Resource-Driven Encounters and the Induction of Disease Among Consumers

Submitted Manuscript 2016. Territorial animals share a variety of common resources, which can be a major driver of conspecific encounter rates. We examine how changes in resource availability influence the rate of encounters among individuals in a consumer population by implementing a spatially explicit model for resource visitation behavior by consumers. Using data from 2009 and 2010 in Etosha National Park, we verify our model's prediction that there is a saturation effect in the expected number of jackals that visit a given carcass site as carcasses become abundant. However, this does not directly imply that the overall resource-driven encounter rate among jackals decreases. This is because the increase in available carcasses is accompanied by an increase in the number of jackals that detect and potentially visit carcasses. Using simulations and mathematical analysis of our consumer-resource interaction model, we characterize key features of the relationship between resource-driven encounter rate and model parameters. These results are used to investigate a standing hypothesis that the outbreak of a fatal disease among zebras can potentially lead to an outbreak of an entirely different disease in the jackal population, a process we refer to as indirect induction of disease.

have as a result of temporarily available resources. 84 For the sake of simplicity, and because we believed the choice was reasonable 85 for the jackal population in ENP, we choose the time parameters to be the same, 86 τ 1 = τ 2 = τ = one week. We use O to denote the spatial region we are studying. To model the consumer's limited ability to detect resources and/or travel to 99 resources that have been detected, we assume there is a maximum distance 100 within which a given consumer will detect resources. Moreover, we assume that 101 consumers will detect all resources within a surrounding circle of radius and characteristics (e.g. latent periods, population turnover, and acquired immunity), 160 our present focus is on how consumer-resource interactions modulate transmis-161 sion dynamics in the early introduction phase. We are specifically interested in the 162 probability that the level of infection can reach an endemic state in the target pop-163 ulation before the period of resource increase dissipates. Given our context that 164 the disease dynamics take place over a large area and the pathogen introductions 165 are relatively rare, we introduce a fourth assumption:

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-each pathogen introduction resolves itself independently in the target pop-167 ulation (either to extinction or invasion).

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Mathematically, this is tantamount to omitting the γ spillover term in the transition 169 rate formulas and treating each pathogen introduction event independently. The 170 "endemic equilibrium" is the minimum size for the infectious population such 171 that the rate of increase equals the rate of decrease. We consider a pathogen 172 introduction to be "successful" if the size of the resulting infectious population 173 eventually exceeds the endemic equilibrium value: We then study the continuous-time Markov chain {I(t)} t≥0 with the transition rates I → I + 1 at rate λ(I) = bI 1 − I N I → I − 1 at rate µ(I) = νI, 178 and compute the probability of successful invasion assuming that a pathogen has 179 been introduced at time zero: In a sense made rigorous by Kurtz [20], when N is large this stochastic system 182 behaves more and more like an associated ODE,

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where we interpret y(t) as the proportion of the population that is infectious at 185 time t. If b > ν and y(0) > 0, then y(t) converges to the equilibrium value model there is always a chance that an infectious lineage will go extinct before it 191 reaches an endemic state. In Figure 3 we display ten stochastic SIS paths with a 192 population size of 50 with b = 2 and ν = 1. Some of these paths quickly go extinct, 193 while others reach the endemic state. Overlaid on the stochastic paths is Ny(t), 194 the rescaled solution to the associated ODE (2), with initial condition Ny(0) = 1.

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Just as it is for the ODE model, the reproductive ratio is a critical dimensionless 196 parameter in the stochastic model.
. On the other hand, when R 0 > 1, then as N → ∞, the probability of invasion 198 is strictly greater than zero. As described in Appendix A.4, p invasion is commonly 199 approximated by computing the complement of an extinction probability for an 200 associated branching process [24]. This gives the approximation With this in hand, we can estimate the probability that there is a success-203 ful pathogen invasion in the target population during the period of increased 204 resource availability, t ∈ [0, T]. For each pathogen introduction, we label it "suc-205 cessful" with probability p invasion and then note that, from Markov chain theory

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[25] the arrival of successful introductions is also a Poisson process, but with a 207 "thinned" intensity γ spillover p invasion . This implies that the time of the arrival of 208 the first successful spillover has an exponential distribution with rate parameter 209 γ spillover p invasion . Therefore, the probability of a successful invasion occurring dur- out of 244 zebra carcass sites). These data are displayed in Figure 6. less than 100m, we classified the event as a resource visit.

