Model Overview
This model is fairly big and has many parts that can be turned on or off depending on parameter settings. The model allows for 3 types of transmission: direct, through an environmental stage, and through a vector stage. The (human) host is modeled in some detail, the environment and vectors are modeled with 1 and 2 compartments. The following compartments are included:
- 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.
- E - pathogen in the environment
- SV - susceptible vectors
- IV - infected/infectious vectors
The included processes/mechanisms are the following:
- 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, bP, bA, and bI.
- Susceptible individuals (S) can also become infected by contact with the environment or infected vectors, at rates bE and bv.
- Susceptible vectors (Sv) can become infected by contact with symptomatic hosts at rate bh.
- All infected hosts first enter the presymptomatic stage. They remain there for some time (determined by rate gP, the inverse of which is the average time spent in the presymptomatic stage). A fraction f of presymptomatic hosts move into the asymptomatic category, and 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 lose their immunity at rate w and return to the susceptible compartment.
- Symptomatic and asymptomatic hosts shed pathogen into the environment at rates pA and pI. The pathogen in the environment decays at rate c.
- New susceptible hosts and vectors enter the system (are born) at rates eh and ev. Natural death for hosts and vectors occurs at rates nh and nv. (In the code, those rates are called birthi and deathi).
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.)
Also, note that we made several simplifications to keep the model from getting too complex. For instance, presymptomatic individuals do not shed into the environment, and only symptomatic hosts are assumed to be able to infect vectors. Further details relaxing these assumptions could, of course, be included, at the expense of a larger and more complex model.
Model Implementation
The flow diagram and equations describe the model implemented in this app:
\[\dot S = e_h - S (b_P P + b_A A + b_I I + b_E E + b_v I_v) + wR - n_h S \] \[\dot P = S (b_P P + b_A A + b_I I + b_E E + b_v I_v) - g_P P - n_h P\] \[\dot A = f g_P P - g_A A - n_h A\] \[\dot I = (1-f) g_P P - g_I I - n_h I \] \[\dot R = g_A A + (1-d) g_I I - wR - n_h R\] \[\dot D = d g_I I \] \[\dot E = p_I I + p_A A - c E \] \[\dot S_v = e_v - b_h I S_v - n_v S_v \] \[\dot I_v = b_h I S_v - n_v I_v \]
Births and natural deaths are not drawn to keep the diagram from getting too cluttered.