where the variables are s q , the fraction of susceptibles in patch q; x q,i , the fraction of primary infectives with strain i in patch q; c q,i , the fraction of individuals in patch q with partial cross immunity to strain i; r q,i , the fraction of individuals in patch q that are recovered from a primary infection with strain i and no longer have cross immunity to the other strains; and x q,ij , the fraction of individuals in patch q recovered from strain i and currently infected with strain j . The parameters are the number of strains n, the contact rate in patch q β q , the recovery rate σ , the ADE factor φ, the strength of cross immunity , the rate θ for cross immunity to wear off, the birth and mortality rate in patch q μ q , and the migration rate between patches ν. A list of parameters appears in Table 1 .