Treating symptomatic infections and the co-evolution of virulence and drug resistance

Abstract

Antimicrobial therapeutic treatments are by definition applied after the onset of symptoms, which tend to correlate with infection severity. Using mathematical epidemiology models, I explore how this link affects the coevolutionary dynamics between the virulence of an infection, measured via host mortality rate, and its susceptibility to chemotherapy. I show that unless resistance pre-exists in the population, drug-resistant infections are initially more virulent than drug-sensitive ones. As the epidemic unfolds, virulence is more counter-selected in drug-sensitive than in drug-resistant infections. This difference decreases over time and, eventually, the exact shape of genetic trade-offs govern long-term evolutionary dynamics. Using adaptive dynamics, I show that two types of evolutionary stable strategies (ESS) may be reached in the context of this simple model and that, depending on the parameter values, an ESS may only be locally stable. In general, the more the treatment rate increases with virulence, the lower the ESS value. Overall, both on the short-term and long-term, having treatment rate depend on infection virulence tend to favour less virulent strains in drug-sensitive infections. These results highlight the importance of the feedbacks between epidemiology, public health policies and parasite evolution, and have implications for the monitoring of virulence evolution.

Figure 1. Epidemiological model flow diagram.

Drug-resistant infections (in red) can originate from direct transmission or treatment failure in drug-sensitive infections (in blue). Dashed lines show infection events, arrows transition between states and lines with a circle death or recovery events. Individuals who die or recover from the infection are removed from the system. Parameter notations are detailed in Table 1.

Figure 2. Short term evolutionary dynamics.

A) Dynamics of the densities of susceptible hosts (dashed black line), of drug-susceptible infections (plain coloured lines) and drug-resistant infections (dotted coloured line). Each colour corresponds to one of the n = 20 strains. B) Same as A but the blue line shows the total density of drug-susceptible infections and the red line the total density of drug-resistant infections. The dashed lines are the predictions from the Price equation system. C) Average virulence in the drug-sensitive (blue) and drug-resistant infections (red) for numerical multi-strain simulations (plain line) and the Price equation system (dashed line). D) Same as panel C for transmission rate. E) Same as panel C for treatment rate. F) Fraction of the infections that are drug-resistant in the multi-strain simulation (plain line) or using the Price equation (dashed line). We assume no fitness cost (a = b = 1) and a transmission-virulence trade-off ($\beta (\alpha )={10}^{-4}\sqrt{\alpha }$). Other parameter values are λ = 0.02, μ = 4.5 × 10⁻⁵, ν = 0, a = b = 1, ρ = 0.1, Cov(α, γ) = 0.1, Var(γ) = 1/12, Var(α) = 1/6, $\overline{\alpha }(0)=1/2$, $\overline{\gamma }(0)=1/4$, S(0) = 10⁴, I_i = 5, and $I_i^R=0$. The default time unit is the week (but it can be rescaled without affecting the results qualitatively).

Figure 3. Long term evolutionary dynamics.

A) Dynamics of the densities of susceptible hosts (black), drug-susceptible infections (blue) and drug-resistant infections (red). B) Average virulence in the drug-sensitive (blue) and drug-resistant infections (red) for numerical multi-strain simulations. C) Dynamics of the densities of susceptible hosts (dashed black line), of drug-susceptible infections (plain coloured lines) and drug-resistant infections (dotted coloured line). Each colour corresponds to one of the n = 20 strains. D) Fraction of drug-resistant infections. Parameter values are the same as in Figure 2 for the multi-strain simulation.

Figure 4. Fitness of drug-resistant (plain red) and drug-sensitive (dotted blue and yellow) infections as a function of virulence.

This illustration assumes trade-offs between virulence and transmission rate (β(α) = β₀α^p) and treatment rate and virulence (γ(α) = g α^q). In the blue dashed case, treatment rate is more intense against virulent infections (g = 10) than in the dotted yellow case (g = 0.1). Vertical lines show the ESS values. Parameter values are β₀ = 10⁻⁴, p = 0.4, q = 0.75, μ + ν = 0.02, a = 0.1 and b = 0.5.