We develop a mathematical model that allows for the study of many strains and various levels of cross-immunity among them. For instance, partial cross-immunity to next-to-kin strains and differential cross-immmunity are among the strain interactions that can be explored. The immune status of the host population is captured by an index-set notation where the index specifies the immune-competence level against a particular strain. Here we assume that individuals with a previous exposure to other strains have acquired some levels cross-immunity against future infections. In contrast to previous modeling frameworks, the population here is structured into non-intersecting subclasses. Since multiple infection with influenza strains is unlikely to occur, we do not imbed superinfection with the same or different strains as part of our model. To illustrate the impact of strain structure and cross-immunity, we study a two-strain and three-strain model and provide conditions for the invasion or coexistence of strains. We provide conditions for the existence of both disease-free equilibrium (DFE) and endemic equilibrium
(EE) and show that DFE is locally and globally stable.
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