An immuno-epidemiological model with waning immunity after infection or vaccination

Abstract

In epidemics, waning immunity is common after infection or vaccination of individuals. Immunity levels are highly heterogeneous and dynamic. This work presents an immuno-epidemiological model that captures the fundamental dynamic features of immunity acquisition and wane after infection or vaccination and analyzes mathematically its dynamical properties. The model consists of a system of first order partial differential equations, involving nonlinear integral terms and different transfer velocities. Structurally, the equation may be interpreted as a Fokker-Planck equation for a piecewise deterministic process. However, unlike the usual models, our equation involves nonlocal effects, representing the infectivity of the whole environment. This, together with the presence of different transfer velocities, makes the proved existence of a solution novel and nontrivial. In addition, the asymptotic behavior of the model is analyzed based on the obtained qualitative properties of the solution. An optimal control problem with objective function including the total number of deaths and costs of vaccination is explored. Numerical results describe the dynamic relationship between contact rates and optimal solutions. The approach can contribute to the understanding of the dynamics of immune responses at population level and may guide public health policies.

Keywords: 92-10; 92D30; 93C20; 93C95.

Fig. 1

Characteristic lines and illustration of the notations

Fig. 2

Evolution of epidemiological population groups without vaccination

Fig. 3

Evolution of the normalized densities of newly infected and newly recovered individuals

Fig. 4

Evolution of epidemiological groups with no mortality rate

Fig. 5

Evolution of epidemiological population groups with constant vaccination

Fig. 6

Saved lives per vaccine depending on the initial date of implementing vaccination and the vaccination rate v

Fig. 7

Evolution of epidemiological population groups with optimal vaccination applied

Fig. 8

Optimal Strategy for the baseline scenario: Administered vaccines, i.e. newly vaccinated individuals and comparison with the average immunity level of the susceptible group

Fig. 9

Dependence of the optimal vaccination policy on the time horizon [0, t], $t=400,500,600$

Fig. 10

The left plot presents the efficient frontier for the three control scenarios; the red dots on the Pareto curves indicate the Pareto points corresponding to the baseline case. The right plot depicts the time-dependence of the number of newly vaccinated individuals in the three control scenarios in the baseline case

Fig. 11

Objective function components for different contact rates