Fig. 1

Phase diagram in the (λ,σ) plane for pair approximation model of two strain SIRS model. Other parameters: κ = 4, μ = 0.0005, and ψ = 0.0. The boundary of the one strain model of [16] is included for comparison. Region I is the constant endemics phase where the number of new infections is balanced by the number of recoveries and region II the oscillatory epidemics phase. The critical infection rate that separates disease-free phase and region I is λ_c ≈ 0.5. Data for the four childhood infectious diseases are from [16]. As the infectious period is used as the unit of time, the birth and death rate of μ = 0.0005 is equivalent to a life span of about 50 years in the model system for infectious diseases which have infectious periods of 1-2 weeks (such as Measles, chickenpox, rubella). It is worth mentioning that our predicted threshold waning rate of immunity in one strain SIRS model with μ = 0.0005 (i.e., the dashed line) are only slightly smaller than that presented in [16] who do not consider the birth rate