Human-vector malaria transmission model structured by age, time since infection and waning immunity

Malaria is one of the most common mosquito-borne diseases widespread in tropical and subtropical regions, causing thousands of deaths every year in the world. Few models considering a multiple structure model formulation including (i) the chronological age of human and mosquito populations, (ii) the time since they are infected, and (iii) humans waning immunity (i.e. the progressive loss of protective antibodies after recovery) have been developed. In this paper we formulate an age-structured model containing three structural variables. Using the integrated semigroups theory, we first handle the well-posedness of the model proposed. We also investigate the existence of steady-states. A disease-free equilibrium always exists while the existence of endemic equilibria is discussed. We derive the basic reproduction number R0 which expression highlights the effect of the above structural variables on key important epidemiological traits of the human-vector association such as vectorial capacity (i.e., vector daily reproduction rate), humans transmission probability, and survival rate. The expression of R0 obtained here generalizes the classical formula of the basic reproduction number. Next, we derive a necessary and sufficient condition that implies the bifurcation of an endemic equilibrium. In the specific case where the age-structure of the human population is neglected, we show that a bifurcation, either backward of forward, may occur at R0=1 leading to the existence, or not, of multiple endemic equilibrium when 0≪R0<1. Finally, the latter theoretical results are enlightened by numerical simulations.