Demographic Processes Drive Increases in Wildlife Disease following Population Reduction

Population reduction is often used as a control strategy when managing infectious diseases in wildlife populations in order to reduce host density below a critical threshold. However, population reduction can disrupt existing social and demographic structures leading to changes in observed host behaviour that may result in enhanced disease transmission. Such effects have been observed in several disease systems, notably badgers and bovine tuberculosis. Here we characterise the fundamental properties of disease systems for which such effects undermine the disease control benefits of population reduction. By quantifying the size of response to population reduction in terms of enhanced transmission within a generic non-spatial model, the properties of disease systems in which such effects reduce or even reverse the disease control benefits of population reduction are identified. If population reduction is not sufficiently severe, then enhanced transmission can lead to the counter intuitive perturbation effect, whereby disease levels increase or persist where they would otherwise die out. Perturbation effects are largest for systems with low levels of disease, e.g. low levels of endemicity or emerging disease. Analysis of a stochastic spatial meta-population model of demography and disease dynamics leads to qualitatively similar conclusions. Moreover, enhanced transmission itself is found to arise as an emergent property of density dependent dispersal in such systems. This spatial analysis also shows that, below some threshold, population reduction can rapidly increase the area affected by disease, potentially expanding risks to sympatric species. Our results suggest that the impact of population reduction on social and demographic structures is likely to undermine disease control in many systems, and in severe cases leads to the perturbation effect. Social and demographic mechanisms that enhance transmission following population reduction should therefore be routinely considered when designing control programmes.

The first equation represents the rate of change of the susceptible population in terms of three processes: birth of susceptibles at rate rN (1 − N/c) (note all individuals are assumed susceptible at birth since we ignore vertical and pseudo-vertical transmission); death of susceptibles at rate dS and infection of susceptibles at rate βSI. Similarly, the second equation represents the dynamics of the infected population which increases as susceptibles become infected at rate βSI, and decreases as infectives die due to the effects of the disease at rate eI and due to other natural causes at rate dI.
As described in the main text we model the effect of simple population reduction by introducing an additional death rate p. Thus the death rates become (d + p)S and (d + e + p)I for susceptibles and infectives respectively. To account for the impact of changes in host susceptibility and behaviour induced by population reduction we add kp to the horizontal disease transmission rate which therefore becomes (β + kp)SI. The resulting equations are

Appendix S1.3 Disease free and endemic equilibria
Solving Eqn. 1 for biologically realistic steady states, whereṠ =İ = 0, and S, I ≥ 0, gives: is often referred to as the population equilibrium, K.
3. The endemic equilibrium, at When e = 0, provided that d + p < 1 (i.e. where the population persists because the birth rate, rescaled to 1, exceeds the combined mortality and removal rate), the endemic equilibrium simplifies to and when p = 0, the endemic equilibrium simplifies further to Appendix S1.4 Disease stability Equilibria are stable (i.e. attract trajectories) when all eigenvalues have negative real parts, unstable (i.e. repel trajectories) when all eigenvalues have positive real parts, or a saddle node when some are positive and the rest are negative (e.g. this may allow the disease free equilibrium to be stable in the absence of infectives, but unstable in their presence). Note that while an equilibrium can be mathematically negative, it directs a biological process, which must remain non-negative, and thus a negative equilibrium cannot be reached. We examine the eigenvalues of Eqn. 1 (main text), with e = 0 for tractability. These are: Evaluating the eigenvalues for the endemic equilibrium gives: Evaluating the eigenvalues for the disease free equilibrium gives: (required for non-population extinction), therefore the endemic equilibrium is stable, and the disease free equilibrium is a saddle node if λ 2 < 0, which occurs when: otherwise the stabilities are reversed. For p = 0, this becomes β > d/ (1 − d).
Assuming the endemic equilibrium is stable, it can only persist if I * > 0. This is equivalent to S * < S * DF , provided that e ≥ 0, which is an easier calculation. Therefore Bounds under which the disease can persist may be obtained for each of the parameters by rearranging the above inequality as follows: In the absence of population reduction, when p = 0 these become: Appendix S1.5 Conditions for the deterministic perturbation effect Π eqm > 0 The size of the persistent PE is given by: Focussing on the algebraically tractable case without disease induced mortality, e = 0, and on situations where disease is still present with population reduction, I * (p) > 0, we examine two cases: Note that Π 2 = S * DF − S * , and so for case 2, the values are exactly the same as for Eqns. S2 in Section Appendix S1.4. Also note that Π 1 = Π 2 when the stability changes, i.e. when I * (0) = 0.
Next, we look for conditions on the perturbation effect under Case 1, by solving Π 1 > 0 for different parameters (conditions for Case 2 correspond to the bounds for stability given by Eqns. S2).
• Natural mortality rate d. Rearranging Π 1 > 0 for d, we obtain: • Population reduction rate p. First, observe that if the number of infectives prior to population reduction is greater than the total population size under persistent population reduction, then there is no room for I to increase. Therefore However, rearranging Π 1 > 0 gives: For which Eqn. S3 is the limit as k → ∞.
• Behaviour change k. We look for conditions on the perturbation effect under both cases: Case 2: Note that k 2 is the minimum k for which the disease is able to persist(i.e. population reduction maintains the disease when k > k 2 ). Also note that k 1 = k 2 when I * (0) = 0.
This demonstrates that β is bounded below in order for the disease to persist, 0 < I * (p), however it also bounded above in order for the disease to increase with population reduction, I * (0) < I * (p). If both of these bounds are satisfied, then 0 < I * (0) < I * (p) (which gives Case 1). Note that this interval increases with k.

