Disease Control in Age Structure Population

We combine the Leslie model and its derivatives with the classical compartmental SIRS models to build a model of transmission of infected diseases, in a population of hosts, whether opened or closed systems. We calculate the basic reproductive rate R0. Under certain conditions, when , there is a disease-free equilibrium that is locally asymptotically stable. In contrast, when , this equilibrium is unstable. Then, through an example, we show how we can define public health strategies to tackle an endemic. Finally we carry a global sensitivity analysis based on this basic reproduction rate to exhibit the most influential parameters of our model that are applied to influenza. 1. Introduction During a certain period it was believed to have vanquished a large number of infectious diseases, until they redo their appearance, sometimes more dramatically (e.g., foot and mouth disease in the United Kingdom in 2001, highly pathogenic avian influenza pathogen in Europe in 2006) [1]. The same observation was made in Africa where, in the 70s, it was believed to have neutralized human African trypanosomiasis, long fought by the WHO until it revives later and often in the same historic foci [2]. This shows how much the prevention and control of transmission of infection diseases in the community requires constant vigilance. Many scientific disciplines have addressed this problem. Among these, the contribution of mathematical models has been very helpful. Specifically, mention may be made of introduction of the concept of and its consequences. Generally, , the basic reproductive ratio, defined as the expected number of secondary infections that occur when one infective is introduced into a completely susceptible host population, is used to characterize the nature of the disease [3–9]. So, it is a threshold used to evaluate the conditions for starting an epidemic in a given area and for a specific disease. In general, when this threshold is less than 1, the disease will disappear. By cons, when it is greater than 1, the disease spreads permanently. Knowledge of the was used to evaluate the critical vaccination coverage, that is the minimum proportion of the population that must be immunized in order for the infection to die out in the population ( ). As an example, when , the vaccination of the third of population is enough to stop the spread of the disease [10]. This threshold is even more used in most epidemiological models through compartmental representations [11–13]. In such a representation, compartments are discretized, each corresponding to all individuals in the