For simplicity, the birth and death rates in a patch are set equal to each other so that the total population of each patch is constant. The model of (1)-(5) allows for one reinfection. Tertiary infections are not considered (Nisalak et al. 2003) . The parameter determines how susceptible the cross immune compartments c i are to (Bianco et al. 2009 ). Note the reduction of susceptibility to a secondary infection through the cross immunity factor (1 − ) and the enhancement of secondary infectiousness due to the ADE factor φ. Mortality terms for each compartment are not included in the diagram for ease of reading Ferguson et al. (1999b) β, transmission coefficient, years −1 ∼200 Ferguson et al. (1999b) σ , recovery rate, years −1 50 Rigau-Perez et al. (1998) θ, rate to leave the cross 2 Wearing and Rohani (2006) immune compartment, years −1 φ, ADE factor ≥1 S c h w a r t z e t a l . ( 2005) , strength of cross immunity 0-1 Bianco et al. (2009) ν, migration rate, years −1 0-0.05 n, number of strains 4 a secondary infection, where = 0 means no cross immunity (the infection rate is identical for compartments c i and r i ) and = 1 confers complete cross immunity (cross immunes are immune to a secondary infection for an average time θ −1 ). We allow to take any value between 0 and 1. The ADE factor φ is the enhancement in the infectiousness of secondary infectives. φ = 1 means that secondary infectives are as infectious as primary infectives, while φ = 2 means that secondary infectives are twice as infectious as primary, and so forth. In contrast to Aguiar and Stollenwerk (2007), we will not consider values of φ smaller than 1. The migration rate from patch q to q , and from patch q to q, is ν. For simplicity, we assume that all individuals migrate with equal probability, independent of their infection status. This assumption may be relaxed in a future study. The migration rate is assumed to be slow compared to the infection spread. For convenience, we put it on the same order as the birth/death rate. We assume that the social parameters, which are the contact rate β q and birth/death rate μ q , may vary between patches, while the epidemic parameters are the same in all regions. Because the social parameters depend on human factors (and in the case of the contact rate also include mosquito levels, which are weather-dependent), these parameters are the most likely to be different in adjacent regions. We use parameter values compatible with dengue fever, which are summarized in Table 1 . Our contact rate β corresponds to a reproductive rate of infection R 0 of 3.2 − 4.8, which is consistent with previous estimates (Ferguson et al. 1999b; Nagao and Koelle 2008) .