Transient perturbation effect Appendix S2.1 Analysis
While intermediate behaviour of I(t) prior to the disease reaching a new equilibrium can only be solved numerically, both the long term and initial behaviour can be analysed algebraically. Here we examine the initial behaviour of Π(t; p) by linearising the system, and measuring the difference betweenİ(t; p) anḋ I(t; 0) at time t = 0. Substituting I(t) from Eqn. 1 into Eqn. 3 (see main text), and differentiating with respect to time givesΠ Note that since S ∈ (0, 1), this requires that k t > 1, therefore the transient perturbation effect does not occur in the absence of a change in behaviour. It is immediately clear that the transient perturbation effect depends directly on, and increases with, p and k, however it also depends on initial conditions S(0) and I(0), which in turn depend on the remaining parameters. Intermediate behaviour must be found numerically, but some further insight can be obtained by observing howΠ(t) depends on S(t) and I(t).
Substituting S = N − I intoΠ gives a quadratic in I, which can be solved to show thatΠ 0 > 0 when I ∈ (0, N − 1/k), and is maximised when S = N/2 + 1/2k and I = N/2 − 1/2k. Therefore the transient perturbation effect increases fastest when S and I are near these values (i.e. roughly equal), and larger k permits a greater range of I for which the transient perturbation effect is possible.

Analysis and simulation of the perturbation effect in the stochastic spatial model Appendix S3.1 Robustness of transient perturbation effect in spatial model
We examine the impact of varying certain mechanisms on the transient perturbation effect; in particular, the density dependent dispersal. In the main body we considered the case where f (N j ) = 1 when N j < αN * DF (where N * DF is the disease free equilibrium), and 0 otherwise. The threshold parameter α = 0.7 was arbitrarily chosen, however Fig. S1 shows that the transient perturbation effect occurs for a range of α that determines how sensitive the rate of dispersal is to local reductions in the size of the population in the destination site.  Table 1, and one parameter is varied at a time. 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.
In addition, P I (t) should be allowed to fluctuate freely, with an non-negligible chance of the disease becoming extinct within any individual sub-population, and of the disease transferring to neighbouring disease free groups; thus, from any set of initial conditions, the system should reach equilibrium within a reasonable time period. Only a very narrow parameter range allows for this situation, while still allowing the perturbation effect to occur. A set of parameters that allowed the perturbation effect to occur, while still allowing P I (t) to fluctuate freely and that P I (t) → P * I ∈ (0, 1) in a reasonable time period are: c = 20, r = 1, d = 0.08, e = 0.46, m = 0.1, α = 0.7, β w = 0.8, β b = 0. See Fig. S2 for results. These parameters were chosen to provide an equilibrium where P I (t) fluctuates, but remains roughly stable and between 0 and 1, and the perturbation effect still occurs. The system is given 2000 years to stabilise, then population reduction is applied for 2000 years, during which a new equilibrium appears to be reached. Afterwards, the system eventually returns to the original equilibrium.