IJPAM: Volume 106, No. 3 (2016)

THE INFLUENZA AH1N1: A MATHEMATICAL MODEL
WITH APPLICATION OF OPTIMAL CONTROL

Elmer De la Pava$^1$, Beatriz Salguero$^2$, Lilian Sofía Sepúlveda$^3$
$^{1,2,3}$Department of Mathematics
Universidad Autónoma de Occidente
Cali, COLOMBIA


Abstract. Human influenza is transmitted directly from an ill person to a healthy one, by air, during the symptomatic period of the disease. The virulence and antigenicity of the virus, host immunity and environment, interact with each other affecting the transmission of the virus person-person. Since the alert in 2009 of the influenza due to the H1N1 virus, the number of new cases increased despite the control measures implement, such as wearing masks, and other recommendations made by the World Health Organization. In Colombia it was found that the first case of AH1N1 coincided with a person from Mexico. Therefore, and considering that all travelers from this country are a suspected case of the disease, it makes the transit from one country to another a possible route of transmission. Hence, the approach to the question: What has been the impact in Colombia of the outbreak caused by the migration of people from Mexico infected with the H1N1 influenza virus?

The porpuse of this paper is, in a certain way, to respond this question using a mathematical model that studies the transmission of this disease in both immigrant and local populations. Population $N$ is divided in $N_E$ immigrant populations and $N_L$ local population, where $N = N_E + N_L$, according to the natural history of H1N1. Each subpopulation is divided into three classes, susceptible $ S $, infectious $I$ and recovered $R$, resulting in the six compartments $S_E, I_E, R_E $ and $S_L, I_L, R_L$. The equilibriums and their qualitative analysis were calculated. Besides, the basic reproductive number representing the classical measure of transmission of infectious diseases is estimated and, from a biological point of view, it is defined as the number of secondary cases produced by a typical infected individual when introduced in a fully susceptible host population, during its effective period of infectivity. If ${\cal R}_0 <1$ the disease goes out and if ${\cal R}_0>1$ an outbreak occurs.

Received: January 13, 2016

AMS Subject Classification: 34H05, 49J15, 49K15, 93C15

Key Words and Phrases: model, control, influenza, infectious

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DOI: 10.12732/ijpam.v106i3.15 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 106
Issue: 3
Pages: 893 - 907


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