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

SIR model with carrier state (page 44)

Although the SIR and SEIR model paradigms are a good approximation to the epidemiological characteristics of many infectious diseases, such as measles or influenza, other infections have a more complex natural history. As an example of how such complexities can be accommodated in the model, will we consider infections such as hepatitis B, herpes, or chickenpox, where a proportion of infected individuals may become chronic carriers, transmitting infection at a low rate for many years.
For diseases with carrier states, susceptible individuals can be infected by either carriers or acutely infectious individuals. It is generally assumed that the progress of infection within an individual is independent of their source of infection; that is, those infected by acutely infectious individuals and those infected by carriers are indistinguishable. A recently infected individual is acutely (highly) infectious for a given period and then either recovers completely or moves into the carrier class. Such dynamics lead to the following 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.
is the proportion reduction in transmission from carriers compared to standard infectious individuals
is the proportion of infected individuals that become carriers rather than fully recover
is the recovery rate associated with carriers; hence the reciprocal (1/Γ) is the average time an individual is in the carrier class
S(0) is the initial proportion of the population that are susceptible.
is the initial proportion of the population that are infectious
is the initial proportion of the population that are carriers
All rates are specified in days.

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

C++ ProgramPython ProgramFortran ProgramParametersMATLAB Code.

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