A symbiotic SIR process

TL;DR

The paper addresses how concurrent SIR epidemics with explicit co-infection evolve on networks when co-infected individuals recover at a rate $\bar{\mu}$ distinct from the single-infection rate $\mu$. It combines a mean-field network framework with a next-generation approach to show the invasion threshold $\mathcal{R}_0$ equals the maximum of the two independent thresholds, $\mathcal{R}_{0,A}$ and $\mathcal{R}_{0,B}$, and leverages an exchange symmetry in the symmetric regime $\lambda_1=\lambda_2$ to reduce the system to an invariant subspace for analytic insight. The core findings prove that slower co-infection recovery ($\bar{\mu}$ smaller) monotonically increases the instantaneous and cumulative co-infection burden and lowers a bound on epidemic duration that scales as $1/\bar{\mu}$; numerically, reduced $\bar{\mu}$ can also elevate the epidemic peak in high-transmission settings, supported by a sensitivity-equation analysis showing negative sensitivity near the peak. Overall, the results demonstrate that co-infection-specific recovery dynamics can substantially alter transient epidemic behavior, even in the absence of cross-immunity or endemic equilibria, with implications for understanding and mitigating co-circulating pathogens.

Abstract

We study a symmetric two-disease SIR co-infection model on networks in which co-infected individuals recover at a rate distinct from that of single infections. The model explicitly represents all co-infection states and features absorbing recovered compartments for both diseases. Within a mean-field network approximation, we derive the basic reproduction number of the coupled system and show that invasion thresholds coincide with those of two independent SIR processes. Exploiting an exchange symmetry in the equal-transmission regime, we reduce the dynamics to a lower-dimensional invariant subsystem and analyze the impact of the co-infection recovery rate. We prove that slower recovery of co-infected individuals monotonically increases the co-infection burden and yields a lower bound on epidemic duration that grows as the co-infection recovery rate decreases. Numerical simulations further indicate that reduced co-infection recovery can increase the epidemic peak, an effect supported by a sensitivity-equation analysis. Together, these results highlight how co-infection-specific recovery dynamics can substantially alter transient epidemic behavior, even in the absence of endemic equilibria.

Figures (4)

• Figure 1: Coinfection dynamics for different values of $\bar{\mu}$, with $\langle k\rangle=10, \lambda = 0.25, \mu = 1$.
• Figure 2: Ratio of duration of the epidemic with $\bar{\mu} < \mu = 1$ for different values of $\lambda$
• Figure 3: Maximum value of $J(t)$ for $\lambda = 0.2, 0.24, \dots, 0.6$ and $\bar{\mu} = 0.1, \dots,1$, with $\langle k\rangle = 10$ and $\mu = 1$.
• Figure 4: Sensitivity $\partial_{\bar{\mu}}J(t)$ and $J(t)$ in a numerical solution for the case $\langle k \rangle = 10, \lambda_1 = \lambda_2 = 0.3, \mu = 1, \bar{\mu} = 0.5$

Theorems & Definitions (2)

• Proposition 1
• proof