Table 1. Transitions in the two-pathogen stochastic models.
The prevalence of uninfected host is J∅, the prevalence of each class of singly infected hosts is Ji (for i∈[1,2]), and the prevalence of coinfected host is J1,2. The net force of infection of pathogen i is Fi = βiIi/N = βi(Ji+J1,2)/N (note the scaling by the population size N relative to the forces of infection as used in the deterministic version of the model). To ensure a constant host population size, we have made the simplifying assumption that removal and replacement occur simultaneously; this has no effect on our qualitative results.
| Event Number | Event | Rate | Change(s) to state variable(s) (ΔX) |
|---|---|---|---|
| 1 | Infection of uninfected host by pathogen 1 | F1J∅Δt+o(Δt) |
J∅→J∅−1 J1→J1+1 |
| 2 | Infection of uninfected host by pathogen 2 | F2J∅Δt+o(Δt) |
J∅→J∅−1 J2→J2+1 |
| 3 | Infection by pathogen 1 of host singly infected by pathogen 2 | F1J2Δt+o(Δt) |
J2→J2−1 J1,2→J1,2+1 |
| 4 | Infection by pathogen 2 of host singly infected by pathogen 1 | F2J1Δt+o(Δt) |
J1→J1−1 J1,2→J1,2+1 |
| 5 | Death of host singly infected by pathogen 1 and replacement with an uninfected host | μJ1Δt+o(Δt) |
J1→J1−1 J∅→J∅+1 |
| 6 | Death of host singly infected by pathogen 2 and replacement with an uninfected host | μJ2Δt+o(Δt) |
J2→J2−1 J∅→J∅+1 |
| 7 | Death of coinfected host and replacement with an uninfected host | μJ1,2Δt+o(Δt) |
J1,2→J1,2−1 J∅→J∅+1 |