The Model

Model Overview

This model has the same compartments as the one we saw previously. They are listed and described again:

We include the following processes in this model:

Note that we only track people that die due to the disease in our \(D\) compartment. All hosts dying due to other causes just “exit the system” and we don’t further keep track of them (though we could add another compartment to “collect” and track all individuals who died from non disease related causes.)

Model Implementation

The flow diagram and the mathematical model which are implemented in this app are as follows:

Flow diagram for this model.

Flow diagram for this model.

\[b_P^s = b_P(1+\sigma \sin(2\pi t /12))\] \[b_A^s = b_A(1+\sigma \sin(2\pi t /12))\] \[b_I^s = b_I(1+\sigma \sin(2\pi t /12))\] \[\dot S = \lambda - S (b_P^s P + b_A^s A + b_I^s I) + wR - n S \] \[\dot P = S (b_P^s P + b_A^s A + b_I^s I) - g_P P - n P\] \[\dot A = f g_P P - g_A A - n A\] \[\dot I = (1-f) g_P P - g_I I - n I \] \[\dot R = g_A A + (1-d) g_I I - wR - n R\] \[\dot D = d g_I I \]

Since we do not track people dying due to non-disease causes, all the “n - arrows” are not pointing to another compartment, instead those individuals just “leave the system”. Similarly new susceptibles enter the system (are born) from “outside the system”.

Also note that the transmission rates, bi, can be time varying as described above.