Multi-strain SIS dynamics with coinfection under host population structure

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.

💡 Research Summary

This paper develops a comprehensive analytical framework for the dynamics of multiple interacting pathogen strains that can co‑infect hosts, extending the classic SIS (susceptible–infected–susceptible) model to incorporate both strain diversity and host population structure. The authors consider a host population divided into a finite set of classes (e.g., species, vaccination status, contact degree) indexed by (k\in K). Each class has its own epidemiological parameters: transmission rate (\beta_k), clearance rate (\gamma_k), birth/death rate (r_k), and a second‑infection scaling factor (\sigma_k) that governs the rate at which a singly‑infected host acquires a second strain. Inter‑class contacts are encoded in a non‑negative, irreducible matrix (Q=(q_{ks})); the entry (q_{ks}) can be interpreted as the probability that a host of class (k) contacts a host of class (s).

In the absence of strain structure (Section 2) the model reduces to a classical SIS system with classes. The next‑generation matrix is (\mathrm{diag}(R)Q) where (R_k=\beta_k/(r_k+\gamma_k)). Its spectral radius (R_0) serves as the basic reproduction number: if (R_0>1) the disease persists and a unique endemic equilibrium exists, characterised by a class‑specific infection prevalence (\Theta_k) that satisfies (\Theta = Q T) with (T_k=I_k+D_k).

Section 3 introduces (N) pathogen strains. For each strain (i) and class (k) the model tracks singly‑infected hosts (I_{ik}) and co‑infected hosts (D_{ijk}) (where a host carries strain (i) and a second strain (j)). Strain interactions are captured by small perturbation parameters (\varepsilon_{ij}) (e.g., cross‑immunity, competition). The key “strain similarity” assumption posits that all strains share the same baseline transmission and clearance parameters; differences arise only through the (\varepsilon_{ij}) terms, which are assumed to be of order (\varepsilon\ll1).

Under this assumption the high‑dimensional ODE system can be decomposed into fast (neutral) and slow (non‑neutral) subsystems via singular‑perturbation theory. The fast subsystem collapses all strains onto a common manifold where the total infected proportion in each class follows the class‑level SIS dynamics described above. On this manifold the strain frequencies (z_i) (proportion of total infection attributable to strain (i)) evolve on a slower time scale.

The authors derive a reduced “global replicator equation” for the strain frequencies: \