Geodesic
Epidemic dynamics in a heterogeneous incompletely isolated population with allowance for seasonal variations in the infection rate
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 8, pp. 1488-1499

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Mathematical parasite-host models are generalized to the case when the population members differ in susceptibility and contagiousness, there is an external source of infection, and the model parameters depend periodically (seasonally) on time. The model is proved to have a periodic solution that is unique and exponentially stable for sufficiently small periodic oscillations of the coefficients. 
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     author = {A. N. Gerasimov and V. N. Razzhevaikin},
     title = {Epidemic dynamics in a heterogeneous incompletely isolated population with allowance for seasonal variations in the infection rate},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     publisher = {mathdoc},
     volume = {48},
     number = {8},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_8_a9/}
}
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A. N. Gerasimov; V. N. Razzhevaikin. Epidemic dynamics in a heterogeneous incompletely isolated population with allowance for seasonal variations in the infection rate. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 8, pp. 1488-1499. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_8_a9/