Optimal vaccination policy to prevent endemicity: a stochastic model

Abstract

We examine here the effects of recurrent vaccination and waning immunity on the establishment of an endemic equilibrium in a population. An individual-based model that incorporates memory effects for transmission rate during infection and subsequent immunity is introduced, considering stochasticity at the individual level. By letting the population size going to infinity, we derive a set of equations describing the large scale behavior of the epidemic. The analysis of the model's equilibria reveals a criterion for the existence of an endemic equilibrium, which depends on the rate of immunity loss and the distribution of time between booster doses. The outcome of a vaccination policy in this context is influenced by the efficiency of the vaccine in blocking transmissions and the distribution pattern of booster doses within the population. Strategies with evenly spaced booster shots at the individual level prove to be more effective in preventing disease spread compared to irregularly spaced boosters, as longer intervals without vaccination increase susceptibility and facilitate more efficient disease transmission. We provide an expression for the critical fraction of the population required to adhere to the vaccination policy in order to eradicate the disease, that resembles a well-known threshold for preventing an outbreak with an imperfect vaccine. We also investigate the consequences of unequal vaccine access in a population and prove that, under reasonable assumptions, fair vaccine allocation is the optimal strategy to prevent endemicity.

Keywords: Age-structured model; Endemicity; Heterogeneous vaccination; Mitigation; Non-Markovian model; Recurrent vaccination; Varying infectiousness and susceptibility; Waning immunity.

Fig. 1

Typical evolution of the susceptibility $\sigma$ (in , on top) and infectiousness $\lambda$ (in , below) of an individual (color figure online)

Fig. 2

Independent simulations of the model (colored lines) for $N=500$ and of its deterministic limit, as $N\to \infty$ (solid black line). All parameter values are given in Table 1 (Appendix A). The simulations are initialized with a fraction $I_0=0.1$ of infectious individuals (color figure online)

Fig. 3

Bifurcation diagram for Eq. (5). For each value of $R_0$, the value of ${\int }_0^{\infty }I(t,a)\text{d}a$ is reported, for a large time $t=5000$. The simulations are initialized with a fraction $I_0=0.1$ of infectious individuals, all other parameter values are given in Table 1 (Appendix A). The dashed vertical grey line indicates the endemic threshold $1/\mathrm{\Sigma }$ computed from (18), above which we expect to see existence of a stable endemic equilibrium. In the two insets ${\int }_0^{\infty }I(s,a)\text{d}a$ is plotted as a function of time $s\le t$ for $R_0=2$ and $R_0=5$ (color figure online)

Fig. 4

Left: Solutions of the PDE (5) for three values of $R_0$ and ${\theta }_{\sigma }$. The parameters correspond to the grey dots on the right plot. All other parameters are given in Table 1 (Appendix A). Right: Bifurcation diagram of Eq. (5), as a function of $R_0$ and ${\theta }_{\sigma }$ (scale parameter, defined in Sect. A.1). Each point of the heatmap represents the value of ${\int }_0^{\infty }I(t,a)\text{d}a$ for a large time $t=300$. The grey curve is the endemic threshold $1/\mathrm{\Sigma }$ defined in (18), as a function of ${\theta }_{\sigma }$ (color figure online)

Fig. 5

Left: Solutions of the PDE (5) for three values of $R_0$ and ${\theta }_{\sigma }$. The parameters correspond to the grey dots on the right plot. All other parameters are given in Table 1 (Appendix A). Right: Bifurcation diagram of Eq. (20), as a function of $R_0$ and ${\theta }_{\sigma }$ (scale parameter, defined in Sect. A.1). The population is made of three subpopulations with contact matrix and vaccination parameters given in Table 2. Each point of the heatmap represents the value of the total fraction of infectious individuals ${\int }_0^{\infty }{I}_1(t,a)+I_2(t,a)+I_3(t,a)\text{d}a$ for a large time $t=300$. The grey curve is the endemic threshold $1/\rho$ as a function of ${\theta }_{\sigma }$, where $\rho$ is the leanding eigenvalue of M defined in (22) (color figure online)

Fig. 6

Behavior of $1/\rho$ as a function of the fairness parameter $\beta$, for different values of $\alpha$. The random variable T in (28) has a Gamma distribution with shape and scale parameters given by ${\kappa }_V$ and ${\theta }_V$ respectively, as in Table 1 (Appendix A). The population is assumed to be made of two groups of the same size, $p_1=p_2=\frac{1}{2}$. All other parameters are given in Table 1 (Appendix A) (color figure online)

Fig. 7

Numerical approximation of the function $F_{\text{e}}$ in (16). The susceptibility $\sigma$ is given by (40), where the parameters ${\kappa }_{\sigma }$ and ${\theta }_{\sigma }$ are given in the legend. The parameters of $T_V$ are given in Table 1 (Appendix A), and we assumed $\mathbb{E}[T_I]=0$ (color figure online)