On a Mathematical Model for Co-infection (with HCV/HIV)

Abstract

In many developing countries, with very high population densities, co-infection with both HIV and HCV is fairly common [1-3]. This is perhaps because, in such situations, people live in close proximity and generally, are not well cared for. Medical facilities are minimal, so that infections spread easily and persist. Also, people who are already infected with HIV or HCV, are likely to share common "risky" behavior (risk factors for both HIV and HCV are the same) so that contacts are fairly common (sexual, social, sharing needles and other). Considering all this, one may assume that a certain fraction of people suffering from HIV are also suffering from HCV and vice versa. We present a mathematical model which describes the development of co-infection with HCV and HIV, in such situations. We assume that, while susceptible people become infected with HIV and/or HCV through physical contact (sexual, sharing needles, and so on), co-infection occurs simply because they live in close proximity to each other. Accordingly, we assume that a certain percentage of people who are infected with HCV in a densely populated environment are also infected with HIV, and vice versa. We give several illustrative examples. These examples are to show how the model works and are not necessarily disease specific. We prove the important result that our model has AT MOST one solution other than the disease free one. Most infection models share this property [4-6]. We also calculate the Basic Reproduction Number of the model in two different ways and give an example to show that these two ways are “equivalent”. If either number is less than one, then so is the other and there is no infection, and if either number is greater than one, then so is the other, and the infection is endemic. It should also be pointed out that this is simply a model for coinfection, and it should work for other similar coinfections as well.