Age-dependence in epidemic models of vector-borne infections

We develop several age-structured epidemic models for the spread of vector-borne diseases such as malaria and Dengue fever. These models can be grouped into two main types---continuous age structure and discrete age structure. In both cases we allow for disease-induced death in the human population. We use a susceptible-exposed-infectious-recovered (SEIR ) disease progression for the human population and a susceptible-exposed-infectious (SEI) progression for the vector population. The continuous age-structured model consists of a system of coupled nonlinear partial and ordinary differential equations. An infection-free age distribution is obtained. The threshold condition that determines the local stability of this solution is presented in terms of the basic reproduction number which is obtained through a linearization analysis. A special case of this model is the discrete age-structured model. This continuous age group model consists of a system of nonlinear ordinary differential equations. We use standard linearization techniques to determine the stability of the infection-free equilibrium. We also obtain the stability result using the next-generation operator. The threshold condition for stability is given in terms of the reproduction number. Conditions on the existence and stability of an endemic equilibrium are also presented in terms of this parameter. Numerical examples for both Dengue fever and malaria are presented. Also, the sensitivity of the reproduction number to changes in the parameters is discussed.