The Model

Model Overview

This model has the same compartments as the Characteristics of ID model:

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 equations describe the model implemented in this app:

Flow diagram for this model.

Flow diagram for this model.

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

Since we do not track people dying due to non-disease causes, all the “n - arrows” are not pointing to another compartment, instead of 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. The parameter T is set depending on the time units chosen for the model. For example if you want to run the model in units of days, the underlying simulation code will set T=365, similarly, for weeks it will be T=52. This ensures that the seasonal variation always has a period of a year.