Modeling Infectious Diseases in Humans and Animals
Matt J. Keeling & Pejman Rohani

SIR model with births and deaths (page 27)

If we are interested in exploring the longer-term persistence and endemic dynamics of an infectious disease, then clearly demographic processes will be important. The simplest and most common way of introducing demography into the SIR model is to assume there is a natural host “lifespan”, 1/μ years. Then, the rate at which individuals (in any epidemiological class) suffer natural mortality is given by μ. It is important to emphasize that this factor is independent of the disease and is not intended to reflect the pathogenicity of the infectious agent. Historically, it has been assumed that μ also represents the population’s crude birth rate, thus ensuring that total population size does not change through time (dS/dt + dI/dt + dR/dt = 0). This framework is very much geared toward the study of human infections in developed nations—our approach would be different if the host population exhibited its own “interesting” dynamics (as is often the case with wildlife populations; see Chapter 5). Putting all these assumptions together, we get the generalized SIR model:
μ is the per capita death rate, and the population level birth rate.
β is the transmission rate and incorporates the encounter rate between susceptible and infectious individuals together with the probability of transmission.
γ is called the removal or recovery rate, though often we are more interested in its reciprocal (1/γ) which determines the average infectious period.
S(0) is the initial proportion of the population that are susceptible.
I(0) is the initial proportion of the population that are infectious.
All rates are specified in days.

All parameters must be positive, and S(0)+I(0) ≤ 1

C++ ProgramPython ProgramFortran ProgramParametersMATLAB Code.

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