Stability Switches Induced by Immune System Boosting in an SIRS Model with Discrete and Distributed Delays

Abstract

We consider an epidemiological model that includes waning and boosting of immunity. Assuming that repeated exposure to the pathogen fully restores immunity, we derive an SIRS-type model with discrete and distributed delays. First we prove usual results, namely that if the basic reproduction number, R 0 , is less or equal than 1, then the disease free equilibrium is globally asymptotically stable, whereas for R 0 > 1 the disease persists in the population. The interesting features of boosting appear with respect to the endemic equilibrium, which can go through multiple stability switches by changing the key model parameters. We construct two-parameter stability charts, showing that increasing the delay can stabilize the positive equilibrium. Increasing R 0 , the endemic equilibrium can cross two distinct regions of instability, separated by Hopf-bifurcations. Our results show that the dynamics of infectious diseases with boosting of immunity can be more complex than most epidemiological models, and calls for careful mathematical analysis.

Temporal evolution of immunity distributions in a population with waning and boosting

“…On different meshes, we compute numerically the stationary solution of the MAXboost system (3)-(4), (9)- (10). Although, we know the explicit solution for the stationary distributionr, there is no explicit analytical formulation for the endemic equilibrium I * , which can only be determined numerically (see also [3]). We fix I * to the value computed on the finest mesh (M = 14000 equidistant mesh points) and we insert this value into the formula (14) of the stationary distribution.…”

“…To investigate the temporal evolution of how the distribution of these immunity levels change in the population, we propose a mathematical modeling framework along the lines of our previous works [4,3]. The core of the model is a hybrid system of equations of SIRS type, in which the immune population is structured by the level of immunity, whereas the susceptible and the infective populations are non-structured.…”

“…The core of the model is a hybrid system of equations of SIRS type, in which the immune population is structured by the level of immunity, whereas the susceptible and the infective populations are non-structured. In [4] we investigated the well-posedness of the general model and its basic qualitative properties, whereas in [3] we considered a special case of the hybrid system in form of delay differential equations (DDEs) with constant and distributed delay. Here we focus on the immune response identifying several possible scenarios for immune system boosts.…”

Temporal Evolution of Immunity Distributions in a Population with Waning and Boosting

“…To investigate the temporal evolution of how the distribution of these immunity levels changes in the population, we propose a mathematical modeling framework along the lines of our previous works [7,8]. The core of the model is a hybrid system of equations of SIRS type, in which the immune population is structured by the level of immunity, whereas the susceptible and the infective populations are nonstructured.…”

“…The core of the model is a hybrid system of equations of SIRS type, in which the immune population is structured by the level of immunity, whereas the susceptible and the infective populations are nonstructured. In [8], we investigated the well-posedness of the general model and its basic qualitative properties, whereas in [7], we considered a special case of the hybrid system in form of delay differential equations (DDEs) with constant and distributed delay. Here, we focus on the immune response identifying several possible scenarios for immune system boosts.…”