Multi-strain SIS dynamics with coinfection under host population structure

TL;DR

The paper develops an $N$-strain SIS coinfection model on structured host populations and derives a reduced, analytically tractable description of multi-strain dynamics. By leveraging strain similarity and slow–fast dynamics, it obtains a global replicator equation for strain frequencies that integrates host population structure into invasion fitness and pairwise coexistence criteria. The work provides neutral and quasi-neutral regimes, explicit expressions for invasion fitness, and three concrete applications: two-host-class systems, vaccination-structured populations, and heterogeneous contact networks, illustrating how structure shapes prevalence and strain selection. This framework offers a powerful tool for understanding persistence and selection in complex endemic ecosystems and suggests avenues for intervention design and network-aware epidemiology. Its significance lies in reducing high-dimensional, structured, multi-strain dynamics to a single replicator system that preserves key dependencies on host heterogeneity, contact networks, and coinfection interactions.

Abstract

Coinfection phenomena are common in nature, yet there is a lack of analytical approaches for coinfection systems with a high number of circulating and interacting strains. In this paper, we investigated a coinfection SIS framework applied to N strains, co-circulating in a structured host population. Adopting a general formulation for fixed host classes, defined by arbitrary epidemiological traits such as class-specific transmission rates, susceptibilities, clearance rates, etc., our model can be easily applied in different frameworks: for example, when different host species share the same pathogen, in classes of vaccinated or non-vaccinated hosts, or even in classes of hosts defined by the number of contacts. Using the strain similarity assumption, we identify the fast and slow variables of the epidemiological dynamics on the host population, linking neutral and non-neutral strain dynamics, and deriving a global replicator equation. This global replicator equation allows to explicitly predict coexistence dynamics from mutual invasibility coefficients among strains. The derived global pairwise invasion fitness matrix contains explicit traces of the underlying host population structure, and of its entanglement with the strain interaction and trait landscape. Our work thus enables a more comprehensive study and efficient simulation of multi-strain dynamics in endemic ecosystems, paving the way to deeper understanding of global persistence and selection forces, jointly shaped by pathogen and host diversity.

Figures (4)

• Figure 1: Model illustration for 2-strain SIS system outcomes on a two-class host population, each with a different basic-reproduction number and mean coinfection vulnerability. The strain 1 is the only survivor in both classes $A$ and $B$ ($\lambda_1^2>0,\lambda_2^1<0$ in both extremes $q_A=0,q_A=1$). By continuity, this is true when $q_A\approx 0$ or $q_A\approx 1$ (filled in blue). For intermediate values of $q_A$, there are 3 possibilities (bottom panel): i) strain 1 still is the only survivor (blue-shaded region), ii) both strains coexist (filled in gray) or iii) strain 2 is the only survivor (filled in red). The shifts between these qualitative regimes occur exactly at the values when $\mu$ of the heterogeneous population intersects with $\mu_{crit}^1$ and $\mu_{crit}^2$ (top panel). The epidemiological parameters are $\mathcal{R}_A=1.5$, $\mathcal{R}_B=5$, $\sigma_A=8$ and $\sigma_B=1$ so that $\mu_A=\mu_B=\frac{1}{4}$. The perturbations from neutrality among strains are in pairwise vulnerabilities to coinfection madec2020predicting: $(\alpha_{ij})=5.14-1-2.7.$
• Figure 2: Vaccination-induced heterogeneity in $\beta_k=\beta_0(1-v_k)$ may affect strain selection through its effect on $\mu$ given in \ref{['eq:mu_app']}. The heterogeneity of $\mathcal{R}_k=\beta_k(r+\gamma)^{-1}$ may result in a smaller or larger $\mu$ than the homogeneous value $\mu^0$. The relative effect depends on the coinfection vulnerability factor $\sigma$, on the mean vaccine efficacy relative to $q$ ($v=\mathbb{E}_q(v_k)$), and on the heterogeneity of the vaccine-effect distribution $\mathrm{std}_q(v_k)=\mathbb{E}_q(v_k^2)-v^2$. On average, small values of $\sigma$ and large values of $v$ lead to smaller $\mu$ under vaccination, while large $\sigma$ and small $v$ lead to larger $\mu$. The effect strengthens as $\mathrm{std}_q(v_k)$ increases. Hence, transmission heterogeneity induced by vaccination may alter $\mu$ and thereby affect the dynamics and final outcomes of interacting pathogens. Here $\beta_0=10$.
• Figure 3: Vaccination-induced heterogeneity alters strain selection through its effect on $\mu$ given in \ref{['eq:mu_app']}. We illustrate how $\mu$ pre- and post-vaccine shapes four-strain replicator dynamics. All simulations use $\beta_0=10$, deviation from neutrality $\alpha$ defined in \ref{['alphavaccin']} and mean vaccine efficacy of 50%: $\mathbb{E}_q(v_k)=0.5$. The first row corresponds to a multi-strain system with relatively more competition among strains $\sigma=0.1$ (lower coinfection vulnerability factor $<1$), and the second row to a system with relatively more facilitation among strains $\sigma=4$ (high coinfection vulnerability factor $>1$). Left panels show dynamics under a homogeneous universal vaccine; right panels show dynamics under a universal vaccine with the same mean efficacy but variable protection across individuals $\mathrm{std}_q(v_k)=0.4$. a-b. Heterogeneity in vaccine effect decreases $\mu$ for small $\sigma$ (low coinfection propensity), and changes drastically the outcome of strain selection. c-d. The same heterogeneity in vaccine effect can markedly increase $\mu$ for large $\sigma$ (higher coinfection propensity), again significantly modifying strain selection.
• Figure 4: Effect of host contact network heterogeneity on $\mu$ — a key driver of strain selection in the replicator.a. We set and the per contact rate of infection $\rho=0.6$. We compare the values of $\mu(\mathcal{N})$ and $\mu(\mathcal{N}_0)$ for 1000 randomly generated networks and random coinfection susceptibility factor $\sigma\in[0,5]$. For each network we draw one marker with coordinates $\left(\mathrm{std}(\mathcal{N}), \mu(\mathcal{N})/\mu(\mathcal{N}_0)\right)$ and a color corresponding to the value of $\sigma$. We fix $\rho=0.6$ and the mean contact degree $\mathbb{E}_p(k)=5$ yielding a fixed reproduction number in the homogeneous situation, $\mathcal{R}(\mathcal{N}_0)=\rho\,\mathbb{E}_p(k)=3.$. We observe that the ratio — and thus the effect of network heterogeneity on strain dynamics — increases with the heterogeneity $\mathrm{std}(\mathcal{N})$ and decreases with the epidemiological parameters $\sigma$. For a given deviation from neutrality $\alpha's$, a decrease in $\mu$ stabilizes the dynamics as shown in the two other sub-figures. b. Here we have set, $\sigma=0.4$ and $\rho=0.3$ thus $\mu(\mathcal{N}_0)=\left(\sigma (\rho\mathbb{E}_p(k)-1\right)^{-1}=5$ and $(\alpha_{ij})_{1\leq i,j\leq 3}=0.15-0.080.49-0.07-0.16-0.420.32-0.38-0.37$ and simulate the corresponding replicator system starting from uniform initial conditions among 3 strains. c. Same as in b but assuming a random contact network for the host population resulting in the new lower value of $\mu(\mathcal{N}).$

Theorems & Definitions (44)

• Proposition 2.1
• Remark 2.1: An important particular case
• Remark 2.2
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Theorem 2.4
• proof
• Corollary 2.5