Competitive Exclusion In A Multi-strain Immuno-epidemiological Flu Model With Environment Transmission

In this paper, a multi-strain model that links immunological and epidemiological dynamics across scales is formulated. On the within-host scale, the n strains eliminate each other with the strain with the largest immunological reproduction persisting. However, on the population scale, we extend the competitive exclusion principle to a multi-strain model of SI-type for the dynamics of highly pathogenic flu in poultry that incorporates both the infection age of infectious individuals and biological age of pathogen in the environment. The two model are linked through the age-since-infection structure of the epidemiological variables. In addition the between-host transmission, the shedding rate of individuals infected by strain j and the disease-induced death rate depend on the within-host viral load. The immunological reproduction numbers Rj and the epidemiological repro-duction numbers Rj are computed. By constructing a suitable Lyapunov function, the global stability of the infection-free equilibrium in the system is obtained if all reproduction numbers are smaller or equal to one. If Rj, the reproduction number of strain j is larger than one, then a single-strain equilibrium, corresponding to strain j exists. This single strain equilibrium is locally stable whenever Rj>1and Rj is the unique maximal reproduction number. Without loss of generality, we assume that R1>1is the maximal reproduction number. Using a Lyapunov function, we establish that the corresponding single-strain equilibrium ε1is globally stable. That is, strain one eliminates all other strains, independently of their reproduction numbers as long as they are smaller than R1.