On latencies in malaria infections and their impact on the disease dynamics

In this paper, we modify the classic Ross-Macdonald model for malaria disease dynamics by incorporating latencies both for human beings and female mosquitoes. One novelty of our model is that we introduce two general probability functions (P 1 (t) and P 2 (t)) to reflect the fact that the latencies differ from individuals to individuals. We justify the well-posedness of the new model, identify the basic reproduction number R0 for the model and analyze the dynamics of the model. We show that when R 0 <1, the disease free equilibrium E0 is globally asymptotically stable, meaning that the malaria disease will eventually die out; and if R 0 >1, E 0 becomes unstable. When R 0 >1, we consider two specific forms for P 1 (t) and P 2 (t): (i) P 1 (t) and P 2 (t) are both exponential functions; (ii) P 1 (t) and P 2 (t) are both step functions. For (i), the model reduces to an ODE system, and for (ii), the long term disease dynamics are governed by a DDE system. In both cases, we are able to show that when R 0 >1 then the disease will persist; moreover if there is no recovery (γ1=0), then all admissible positive solutions will converge to the unique endemic equilibrium. A significant impact of the latencies is that they reduce the basic reproduction number, regardless of the forms of the distributions.