Models for Endemic Diseases

The classical epidemic models derived from the Kermack-McKendrick model consider the prevalence of the disease only. The host population is partitioned into classes of susceptibles S, infectious I, and recovered R. The transition between these classes is described by ordinary differential equations. The following equations take into account the recruitment of new individuals into the class of susceptibles by birth and a death rate depending on the class. The equations read (1)$$\begin{gathered} \dot s = - \beta SI + \gamma R - {\delta _1}S + \left( {{\delta _1}S + {\delta _2}I + {\delta _3}R} \right) \hfill \\ \dot I = \beta SI - \alpha I - {\delta _2}I \hfill \\ \dot R = \alpha I - \gamma R - {\delta _3}R \hfill \\ \end{gathered}$$ Of course these equations are independent of δ₁, and P = S + I + R is an invariant. Thus the equations simplify to (2)$$\begin{gathered} <Emphasis Type="Bold">\dot s = - \beta SI + \left( {\gamma + {\delta _3}} \right)\;\left( {P - S} \right) - \left( {\gamma - {\delta _2} + {\delta _3}} \right)I \hfill \\ \dot I = \beta SI - \left( {\alpha + {\delta _2}} \right)I \hfill \\</Emphasis> <Emphasis Type="Bold">\end{gathered}</Emphasis>$$ The “natural” parameters of the system are β, γ + δ₃, α + δ₂, α + γ + δ₃. In the following we consider only the endemic case γ + δ₃ > 0. The state space of the system (2) is the triangle T = {S ≧ 0, I ≧ 0, S + I ≦ P} which is positively invariant with respect to the flow.