Model Overview
This model has the same compartments as the one we saw previously. They are listed and described again:
- S - uninfected and susceptible individuals
- P - individuals who are infected and do not yet show symptoms. Those individuals can potentially be infectious
- A - individuals who are infected and do not show symptoms. Those individuals can potentially be infectious
- I - individuals who are infected and show symptoms. Those individuals are likely infectious, but the model allows to adjust this, including no infectiousness.
- R - recovered/removed individuals. Those individuals have recovered and are immune. They can loose their immunity in this model.
- D - individuals who have died due to the disease.
We include the following processes in this model:
- Susceptible individuals (S) can become infected by pre-symptomatic (P), asymptomatic (A) or symptomatic (I) hosts. The rates at which infections from the different types of infected individuals (P, A and I) occur are governed by 3 parameters, \(b_P\), \(b_A\) and \(b_I\).
- All infected individuals first enter the presymptomatic stage. They remain there for some time (determined by rate \(g_P\), the inverse of which is the average time spent in the presymptomatic stage). A fraction f of presymptomatic hosts move into the asymptomatic category, the rest become symptomatic infected hosts.
- Asymptomatic infected hosts recover after some time (specified by the rate gA). Similarly, the rate gI determines the duration the symptomatic hosts stay in the symptomatic state. For symptomatic hosts, two outcomes are possible. Either recovery or death. The parameter \(d\) determines the fraction of hosts that die.
- Recovered individuals are initially immune to reinfection. They can loose their immunity at rate \(w\) and return back to the susceptible compartment.
- New susceptibles enter the system/model at a fixed rate \(\lambda\). From each compartment (apart from the dead compartment), hosts “leave” after some time (i.e. they die) at rate \(n\). The inverse of this rate is the average lifespan of a host.
- The rates of transmission, \(b_P\), \(b_A\) and \(b_I\) can vary seasonally/annually, modeled as a sinusoidal function. The strength of this seasonal variation is controlled by the parameter \(\sigma\).
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:
\[b_P^s = b_P(1+\sigma \sin(2\pi t /12))\] \[b_A^s = b_A(1+\sigma \sin(2\pi t /12))\] \[b_I^s = b_I(1+\sigma \sin(2\pi t /12))\] \[\dot S = \lambda - S (b_P^s P + b_A^s A + b_I^s I) + wR - n S \] \[\dot P = S (b_P^s P + b_A^s A + b_I^s I) - g_P P - n P\] \[\dot A = f g_P P - g_A A - n A\] \[\dot I = (1-f) g_P P - g_I I - n I \] \[\dot R = g_A A + (1-d) g_I I - wR - n R\] \[\dot D = d g_I I \]
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.