# Program 3.4 SEIR model with n age groups and yearly aging

This program is a clear step away from simple theoretical models (which are ideal for illustrating principles) to a more complex applied model which includes many more realistic features. We consider the SEIR model with four age-classes and yearly aging, closely matching the implications of grouping individuals into school cohorts. The four age-classes modelled are 0-6, 6-10, 10-20 and 20+ years old.

Given that we are aiming for a mechanistic model, we have fixed the basic demography and only allow the epidemiological parameters to be altered (although by now you may have enough confidence in your programming to alter the terms that are fixed within the code).

Key to this model are two basic assumptions:

1) Only individuals in the adult class give birth, and only individuals in the adult class die.

2) Births and deaths are continuous, but aging only happens once per year.

The continuous time dynamics are given by:

Given that we are aiming for a mechanistic model, we have fixed the basic demography and only allow the epidemiological parameters to be altered (although by now you may have enough confidence in your programming to alter the terms that are fixed within the code).

Key to this model are two basic assumptions:

1) Only individuals in the adult class give birth, and only individuals in the adult class die.

2) Births and deaths are continuous, but aging only happens once per year.

The continuous time dynamics are given by:

with the annual aging being given by:

**Parameters**

β | is the matrix of transmission rates and incorporates the encounter rate between susceptible and infectious individuals together with the probability of transmission. |

σ | is the rate at which individuals move from the exposed to the infectious classes. Its reciprocal (1/σ) is the average latent (exposed) period. |

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. |

μ | is the death rate in each age group; it is assumed that only adults die. |

ν | is the birth rate into the childhood class; it is assumed only adults give birth. |

S(0) | is the vector of initial proportions of the population that are both susceptible and in a particular age group. |

E(0) | is the vector of initial proportions of the population that are both exposed and in a particular age group. |

I(0) | is the vector of initial proportions of the population that are both infectious and in a particular age group. |

All rates are specified in days.**Requirements**.

All parameters must be positive, S(0) + E(0) + I(0) ≤ n for each age group.**Files**

C++ Program, Python Program, Fortran Program, Parameters, MATLAB Code.