Figure 2

A diagram of the queuing models, which extend the Ross-Macdonald model by tracking changes in the MOI. The fraction of the whole population with MOI of m, denoted x _m , changes when new infections occur or when existing infections clear, and in any short interval of time, one individual's MOI increments or decrements by one [18]. The rate that new infections arise may depend on MOI, denoted h _m . The rate of loss may depend on MOI, denoted ρ _m , and ρ₁ = r denotes the rate a simple infection is lost. Changes in the fraction of the population that is uninfected are described by an equation: = -h₀x₀ + rx₁. Changes in the fraction of people who are already infected with a given MOI are described by a set of equations: = -h _m x _m + h_m-1x_m-1- ρ _m x _m + ρ_m+1x_m+1. These equations describe a family of queuing models: each queuing model makes different assumptions about infection and clearance. In "infinite strain" models, h _m = h, and in "finite strain" models where M denotes the maximum number of types, h _m = h(1-m/M). Models considered parasite types that cleared independently, or with competition or facilitation. For independent clearance, individual types were unaffected by concurrent infection with other parasite types, so ρ _m = rm. Competition and facilitation were modelled by letting ρ _m = rm^σ, where σ >1 described competition and σ <1 implies facilitation. Compared with independent clearance, per-strain clearance rates are faster with competition (i.e. ρ _m > rm), and slower with facilitation (i.e. ρ _m <rm). Clearance rates per person increase with MOI in all the models; if no new infections occurred, the expected waiting time to lose all the existing infections would be the sum of times to progressively decrement MOI: 1/r+1/ρ₂+1/ρ₃+...+1/ρ _m .