Table 1.
Default event rates for the stochastic SI model.
Table 2.
List of parameters used in the deterministic and stochastic SI models.
Figure 1.
Deterministic simulation of I(t), and algebraic solution of I*(p).
The results of ongoing population reduction are shown for various levels of disease enhancement k in (A) endemic disease, (B) emergent disease (starting near the disease free equilibrium, ). (C) shows the endemic equilibrium for varying β. The lines cut the vertical axis at
, and so the perturbation effect occurs whenever a line rises above this value. Note that for β = 0.2, the equilibrium is negative for small p (which cannot be reached, since only a non-negative number of individuals is biologically possible), and so if any disease is introduced for p = 0, it moves to the disease free equilibrium
, and the perturbation effect does not occur until p is sufficiently high. The dotted line shows βu (see text for details), marking the upper bound of β for given p for which the perturbation effect is possible, and crosses each line at the point where the increase no longer occurs for that value of β (βu is also illustrated in Fig. 2A). Parameters are given in Table 2, except p = 0.2 in (A) and (B).
Figure 2.
Sensitivity analysis of the persistent and transient perturbation effects in the deterministic model.
Parameter values as in Table 2 and marked by a red dot when explicitly varied, the transient perturbation (green) is shown for t = 5, the persistent perturbation (blue) is evaluated at
. The transient perturbation: (A) has an optimum for intermediate β, decays with (B) natural and (C) disease induced mortality, and (D) increases with k. With the exception of (D), the behaviour of the persistent perturbation is more complex. In (A) to the left of
(dashed vertical line) β is low and there is no perturbation (
). To the right of
(dotted vertical line corresponding to upper bound
, see text) the prevalence in the absence of culling is sufficiently high to prevent a perturbation. The central region between
and
is divided by a third vertical line
(dot dashed), independent of k and p, into regions corresponding to case 1 (
) where the disease persists the absence of population reduction, and case 2 (
) where it does not (see text for details). The maximum persistent perturbation occurs at this boundary. Under case 2, population reduction is sufficient to stabilise the endemic equilibrium. In (B) as natural mortality d increases from zero (moving left to right)
decreases, and the pattern seen in (A) is reversed. Here the dotted vertical line
denotes the lower bound
. (C) shows the impact of disease induced mortality e is similar to that of natural mortality, but the chosen parameter values mean that prevalence is never too high to prevent a perturbation effect. Note: dotted and dashed lines are reversed when k is too low for the perturbation effect to occur, leaving no room for cases 1 and 2. See Fig. 4 for analogous spatial model results.
Figure 3.
Time trajectories and heterogeneity for emergent disease in the stochastic model.
(A) Population numbers, S(t), I(t), and N(t). (B) Proportion of sub-populations containing infectives, , effective transmission rate β, and dispersal rate. (C) Distribution of I across sites. (D) Effective transmission rate β for disease transmission vs population reduction coverage p1. Parameters are given in Table 2, and initial conditions are at the disease free equilibrium
, while in 20% of sites randomly chosen, a single individual is infected, resulting in
. Population reduction occurs annually from years 50–69, and in
of sites (chosen randomly each year) the removal rate is set to
, without regard to disease status (equivalent to an overall culling rate of
). An initial reduction in I is rapidly replaced by an increase, which is due to the increased chance of invasion of naïve groups by infectives due to the density dependent dispersal. The CI for the effective transmission rate increases for large
due to the increasing number of simulations where the disease becomes extinct.
Figure 4.
Sensitivity analysis of in the stochastic model.
The size of the perturbation effect, , at time
starting near the disease free equilibrium for (A) Between and within-groups infection rates
and
(log scale). (B) Natural mortality rate d. (C) Disease induced mortality rate e. (D) Dispersal rate m (log scale). Default parameters are given in Table 2, and one parameter is varied at a time. This is analogous to Fig. 2 for the non-spatial case. Initial conditions are such that 20% of sites are randomly chosen to start near the endemic equilibrium (with a minimum of 1 infective), while the remainder begin at the disease free equilibrium.