# Efficient solution of a stochastic SI epidemic system

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## Abstract

One of the side-effects of the climate changes that are upon us is that infectious diseases are adapting, evolving and spreading to new geographic regions. It is, therefore, imperious to develop epidemic models that shed light on the interplay between the dynamics of the spread of infectious diseases and the combined effects of various vaccination and prevention regimens. With this in mind, in this work we propose a epidemic model operating on a large population; we restrict our attention to strains of infectious diseases that resist treatment. The time-dependent epidemic accounts, among others, for the effects of improved sanitation, education and vaccination. Our first main contribution is to derive the time-dependent probability mass function of the number of infected individuals in such a system. Our derivation does not use probability generating functions and partial differential equations. Instead, we develop an iterative solution that is conceptually simple and easy to implement. Somewhat surprisingly, the epidemic model also provides insight into various *stochastic phenomena* noticed in sociology, macroeconomics, marketing, transportation and computer science. Our second main contribution is to show, by extensive simulations, that suitably instantiated, our epidemic model be used to model phenomena describing the adoption of durable consumer goods, the spread of AIDS and the dissemination of mobile worm spread.

## Keywords

Stochastic epidemic models Time-dependent probabilities Waiting-time probabilities Compartmental epidemics model## Notes

### Acknowledgements

We would like to thank six anonymous referees for their helpful comments that have greatly contributed to improve the presentation of the paper. Last, but certainly not least, we wish to extend our thanks to Professor H. Arabnia for his professional handling of our submission.

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