Human-vector malaria transmission model structured by age, time since infection and waning immunity. (February 2022) - Richard, Quentin; Choisy, Marc; Lefèvre, Thierry; Djidjou-Demasse, Ramsès
Abstract: 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 R 0 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 R 0 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 R 0 = 1 leading to the existence, or not, of multiple endemic equilibrium when 0 ≪ R 0 < 1 . Finally, the latter theoretical results are enlightened by numerical simulations.