1. A. D. Cliff P. Haggett J. K. Ord Spatial Aspects of Influenza Epidemics (Pion London 1986).
2. Andreasen V., Lin J., Levin S. A., J. Math. Biol.35, 825 (1997).
3. Gupta S., et al., Nature Med.2, 437 (1996).
4. The proportion immune to a given strain i z i is simply given by
dzidt=(1−zi)λi−μzi
Here λ i represents the force of infection of strain i. We assume for simplicity that immunity is lifelong. We then define an additional compartment w i which represents those immune to any strain j that shares alleles (at the relevant polymorphic loci) with strain i (including i itself). The dynamics of w i are then given by (where j ∼ i means j shares alleles with i )
dwidt=(1−wi)∑j∼i λi−μwi
Individuals who have never been exposed to any strain sharing alleles with strain i (that is 1 – w i ) are completely susceptible to strain i. However those that have been exposed to a strain sharing alleles with i but not exposed to strain i itself (that is w i − z i ) will become infectious with a probability 1 − γ when they are infected by strain i. With σ being the rate of loss of infectiousness of the host the dynamics of the proportion of the population infectious for strain i may therefore be represented as
dyidt [(1−wi)+(1−γ)(wi−zi)]λi−σyi
The impact of genetic exchange on the population structure of infectious disease agents may be examined within this framework by modifying the force of infection term λ to include the assumption that the progeny of parasites within hosts infectious for two or more strains will consist of defined fractions Ω ijk of the various combinations of the different strains j and k that may generate strain i through recombination. Because the proportions of infectious hosts are very small the force of infection of strain i may be approximately represented as λ i = β i ( y i + Σ Ω ijk y j y k ) where β i is a combination of parameters affecting the transmission of strain i. The behavior of the model is largely unaffected by the inclusion or precise functional form of the recombination term.
5. R. M. Anderson and R. M. May Infectious Diseases of Humans: Dynamics and Control (Oxford Univ. Press Oxford 1991).