SIR(S) Disease Model Mathematics
The basic SIR (Susceptible, Infectious, Removed)
and SIRS (Susceptible, Infectious, Recovered, 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, is defined by the three equations below:
- Δs = μ − βs i + σr −
μs
- Δi = βs i − γi − μi
- Δr = γi − σr − μr
Where:
- s is the proportion of the population that is
Susceptible
- i is the proportion of the population
that is Infectious
- r is the proportion of the population
that is Removed from the infectious and susceptible populations,
and therefore cannot be infected.
- μ both the rate of immigration (e.g., by birth) and emigration
(e.g., by death) from the population. These rates are assumed to be equal
over the time period of interest (this simplifies the mathematics).
- β is the disease transmission (infection) rate.
The rate at which infectious individuals infect susceptible individuals. Once
infected, susceptible individuals instantly become infectious themselves.
- γ is the rate at which individuals clear infection. In this model these
individuals cannot be re-infected for some period of time after infection (whether through
immunity or removal from the population).
- σ is the immunity loss rate. This coefficient
determines the rate at which Removed population members lose
their immunity to the disease and become Susceptible again. For an SIR
model, this rate is 0.
Following basically the same derivation as outlined for the
SI
model, these become:
Let
- μi be the Infectious Mortality
Rate. This is the increased rate at which infected population members die
specifically due to the disease.
We modify our model to include this additional rate.
- Δs = μ − βs i + σr −
μ s
- Δi = βs i −
γi − (μ+μi)i
- Δr = γi − σr
− μr
Spatial Adaptation
- Δs Pl= μPl −
βl s i Pl + σ r Pl −
μ s Pl
- Δi Pl = βl
s i Pl − γ i Pl −
(μ+μi)i Pl
- Δr Pl= γiPl
− σr Pl− μr Pl
Let Sl = s Pl be the number of Susceptible
population members at location l. Similarly, let Il
= i Pl be the number of population members at location l
that are Infectious, and let r
Pl be the Recovered population. For readability, we
drop the l subscript and substitute.
Substituting
- ΔS = μPl − βl
S i + σR − μ S
- ΔI = βl S i
− γI − (μ+μi)I
- ΔR= γI − σR
− μR
Continuing with
i = I/Pl
, we have:
- ΔS = μPl − (βl/Pl)
S I + σR − μ S
- ΔI = (βl/Pl)
S I − γI − (μ+μi)I
- ΔR= γI − σR
− μR
Letting
β* = βl/Pl = β
(dl/(APDPl))
gives:
- ΔS = μPl − β*
S I + σR − μ S
- ΔI = β* S I
− γI − (μ+μi)I
- ΔR= γI − σR
− μR
TSF
- TSFl = ((S+I+R)/Areal) /
(P/Area(S+I+R))
- TSFl = (1/Areal) / (P/Area )
- TSFl = Area / (P *Areal )
- TSFl = (1 / P)* (Area/Areal )
Neighboring Infectious Populations
- ΔS = μPl − β*
S (I + Ineighbor() ) + σR − μ S
- ΔI = β* S (I
+ Ineighbor() ) − γI − (μ+μi)I
- ΔR = γI − σR
− μR
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.
- B= μ (S + I + R), is the number of Births
- D = μ S + (μ + μi )I + μR,is the total number of Deaths
- DD= μi I, is the number of
Disease Deaths