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

SIR model with disease induced mortality:
Frequency-dependent transmission (page 36)

Numerous infectious diseases are associated with a substantial mortality risk. Examples include malaria, measles, whooping cough, SARS, and dengue fever, among others. How do we explore the consequences of infection-induced mortality? Specifically, how do we incorporate a mortality probability into the SIR equations? The obvious approach would be to add a term such as −mI to the basic equation, where m is a per capita disease-induced mortality rate for infected individuals. However, this may be tricky to interpret biologically or estimate from data. Instead, it is preferable to think about the probability, ρ, of an individual in the I class dying from the infection before either recovering or dying from natural causes.
We initial consider the case of frequency-dependent transmission, where as total population size N decreases, due to disease-induced mortality, there is no change in the  interaction between hosts. To make the dynamics clearer we switch to considering the number or density (rather than proportion) of individuals.
is the mortality probability. The probability than an infected individual dies from the disease before recovering.
μ is the per capita death rate from natural causes
ν is the population level birth rate, we can equate ν/μ with the concept of a carrying capacity
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.
X(0) is the initial number or density of susceptible individuals.
Y(0) is the initial number or density of infectious individuals.
is the initial population size
All rates are specified in days.

All parameters must be positive. Remember, X, Y and ν all refer to numbers; ρ ≤ 1 as it is a probability.

C++ ProgramPython ProgramFortran ProgramParametersMATLAB Code.

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