iting cases of more general models, we gain the ability to explore the connections between them in ways that might, firstly, help us to understand properties of the models (such as persistence), and that, secondly, can be used to guide modelling decisions about the appro- priateness of particular compartmental structures for specific purposes.
6.5
Persistence thresholds and their problems
In this section, we outline two problems of persistence thresholds in deterministic models. The first of these relates to the application of persistence thresholds to stochastic systems, whereas the first relates to their usual mode of calculation. The first of these problems arises because of the binary nature of persistence thresholds: just above the persistence threshold, population sizes can be arbitrarily small. In stochastic systems, these arbitrarily small popu- lations would be expected to go extinct very rapidly, and as a result, the persistence threshold does not provide a good measure of persistence.
In relation to the second problem, the typical approach to understanding population persis- tence in deterministic models consists of evaluating the stability of steady state solutions. It consists of finding the steady state solutions by solving for dp/dt = 0; the stability of these solutions is then found by considering the derivative with respect to p at the solution and if the derivative is less than zero, the solution is stable. This fact can then be used to establish the parameter range under which the solution is stable (see e.g. Case, 1999, for an accessible explanation). A related approach, based on the Jacobian of the next genera- tion matrix (NGM), can be used in the case where there are a large number of states, or in metapopulation models including the SRLM (see e.g. Feng and DeWoody, 2004).
The application of this approach can be used at any steady state solution. In many models, including the two-state version of the Levins model and the SRLM, only two stable solutions are possible. In this case, the invasion and persistence thresholds are equal and the deriva- tion of the ‘persistence’ threshold often actually proceeds by finding the invasion threshold by showing the conditions under which the trivial solution is unstable (instead of the condi- tions under which the non-trivial solution is stable), presumably because this simplifies the mathematics involved. Although this simplifies the mathematics, it does not necessarily lead to as much biological insight. For example, the SIRS model has only two possible stable solutions; as a result, we can evaluate the invasion threshold as this must be the same as the persistence threshold. For the SIRS model, with β the contact rate between susceptible and infectious individuals, and r the recovery rate, and ν denoting the rate at which recovered in- dividuals return to the susceptible state, the set of differential equations governing the system
6.5. Persistence thresholds and their problems 134 is dS dt = νR− βSI dI dt = βSI − rI dR dt = rI− νR. (6.9)
Invasion occurs if the number of infectious individuals increases, i.e. if dIdt > 0
βSI− rI > 0 ⇐⇒ βS
r > 1. (6.10)
At the beginning of an epidemic, one can assume that the number of susceptibles is ap- proximately equal to zero, so we obtain the condition β/r > 1, equivalent to the expected threshold of R0 > 1, and because there are only two stable solutions, we can also conclude
that disease will persist for R0 > 1. However, although the assumption of a completely
susceptible population is sensible for disease invasion, it is clearly problematic in the case of long-term disease persistence, and deriving the persistence threshold in this manner provides little direct insight into the biological mechanisms of persistence. The result is particularly troubling because although as one would expect, the steady state prevalence is lower in the SIRS model than the SIS model, the persistence threshold is the same for the two models Thus, although it is typically easier to find the persistence threshold by first obtaining the in- vasion threshold (e.g. because of the relative ease of substituting the trivial solution into the derivative), this functions only because of a mathematical property of the model, and little biological insight about persistence is gained. It would therefore be instructive to have an approach that provides more information about persistence and the counterintuitive findings such as the correspondence of the persistence threshold in the SIS and SIRS models. The definition of persistence threshold provided by Hanski and Ovaskainen (2000) considers the persistence threshold directly in relation to the stability of the nontrivial solution, and is therefore more robust to situations in which persistence can occur even when invasion is not possible (e.g. when there is an Allee effect). Nonetheless, presumably for reasons of analytic tractability, most of the work carried out by these authors focuses on the SRLM without an Allee effect. In this situation, they show that the persistence threshold (corresponding to the existence of a non-trivial steady state) can be derived as the leading eigenvalue of a matrix, that they call a ‘landscape matrix’; in other situations, the threshold can be found numerically (Ovaskainen and Hanski, 2001).
6.5.1
Allee effects
First described in the 1930s, an Allee effect can be said to exist when there is a positive relationship between individual fitness and population size (or density) (Allee, 1931). This means that populations go extinct when rare, making initial invasion difficult and persistence
6.5. Persistence thresholds and their problems 135
of small populations unlikely. As noted by Boukal and Berec (2002), the most commonly cited cause is that of the difficulty of finding an appropriate mate in low density populations of sexually-reproducing individuals. However, other causes are reviewed in Berec et al. (2007) and include inbreeding depression, tendency not to be pollinated, or the minimum group size required to raise offspring, search for food or avoid predator attacks (Boukal and Berec, 2002).
