An epidemiological model with waning immunity and reinfection

Abstract

Waning immunity and reinfection are critical features of many infectious diseases, but epidemiological models often fail to capture the intricate interaction between an individual's history of immunity and their current infection status; when they do, the approach is usually overly simplistic. We develop a novel dual-age structured model that simultaneously tracks immunity age (time since the last recovery from infection) and infection age (time since infection) to analyze epidemic dynamics under conditions of waning immunity and reinfection. The model is formulated as a system of age-structured partial differential equations that describe susceptible and infected populations stratified by both immunity and infection ages. We derive basic reproduction numbers associated with the model and numerically solve the system using a second-order Runge-Kutta scheme along the characteristic lines. We further extend the model to explore vaccination interventions, specifically targeting individuals according to their immunity age. Numerical results reveal that higher contact rates produce larger amplitude oscillations with longer interepidemic periods. The relationship between initial infection levels and long-term epidemic behavior is nonmonotonic. Vaccination efficiency depends critically on the viral load profile across immunity and infection age, with more pronounced viral load distributions requiring higher vaccination rates for disease elimination. Most efficient vaccination strategies begin with intermediate immunity ages rather than targeting only fully susceptible individuals. The structured dual-age framework provides a flexible approach to analyzing the dynamics of reinfection and evaluating targeted vaccination strategies based on the history of immunity.

Figures (12)

• Figure 1: Structure of the basic dual-age epidemiological model. Note that immunity age $\tau$ increases only in the susceptible class and infection age $\theta$ increases only in the infected class. Infected individuals with any combination $(\theta,\tau)$ of ages have new immunity age $\tau = 0$ after recovery.
• Figure 2: Structure of the extended dual-age model with separated classes for never infected and first-time infected individuals.
• Figure 3: Two pathogen load functions, $v(\theta,\tau)$ and $v_2(\theta,\tau)$, with the same total load. Both subfigures show the pathogen load depending on infection age $\theta$ and immunity age $\tau$.
• Figure 4: Basic reproduction numbers as functions of the parameter $\theta' \in [0,\bar{\theta}]$: (i) $\hat{{\cal R}}_0$ (corresponding to $\theta' = 0$ and $\tau = \bar{\tau}$ for all the initially infected population) -- lower horizontal line; (ii) ${\cal R}_0$ (corresponding to the worst case initial population) -- upper horizontal line; (iii) ${\cal R}_0[I^0_{\theta'}]$ (corresponding to initial population $I^0_{\theta'}$, $\theta' \in [0, \bar{\theta}]$) -- the curve.
• Figure 5: Left plot: the susceptible population $S(t,\tau)$ for the baseline case. The figure shows the time $t$ dependent evolution of the distribution of the susceptible population $S(t, \tau)$ w.r.t. the immunity age $\tau$. Right plot: the susceptible population for various scenarios for the deviation $\Delta(\tau) := \sigma(\tau) - \sigma(\bar{\tau})$ for $\tau \geq \bar{\tau}$; the lines correspond to $h = \|\Delta \|_{L_\infty}$ as percentage of $\sigma(\bar{\tau}): \;2, \,1, \,0.5, \,0.25, \,0$.

Theorems & Definitions (1)

• Remark 2.1