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
- x be the Infectious Mortality which is the
proportion of the population members who become Infectious
that will eventually die.
- μi be the Infectious Mortality
Rate. This is the rate at which fatally infected population members die
specifically due to the disease.
Thus, we now have two types of
Infectious
population members, those that will eventually recover at rate
γ
, and those that will eventually die at rate
μi
(of course, members in all three states still die at the background rate
μ
).
Let
- i R be the normalized infectious
population that will recover.
- i F be the normalized infectious
population that will die from the disease.
- i = i R + i F be the total
normalized infectious population (as before).
We modify our model to include these additional states and rates.
- Δs = μ − βs i + σr −
μ s
- Δi R = (1-x)βs i −
γi R − μi R
- Δi F = x βs i − (μ +
μi) i F
- Δr = γiR − σr
− μr
Spatial Adaptation
- Δs Pl= μPl −
βl s i Pl + σ r Pl −
μ s Pl
- Δi R Pl = (1-x)βl
s i Pl − γ i R Pl −
μi R Pl
- Δi F Pl = x βls
i Pl − (μ + μi) i F Pl
- Δr Pl= γiRPl
− 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 (both states combined), 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 R = (1-x)βl S i
− γI R − μI R
- ΔI F = x βlS i
− (μ + μi) I F
- ΔR= γIR − σR
− μR
Continuing with
i = I/Pl
, we have:
- ΔS = μPl − (βl/Pl)
S I + σR − μ S
- ΔI R = (1-x)(βl/Pl)
S I − γI R − μI R
- ΔI F = x (βl/Pl)
S I − (μ + μi) I F
- ΔR= γIR − σR
− μR
Letting
β* = βl/Pl = β
(dl/(APDPl))
gives:
- ΔS = μPl − β*
S I + σR − μ S
- ΔI R = (1-x)β* S I
− γI R − μI R
- ΔI F = x β* S I
− (μ + μi) I F
- ΔR= γIR − 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 R = (1-x)β* S (I
+ Ineighbor() ) − γI R − μI
R
- ΔI F = x β* S (I + Ineighbor()
) − (μ + μi) I F
- ΔR= γIR − 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 )IF
+ μIR + μR,is the total number of Deaths
- DD= μi IF, is the number of
Disease Deaths