A distinction in the literature exists between strong and weak Allee effects (see e.g. Brassil, 2001). Strong Allee effects occur when there is a threshold population size or density below which population growth is negative. In the case of weak Allee effects, there is no threshold and growth rate is a positive function of population size, at least at small population sizes. From a mathematical perspective, a strong Allee effect means that the extinction state is a locally stable equilibrium point of the system for all parameter values (McVinish and Pollett, 2013a). Allee effects can also apply to birth or death processes, or both. For example, in- ability to find a mate affects birth rate while the existence of a minimum group size required to avoid predator attacks relates to death processes. Multiple effects may be present simul- taneously. Berec et al. (2007) reviews possible interactions between Allee effects and their combined role in contributing to persistence.
Allee-like effects have also been discussed for metapopulations. In this context an Allee effect refers to a reduction in the colonisation capacity of patches when few patches are oc- cupied. More precisely, the growth rate of the metapopulation due to exports from individual patches is an increasing function of overall occupancy at low levels of occupation (McVin- ish and Pollett, 2013a). Amarasekare (1998) reviews evidence supporting the existence of metapopulation level Allee effects. Zhou and Wang (2004) show that an Allee-like effect in a metapopulation can emerge from an imposed Allee effect at the local population level. While the inclusion of an Allee effect has a negative effect on persistence in general, Brassil (2001) provide an illustration of a model in which the effect is much stronger when applied to metapopulations than in the case of single, fully-mixed population. Further, McVinish and Pollett (2013a) show that habitat degradation can have much stronger effects on populations with a metapopulation level Allee effect than those without. In the case of metapopulations, Allee effects can be implemented as affecting colonisation or extinction processes, or both. Although Allee effects have been discussed most commonly in a conservation biology con- text, they have also been invoked as being important in epidemiology. Most discussion relates to the effect of pathogens on host species subject to an Allee effect. In this con- text, the additional burden of disease can reduce numbers of a species sufficiently to cause extinction (Haydon et al., 2002; Hilker et al., 2009). Although such effects are rarely men- tioned, Allee effects may also be present directly within pathogen and parasite populations. In sexually-reproducing parasites, many of the causes described in Berec et al. (2007) still apply. However, even in asexual pathogen populations, Allee effects may still apply. For
6.5. Persistence thresholds and their problems 136
example, it is likely that they could be caused by reductions in genetic variation when popu- lation sizes are small. This effect could be compounded by the increased probability of gene fixation in small populations due to drift (this mechanism is suggested by its existence in other clonal organisms such as vegetatively reproducing plants Fischer et al., 2008). Other mechanisms in asexual pathogens could be those typically considered in the context of co- operation. For example, slime moulds can reproduce sexually in the amoebae state but also asexually through cellular budding and through sporulation mechanisms. In order to repro- duce through spores, they need to cooperate to form a fruiting body, something that requires a minimum population size. Similar kinds of cooperative behaviours have been invoked in the case of Pseudomonas aeruginosa, a common disease-causing bacterium.
Boukal and Berec (2002) review different functional forms for incorporating Allee effects into population dynamic models. In later sections, we follow Amarasekare (1998) in the incorporation of a strong Allee effect, and Tabares and Ferreira (2011) for the incorporation of a weak Allee effect.
In addition to the persistence threshold, Ovaskainen and Hanski (2001) also discussed the invasion threshold. The metapopulation invasion capacity gives rise to the threshold con- dition λI > δ above which the trivial equilibrium state is unstable, meaning that a single
small local population is able to invade an otherwise empty network. In cases correspond- ing to the Levins model, the invasion threshold is the same as the persistence threshold, and the approach is closely related with the next generation matrix (NGM) approach applied in epidemiology (Diekmann et al., 1990; van den Driessche and Watmough, 2002; Heffernan et al., 2005). The elements of the NGM tell us the mean persistence time of a patch or pop- ulation multiplied by the rate of infection of other patches per unit time, thus the expected number of exported infections during the lifetime of the population infectiousness. The el- ements of matrix M, in contrast, tell us the rate at which a patch becomes infected by a specific other patch (rather than infects another patch), multiplied by the expected time that this patch will remain infectious. Feng and DeWoody (2004) also define a slightly different matrix ˜M that also has λM as its leading eigenvalue. Like the NGM, this approach is based
upon the analysis of the Jacobian matrix at the trivial solution. So although the approaches correspond in the case of Levins-type models in which λI = λM > 0, they diverge when the
assumptions of these models are broken. That is, the definition of metapopulation capacity threshold allows there to be an occupancy level from which the system shrinks, provided it is growing for at least some p. For example, in the case of a strong metapopulation-level Allee effect, there is a threshold below which the metapopulation shrinks.
In the remainder of this chapter, we use the re-parameterised version of the Levins model (Eqn. 6.3), and the SRLM (Eqn. 6.4) in its usual format. All of the models that we use here are deterministic and their relationship with stochastic models is explored in Section 6.7. We follow Amarasekare (1998) in the choice use the re-parameterised Levins model because the