The basic SI (Susceptible, Infectious) and SIS (Susceptible, Infectious Susceptible) disease models assume a uniform population at a single location and that the population members are well "mixed", meaning that they are equally likely to meet and infect each other. This model, for a normalized population, can be defined by the two difference equations below:
Where:
In the first equation, the Susceptible population increases when new members are born. This value is the birth rate μ multiplied by the total population which, since the values are normalized, is 1. It also increases due to Infectious population members recovering. The Susceptible population decreases by members who die. That value is μ, the mortality rate, multiplied by the Susceptible population, s. The Susceptible population also decreases by having members become infected. The product of β and i gives the normalized number of Susceptible population members that would become infected for each Infectious population member assuming all population members are in the Susceptible state. Multiplying that by s, the fraction that actually are Susceptible, gives the normalized amount that become Infectious.
In the second equation, the Infectious population increases by the number of Susceptible population members that become Infectious (the first term). It also decreases by the proportion that Recover from the disease (middle term) and by the proportion that die (last term).
Frequently, being infected by a disease will increase the likelihood that a population member will die. The model above needs to be enhanced to include the likelihood of a fatal infection and a potentially different rate at which infected members die.
Let
We modify our model to include this additional rate
The "SI" disease model computations in STEM enhance these equations by adapting them to populations that are spatially distributed. This relaxes the assumption that the populations are at a single location and opens up the possibility that different locations could have different areas and numbers of population members (i.e., different population densities). To accommodate this situation STEM maintains separate disease state values for each location and uses unnormalized versions of the equations presented above. We develop those below.
To account for population differences at different locations, we define a new parameter Pl which is the number of population members at location l (Note: Pl = Sl + Il). We also need to account for variability in the disease transmission (infection) rate, β, due to potentially different population densities. This modification is based upon the assumption that locations with greater population densities will have a higher effective transmission rates than locations with lower densities (i.e., one value can't be used for all locations). Thus, we need to replace the β in the non-spatial versions of the equations with a β l that is specific to the location.
Making the substitution for β and multiply both sides of the equations by Pl , we obtain:Let Sl = s Pl be the number of Susceptible population members at location l. , let Il = i Pl be the number of population members at location l that are Infectious (both states combined). For readability, we drop the l subscript and substitute.
Computing β* is straightforward. Let TSFl = (dl/(APDPl) be the transmission scale factor at location l.
Thus
Substituting Pl = S + I, APD = P/Area and dl = (S+I)/Areal, where P is the total population for all locations, Areal is the area of location l, and Area is the total area of all locations, we get:
The extension of the non-spatial model into a spatial one in STEM also needs to account for infectious population members that reside in a location's "neighbors". Consider a location with no infections that is physically adjacent to several locations that have large infectious populations. This physical adjacency would naturally lead to population-to-population contact and eventually to disease transmission. We need to further extend the equations we are here to incorporate this aspect of a spatially distributed population.
In STEM, a location has another location as a neighbor Relationship that links it to that location. If the Relationship represents the exchange of of population members (i.e., some kind of transportation relationship like pathways, roads or air travel) then it would be possible for Infectious population members from a neighbor to "visit" a location. We need to account for this potential by increasing the "effective" Infectious population at a location when doing our computations. Each Relationship has a rate at which population members travel from one location to another. It is assumed that the visitors would have the same level of "infectious contact" as an infectious member of the population at the current location (i.e., that they could could be counted as an infectious member of the population at the current location).
The equations become:
Where
Specific statistics on the total number of births, deaths and deaths due to the disease can be computed by adding the appropriate terms of the equations above.