Can treatment increase the epidemic size?

Antiviral treatment is one of the key pharmacological interventions against many infectious diseases. This is particularly important in the absence of preventive measures such as vaccination. However, the evolution of drug-resistance in treated patients and its subsequent spread to the population pose significant impediments to the containment of disease epidemics using treatment. Previous models of population dynamics of influenza infection have shown that in the presence of drug-resistance, the epidemic final size (i.e., the total number of infections throughout the epidemic) is affected by the treatment rate. These models, through simulation experiments, illustrate the existence of an optimal treatment rate, not necessarily the highest possible rate, for minimizing the epidemic final size. However, the conditions for the existence of such an optimal treatment rate have never been found. Here, we provide these conditions for a class of models covered in the literature previously, and investigate the combination effect of treatment and transmissibility of the drug-resistant pathogen strain on the epidemic final size. For the first time, we obtain the final size relations for an epidemic model with two strains of a pathogen (i.e., drug-sensitive and drug-resistant). We also discuss this model with specific functional forms of de novo resistance emergence, and illustrate the theoretical findings with numerical simulations.