A new chaotic attractor in a basic multi-strain epidemiological model with temporary cross-immunity

An epidemic multi-strain model with temporary cross-immunity shows chaos, even in a previously unexpected parameter region. Especially dengue fever models with strong enhanced infectivity on secondary infection have previously shown deterministic chaos motivated by experimental findings of antibody-dependent-enhancement (ADE). Including temporary cross-immunity in such models, which is common knowledge among field researchers in dengue, we find a deterministically chaotic attractor in the more realistic parameter region of reduced infectivity on secondary infection (‘‘inverse ADE’’ parameter region). This is realistic for dengue fever since on second infection people are more likely to be hospitalized, hence do not contribute to the force of infection as much as people with first infection. Our finding has wider implications beyond dengue in any multi-strain epidemiological systems with altered infectivity upon secondary infection, since we can relax the condition of rather high infectivity on secondary infection previously required for deterministic chaos. For dengue the finding of wide ranges of chaotic attractors open new ways to analysis of existing data sets.

💡 Research Summary

The paper investigates a deterministic multi‑strain epidemic model that incorporates two biologically realistic features: a temporary period of cross‑immunity after primary infection and a modified transmissibility for secondary infections. The authors denote the primary transmission rate by β₁ and the secondary transmission rate by β₂, introducing the ratio α = β₂/β₁. In the classic antibody‑dependent enhancement (ADE) scenario α > 1, secondary infections are more infectious and previous studies have shown that chaos can arise only when α is sufficiently large. Here the authors focus on the opposite, “inverse ADE” regime (α < 1), which is more appropriate for dengue fever because individuals experiencing a second infection are more likely to be hospitalized and therefore contribute less to the force of infection.

The model is built on an SIR‑type framework for two serotypes. After a primary infection, individuals enter a cross‑immune class for a fixed duration τ, during which they are completely protected against infection by the other serotype. After τ they become susceptible again, but now as secondary susceptibles whose infection rate is scaled by α. The resulting system of four coupled nonlinear ordinary differential equations captures the interplay of (i) depletion of susceptibles, (ii) temporary immunity, (iii) altered secondary transmissibility, and (iv) recovery.

Using extensive numerical continuation, the authors sweep the parameter space (α ∈