Mathematical analysis of a model describing the number of antibiotic resistant bacteria in a polluted river

In France, 90% of surface water suffer from antibiotic pollution that increases the number of antibiotic resistant bacteria. The first indicator of water quality is revealed by the fish quality. According to the Le Conseil Supérieur de la Pêche, only 15% of rivers in France are considered in good condition, whereas 22% are in very bad condition. The bacterial resistance to antibiotics is a public health problem as it affects humans through drinking water; the treatment of water is costly. Mathematical modeling may estimate and predict the quantity of bacteria in rivers. In this paper, we investigate properties of the mathematical model estimating the number of bacteria in a river presented by Lawrence, Mummert and Somerville. Global analysis of equilibria is presented, using a Lyapunov function. Moreover, the existence of positive periodic solutions is proven. Copyright © 2013 John Wiley & Sons, Ltd.