  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 R0 (the
basic reproduction number). The expression of the R0 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
Cbif given by the parameters, a bifurcation occurs at R0 = 1 that is either
forward or backward. In the former case, it means that there exists a (unique)
endemic equilibrium if and only if R0 > 1. In the latter case, no endemic
equilibrium exists for R0<< 1 small enough, a unique exists if R0 > 1 while
multiple endemic equilibria exist when 0 <<R0 < 1 close enough to 1.
