Cohort Dynamics

Malaria Epidemiology as Ontogeny

Generically, let $\mathbf{X}$ denote a state space. Dynamical systems describing malaria dynamics in a cohort are forced by a trace function $h(a,d)$, the force of infection for a cohort at age $a$ born on day $d$. These models take the form: $$\frac{\textstyle{d \mathbf{X(a)}}}{\textstyle{da}} = F_{\textbf{X}}\left(h, \textbf{X}\right)$$ If we wanted to consider the whole population, and not just a cohort, we could use partial differential equations (PDEs) that track all cohorts as they age over time. Once again, the models are forced by a trace function, with the form $$h(a,t) = F_h(b, \omega(a) E(t)).$$ In this form, the parameter $b$ handles pre-erythrocytic immunity, and $\omega$ is the relative biting rate for a person $$\frac{\textstyle{\partial \mathbf{X(a,t)}}}{\textstyle{da}} + \frac{\textstyle{\partial \mathbf{X(a,t)}}}{\textstyle{dt}} = F_{\textbf{X}}\left(h, \textbf{X}\right)$$ with boundary conditions $$\mathbf{X}(a,t) = B(t),$$ where $B(t)$ is the population birth rate.

To approximate the PDEs, we develop age-structured ordinary differential equation models. The state variables are replicated for each stratum, and Gallerkin methods are used to construct an aging matrix $\mathbf{D}.$ Now the dynamics are with respect to time: $$\frac{\textstyle{d \mathbf{X}}}{\textstyle{dt}} = B(t) + F_{\textbf{X}}\left(h, \textbf{X} \right) + \textbf{D} \cdot \textbf{X}$$ where the term $B(t)$ is zero for all but the birth state.