An SIRS-model considering waning efficiency and periodic re-vaccination

TL;DR

This paper extends the SIRS framework to a SIRSV model that includes waning vaccine efficacy and periodic revaccination by introducing a vaccination-age structure, modeled both as a PDE (continuous age) and as a discretized ODE system. It derives equilibrium conditions, computes the basic reproduction number, and analyzes stability, revealing the possibility of multiple endemic equilibria and bifurcations that create bistable dynamics. Through numerical continuation and optimization, the study demonstrates that combining revaccination with non-pharmaceutical interventions is often necessary to control outbreaks, and it identifies conditions under which optimal vaccination rates emerge, particularly when vaccine waning is present. The work highlights the trade-offs between infection burden and intervention costs, and suggests directions for future extensions to heterogeneous populations and more detailed economic considerations.

Abstract

In this paper, we extend the classical SIRS (Susceptible-Infectious-Recovered-Susceptible) model from mathematical epidemiology by incorporating a vaccinated compartment, V, accounting for an imperfect vaccine with waning efficacy over time. The SIRSV-model divides the population into four compartments and introduces periodic re-vaccination for waning immunity. The efficiency of the vaccine is assumed to decay with the time passed since the vaccination. Periodic re-vaccinations are applied to the population. We develop a partial differential equation (PDE) model for the continuous vaccination time and a coupled ordinary differential equation (ODE) system when discretizing the vaccination period. We analyze the equilibria of the ODE model and investigate the linear stability of the disease-free equilibrium (DFE). Furthermore, we explore an optimization framework where vaccination rate, re-vaccination time, and non-pharmaceutical interventions (NPIs) are control variables to minimize infection levels. The optimization objective is defined using different norm-based measures of infected individuals. A numerical analysis of the model's dynamic behavior under varying control parameters is conducted using path-following methods. The analysis focuses on the impacts of vaccination strategies and contact limitation measures. Bifurcation analysis reveals complex behaviors, including bistability, fold bifurcations, forward and backward bifurcations, highlighting the need for combined vaccination and contact control strategies to manage disease spread effectively.

Figures (5)

• Figure 1: Graphical sketch of the condition $f(z_m)<0$ for the existence of two equilibria.
• Figure 2: Simulation of the epidemiological SIRSV-model \ref{['E:ODEsys-SIRSV']}, calculated using the parameter values and initial conditions provided in Table \ref{['Tab:Parameters']}, with $\omega_{k}=0.5$ constant (no waning effect). The last graph shows time series of $V_{k}$-compartments for $k=0$ (black), $k=45$ (blue) and $k=89$ (red).
• Figure 3: Numerical continuation of equilibrium points of system \ref{['E:ODEsys-SIRSV']} with respect to $\beta$ and $\nu$. Panels (a) and (b) are calculated for the parameter set given in Fig. \ref{['fig-sol-ini']}, while panel (c) is obtained for the same parameters, except for $\nu=0.0003$ and $\alpha=0.01$. Panels (a)--(c) display the infection compartment behavior at equilibrium on the left vertical axis (blue), the second vertical axes (right, in red) represent the basic reproduction number $\mathcal{R}_{0}$. In these plots, solid blue lines indicate stable equilibrium branches, and dashed blue lines represent unstable equilibrium branches. Throughout the calculations, various critical points are identified, including the branching points BP1 and BP2 at $\beta=0.2$, and the fold bifurcation F at $\beta\approx0.13303$. Panel (d) presents time series for the system \ref{['E:ODEsys-SIRSV']} at the test point P1 ($\beta=0.16$), where two stable solutions co--exist.
• Figure 4: Simulation of the epidemiological SIRSV-model \ref{['E:ODEsys-SIRSV']} subject to the contact restrictions \ref{['E:NPI']} and \ref{['eq-rho']}, calculated for the parameter set provided in Table \ref{['Tab:Parameters']} with $\eta=\frac{1}{45}$ and $I_{\hbox{\tiny max}}=6$. The $V_{k}$-compartments follow the same color code as in Fig. \ref{['fig-sol-ini']}. Panel (a) is calculated considering the waning effect, while panel (b) is computed for $\omega_{k}=0.5$ constant (no waning).
• Figure 5: Numerical continuation of the periodic solution shown in Fig. \ref{['fig-sol-control']} with respect to $\eta$ and $\nu$. Panels (a)--(c) show the behavior of the average infection occurrences per period. Panels (d)--(f) display the variation of the political cost (left vertical axis, blue) and vaccination cost (right vertical axis, red) as the control parameter changes. All calculations are performed using the periodic response in Fig. \ref{['fig-sol-control']}(a) (waning effect), except for the curves in panels (c) and (f), which correspond to the solution in Fig. \ref{['fig-sol-control']}(b) (no waning). In these diagrams, both political and vaccination costs are normalized between $0$ and $1$ for better comparison.