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 ${\mathcal{R}}_0$ 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 ${\mathcal{R}}_0$ 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 ${\mathcal{R}}_0=1$ leading to the existence, or not, of multiple endemic equilibrium when $0\ll {\mathcal{R}}_0<1$. Finally, the latter theoretical results are enlightened by numerical simulations.

疟疾是热带和亚热带地区最常见的蚊媒疾病之一，每年在世界上造成数千人死亡。很少有模型考虑多重结构模型公式，包括 (i) 人类和蚊子种群的实际年龄，(ii) 自它们被感染以来的时间，以及 (iii) 人类免疫力下降（即恢复后保护性抗体的逐渐丧失）已被开发。在本文中，我们制定了一个包含三个结构变量的年龄结构模型。使用集成半群理论，我们首先处理所提出模型的适定性。我们还研究了稳态的存在。在讨论地方病平衡的存在时，无病平衡总是存在的。我们推导出基本再生数${\mathcal{电阻}}_0$该表达式突出了上述结构变量对人与媒介关联的关键重要流行病学特征的影响，例如媒介能力（即媒介每日繁殖率）、人类传播概率和存活率。的表达${\mathcal{电阻}}_0$这里得到的推广了基本再生数的经典公式。接下来，我们推导出一个充分必要条件，这意味着地方性平衡的分叉。在忽略人口年龄结构的特定情况下，我们表明分叉，无论是向后还是向前，都可能发生在${\mathcal{电阻}}_0=1$ 导致多重地方性均衡的存在或不存在，当 $0\ll {\mathcal{电阻}}_0<1$. 最后，后面的理论结果受到数值模拟的启发。