# On a family of stochastic SVIR influenza epidemic models and maximum likelihood estimation

## TL;DR

This paper introduces a family of stochastic SVIR influenza models incorporating vaccination effects, and develops maximum likelihood and EM methods for parameter estimation, including the basic reproduction number, supported by numerical simulations.

## Contribution

It presents a novel stochastic SVIR model family with trinomial transitions and derives maximum likelihood and EM estimation methods for key epidemiological parameters.

## Key findings

- Models effectively incorporate vaccination and immunity dynamics.
- Maximum likelihood and EM estimators are derived for model parameters.
- Numerical simulations demonstrate model applicability and estimation accuracy.

## Abstract

This study presents a family of stochastic models for the dynamics of influenza in a closed human population. We consider treatment for the disease in the form of vaccination, and incorporate the periods of effectiveness of the vaccine and infectiousness for the individuals in the population. Our model is a SVIR model, with trinomial transition probabilities, where all individuals who recover from the disease acquire permanent natural immunity against the strain of the disease. Special SVIR models in the family are presented, based on the structure of the probability of getting infection and vaccination at any instant. The methods of maximum likelihood, and expectation maximization are derived for the parameters of the chain. Moreover, estimators for some epidemiological assessment parameters, such as the basic reproduction number are computed. Numerical simulation examples are presented for the model.