Hopf bifurcation in a chronological age-structured SIR epidemic model with age-dependent infectivity

Waning immunity can drive repeated waves of infections

In infectious disease models, it is known that mechanisms such as births, seasonality in transmission and pathogen evolution can generate oscillations in infection numbers. We show how waning immunity is also a mechanism that is sufficient on its own to enable sustained oscillations. When previously infected or vaccinated individuals lose full protective immunity, they become partially susceptible to reinfections. This partial immunity subsequently wanes over time, making individuals more susceptible to reinfections and potentially more infectious if infected. Losses of full and partial immunity lead to a surge in infections, which is the precursor of oscillations. We present a discrete-time Susceptible-Infectious-Immune-Waned-Infectious (SIRWY) model that features the waning of fully immune individuals (as a distribution of time at which individuals lose fully immunity) and the gradual loss of partial immunity (as increases in susceptibility and potential infectiousness over time). A special case of SIRWY is the discrete-time SIRS model with geometric distributions for waning and recovery. Its continuous-time analogue is the classic SIRS with exponential distributions, which does not produce sustained oscillations for any choice of parameters. We show that the discrete-time version can produce sustained oscillations and that the oscillatory regime disappears as discrete-time tends to continuous-time. A different special case of SIRWY is one with fixed times for waning and recovery. We show that this simpler model can also produce sustained oscillations. In conclusion, under certain feature and parameter choices relating to how exactly immunity wanes, fluctuations in infection numbers can be sustained without the need for any additional mechanisms.

Numerical approach to an age-structured Lotka-Volterra model

We study the impact of an age-dependent interaction in a structured predator-prey model. We present two approaches, the PDE (partial differential equation) and the renewal equation, highlighting the advantages of each one. We develop efficient numerical methods to compute the (un)stability of steady-states and the time-evolution of the interacting populations, in the form of oscillating orbits in the plane of prey birth-rate and predator population size. The asymptotic behavior when species interaction does not depend on age is completely determined through the age-profile and a predator-prey limit system of ODEs (ordinary differential equations). The appearance of a Hopf bifurcation is shown for a biologically meaningful age-dependent interaction, where the system transitions from a stable coexistence equilibrium to a collection of periodic orbits around it, and eventually to a stable limit cycle (isolated periodic orbit). Several explicit analytical solutions are used to test the accuracy of the proposed computational methods.