow 0$), taking into account demographic variation, we obtain

$$\begin{aligned} \frac{\partial v}{\partial t} = S(t) \, \beta (x)\, v(t,x) + D\, S(t) \, \frac{\partial ^2 }{\partial x^2}\,\left( \beta (x) \, v(x,t) \right) \, - (\gamma (x) + m(x) + u)\,v(t,x), \end{aligned}$$

where $D = \frac{p(\delta x)^2}{2}$. The Neumann boundary conditions

$$\begin{aligned} \left. \dfrac{\partial }{\partial x}\big ( \beta (x)v(t,x) \big ) \right| _{x=0} = 0 \quad \text { e} \quad \left. \dfrac{\partial }{\partial x}\big ( \beta (x)v(t,x) \big ) \right| _{x=T} = 0 \end{aligned}$$

(18)

are imposed.