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

Abstract

In contrast to the many theoretical studies on the transmission of human-mosquitoes malaria infection, few studies have considered a multiple structure model formulations including (i) the chronological age of humans and mosquitoes population, (ii) the time since humans and mosquitoes are infected and (iii) humans waning immunity (i.e., the progressive loss of protective antibodies after recovery). Such structural variables are well documented to be fundamental for the transmission of human-mosquitoes malaria infections. Here we formulate an age-structured model accounting for the three structural variables. Using integrated semigroups theory, we first handle the well-posedness of the model proposed. We also investigate the existence of model's steady-states. A disease-free equilibrium always exists while the existence of endemic equilibria is discussed. We derive the threshold R 0 (the basic reproduction number). The expression of the R 0 obtained here particularly highlight the effect of above structural variables on key important epidemiological traits of the human-vector association. This includes, humans and mosquitoes transmission probability and survival rates. Next, we derive a necessary and sufficient condition that implies the bifurcation of an endemic equilibrium. In some configuration where the age-structure of the human population is neglected, we show that, depending on the sign of some constant C bif given by the parameters, a bifurcation occurs at R 0 = 1 that is either forward or backward. In the former case, it means that there exists a (unique) endemic equilibrium if and only if R 0 > 1. In the latter case, no endemic equilibrium exists for R 0 1 small enough, a unique exists if R 0 > 1 while multiple endemic equilibria exist when 0 R 0 < 1 close enough to 1.