SIR Disease Model Mathematics
The basic SIR (Susceptible, Infectious, Recovered)
disease model assumes 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 + γr −
μs
- Δi = βs i − σi − μi
- Δr = σi − γr − μr
Where:
- s is the normalized Susceptible population
- i is the normalized Infectious population
- μ is the background mortality rate, and, because it is
assumed that the population was not growing or shrinking significantly
before the onset of the disease, μ is also assumed to be the birth
rate.
- β is the disease transmission (infection) rate.
This coefficient determines the number of population members that
become infected/exposed per population member in the Infectious
state, assuming the entire population is in the Susceptible
state.
- σ is the Infectious recovery rate. This
coefficient determines the rate at which Infectious population
members Recover.
- γ is the immunity loss rate. This coefficient
determines the rate at which Recovered population members lose
their immunity to the disease and become Susceptible again.
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
σ
, 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 + γr −
μ s
- Δi R = (1-x)βs i −
σi R − μi R
- Δi F = x βs i − (μ +
μi) i F
- Δr = σiR − γr
− μr
Spatial Adaptation
- Δs Pl= μPl −
βl s i Pl + γ r Pl −
μ s Pl
- Δi R Pl = (1-x)βl
s i Pl − σ i R Pl −
μi R Pl
- Δi F Pl = x βls
i Pl − (μ + μi) i F Pl
- Δr Pl= σiRPl
− γr 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 + γR − μ S
- ΔI R = (1-x)βl S i
− σI R − μI R
- ΔI F = x βlS i
− (μ + μi) I F
- ΔR= σIR − γR
− μR
Continuing with
i = I/Pl
, we have:
- ΔS = μPl − (βl/Pl)
S I + γR − μ S
- ΔI R = (1-x)(βl/Pl)
S I − σI R − μI R
- ΔI F = x (βl/Pl)
S I − (μ + μi) I F
- ΔR= σIR − γR
− μR
Letting
β* = βl/Pl = β
(dl/(APDPl))
gives:
- ΔS = μPl − β*
S I + γR − μ S
- ΔI R = (1-x)β* S I
− σI R − μI R
- ΔI F = x β* S I
− (μ + μi) I F
- ΔR= σIR − γR
− 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() ) + γR − μ S
- ΔI R = (1-x)β* S (I
+ Ineighbor() ) − σI R − μI
R
- ΔI F = x β* S (I + Ineighbor()
) − (μ + μi) I F
- ΔR= σIR − γR
− 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