A Household Model of German Cockroach Infestations and Their Effects on Symptoms of Atopic Asthma

Asthma is a chronic respiratory condition which affects 25 million people in the United States. Improper extermination of cockroaches and their associated allergens significantly increases the occurrence of atopic asthma, a type of asthma stemming from exposure to allergens. To date, dynamical studies of cockroach infestations and their effects on atopic asthma have not been examined. Here we develop a novel mathematical model, with households as the model unit, to study the dynamics of cockroach infestations within a neighbourhood and how they impact the prevalence of asthma. The focus of this study is to evaluate the impact of extermination and removal of allergens in a household containing an individual with recurrent atopic asthma. Through analysing our model, the basic reproduction number for cockroach infestations was calculated and we were able to determine conditions under which cockroaches and asthma can be eliminated from a community containing infested houses. From the presence of a backward bifurcation, we are able to show that it is