The role of antibody-mediated immunity in shaping the seasonality of respiratory viruses

TL;DR

A novel susceptible-infectious-susceptible (SIS) immuno-epidemiological model of respiratory virus spread is developed, where the susceptible population is stratified by their antibody level against the currently circulating strain of the virus, with this decaying as both antibody waning and antigenic drift occur.

Abstract

In temperate regions, respiratory virus epidemics recur on a yearly basis, primarily during the winter season. This is believed to be induced by seasonal forcing, where the rate at which the virus can be transmitted varies cyclically across the course of each year. Seasonal epidemics can place substantial burden upon the healthcare system, with large numbers of infections and hospitalisations occurring across a short time period. However, the interactions between seasonal forcing and the factors necessary for epidemic resurgence - such as waning immunity, antigenic variation or demography - remain poorly understood. In this manuscript, we examine how the dynamics of antibody waning and antigenic variation can shape the seasonal recurrence of epidemics. We develop a novel susceptible-infectious-susceptible (SIS) immuno-epidemiological model of respiratory virus spread, where the susceptible population is stratified by their antibody level against the currently circulating strain of the virus, with this decaying as both antibody waning and antigenic drift occur. In the absence of seasonal forcing, we demonstrate the existence of two Hopf bifurcations over the effective antibody decay rate, with associated periodic model solutions. When seasonal forcing is introduced, we identify complex interactions between the strength of forcing and the effective antibody decay rate, yielding myriad dynamics including multi-year periodicity, quasiperiodicity and chaos. The timing and magnitude of seasonal epidemics is highly sensitive to this interaction, with the distribution of infection timing (by time of year) varying substantially across the parameter space. Finally, we show that seasonal forcing can produce resonant damping resulting in a cumulative infection incidence that is less than would otherwise be observed.

Figures (23)

• Figure 1: Compartmental flow diagram of our immuno-epidemiological model of respiratory virus transmission. The susceptible class $S_i$ is stratified according to their effective antibody level $c_i$ (which may range from $2^0$ up to $2^a$), with this defining $\omega_i$, the level of immune protection against infection given contact with an infected individual.
• Figure 1: Characteristic dynamics of the immunity-structured model of respiratory virus transmission. Model parameters are as in Table 1. A: Prevalence (i.e. proportion) of infectious individuals $I$ over time. B: Proportion of susceptible individuals in each strata $S_i$ for $0 \leq i \leq k$. C: The mean antibody concentration across the susceptible population, calculated as an arithmetic mean across concentrations (i.e. not a geometric mean).
• Figure 2: Characteristic dynamics of the immunity-structured model of respiratory virus transmission. Model parameters are as in Table \ref{['table-params']}. A: Prevalence (i.e. fraction) of infectious individuals $I$ over time. B: Fraction of susceptible individuals in the minimum antibody strata $S_0$. C: Fraction of susceptible individuals in the strata $S_2$ through to $S_k$ (odd strata not plotted for visual clarity), with each strata plotted as an individual ribbon. D: The mean effective antibody level across the susceptible population, calculated as an arithmetic mean (i.e. not a geometric mean).
• Figure 2: Eigenvalues of the linearised ODE system at the fixed point solution as antibody decay rate $r$ is varied from $0$ to $0.15$. The linearised system was reduced by taking $I = 1-\sum_{i=0}^kS_i$. A: Path of the eigenvalues $\lambda_i$ as decay rate is varied. B: Maximal real part of the eigenvalues. Two Hopf bifurcations occur as this value crosses the x-axis.
• Figure 3: Dynamics of the immuno-epidemiological model for varying values of effective antibody decay rate $r$ (all other model parameters are as in Table \ref{['table-params']}). The system was evaluated across a period of 100 years (36,500 days), following 100 years of burn-in. A: Bifurcation diagram over varying effective antibody decay rate $r$, with lines depicting infection prevalence at the model solutions (noting the log scale on the y-axis). For the periodic solutions, the light blue lines correspond to the maximal or minimal infection prevalence across each solution. For low values of the decay rate (e.g. less than $r = 0.01\;\text{days}^{-1}$), the minimal infection prevalence rapidly approaches values where stochastic effects would be dominant in realistic settings and our deterministic results must be interpreted with care. The separatrix between the stable limit cycle and the stable fixed point for values of $r$ less than approximately $0.005\;\text{days}^{-1}$ is not depicted as it could not be identified numerically. B: The frequency (in years$^{-1}$) of the periodic model solutions for varying effective antibody decay rate $r$. C: The average annual infection incidence at the model solution (across both fixed point and stable periodic solutions) across effective antibody decay rate $r$, calculated as the mean daily infection incidence multiplied by 365. D: Exemplar dynamics in infection prevalence across the three dynamical regimes, corresponding to the vertical dashed lines in A-C. Those which tend towards a stable periodic solution are illustrated in blue while those which tend towards a stable fixed point are black. The solution for $r = 0.003\;\text{days}^{-1}$ (panel i) which tends towards the stable fixed point was initialised by sampling a random point in state space near the fixed point. Note that the x-axis extent differs for panel i.