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Our use of the location datasets for collared jackals to identify "resource vis-

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By way of simulation and analysis, we are able to characterize the most prominent  Furthermore, we provide an approximate formula for the resource density κ * that 305 leads to the maximum number of encounters for a given distance of detection and 306 consumer density.

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Asymptotic results. In Figure 4 we see that E , the expected number of encoun-308 ters for the focal consumer, has a very regular power law behavior in both the correct leading coefficient appears to be larger than ρ, but we were unable to 323 obtain the exact value by mathematical analysis. Characterizing the encounter rate peak. For reasons discussed in Section 3.4, per-325 haps the most important "landmark" of the resource-encounter function is its 326 peak. Unfortunately it is difficult to directly analyze the magnitude of the peak 327 and the corresponding critical resource density. However, there is a natural first-328 order estimate that involves the small-and large-κ approximations. Solving for 329 their intersection yields the estimate κ * ≈ (1/π) −2 and E (κ * ) ≈ ρπ 2 where 330 κ * is the resource intensity that leads to the maximum resource-driven encounter 331 rate. From Figure 4, it is clear that this is an overestimate, but not dramatically 332 so. Using 1000 simulations at an array of κ and values, we found the following 333 estimates using a linear regression: approximations, we were unable to obtain a satisfactory explanation of the leading coefficients through direct mathematical analysis.

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Dependence on distance of detection. In addition to characterizing the encounter 339 rate's dependence on κ, we are also able to obtain an asymptotic understanding of more carcasses were available in that month (right panel of Figure 6).

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To employ a more quantitative statistical test, we fit a Poisson general lin- For example, in an April with i total carcasses, the expected number visitors ob-382 served at a carcass would be exp(β 0 + β 3 + β carc i). Using this statistical model, 383 we found a significant negative correlation between the number of observed car-  Jackal locations were calculated as the average of their GPS pings that occurred between two days before and after the carcass was estimated to be present. Right: Stacked histogram for the times that jackals chose to visit a carcass (teal bars) and the times that jackals refrained from visiting a carcass (gray bars).

The relationship between defendable territory size and the distance of detection
in ENP were estimated to be between 4 km 2 to 12 km 2 . This is comparable to

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The interpretation of the parameter from the data requires some discussion.

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In the mathematical model, is the maximum distance at which a consumer can 427 detect and then respond to a resource. We can think of the model as assuming 428 that the probability of detecting a resource is one within a distance and zero out-429 side that distance. Of course, in reality, this detection probability likely decreases 430 steadily as a function of distance. Rather than identify a specific value that we 431 definitively claim to be the best estimate of , we used the jackal movement data 432 to find a range of reasonable values.

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In Section 2.2, we described our stochastic small-population model for pathogen 450 and further discussion in Appendix A). This reduces our analysis to determining 451 whether the total rate of transmission (which is affected by the resource-driven 452 encounter rate) is greater than the disease-related mortality rate ν. 453 To assess whether a change in the consumer encounter rate is "large" in the 454 context of jackals and rabies, we followed Rhodes et al. [29] in establishing a 455 background rate of pathogen transmission (b = 1 wk −1 ) and a disease-related 456 mortality rate (ν = 1.4 wk −1 ) yielding the reproductive ratio R 0 ≈ 0.7. Since R 0 < 1, rabies is found to be sub-critical. To connect the resource-driven encounter    is that it can be used to predict the qualitative dynamics of a system once certain 507 fundamental parameters are estimated: the consumer density (ρ), the resource 508 density (κ) and the maximum distance of detection and response ( ).

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To work through a specific case study, we used location data for a population

Opportunities for integrating more detailed animal behavior 542
The complex relationship between resource allocation, consumer behavior, and 543 pathogen spread deserves further study. We constructed our model to be detailed 544 enough to examine our primary question, but simple enough to permit rigorous 545 mathematical analysis. While there are many ways to extend the model to account 546 for more nuanced behavior, we highlight a few.

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Resource detection and selection. There are other natural models for the con- as the height of the associated encounter rate peak.

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One major factor that we did not consider is heterogeneity in the resource  The reduction in variance is significant in the following sense. As we report oping a theory for susceptible-infectious encounter rates that considers both types 583 of individuals will be especially important for infections that alter host behavior 584 (e.g. rabies). Specifically, we note that the manner in which a rabid animal de-585 tects and selects resource sites could be much different than that of a susceptible 586 individual.

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Off-site encounters. At present, our model considers the relationship between 588 resource availability and the consumer encounter rate specifically at resource sites.

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However, a change in resource availability will likely influence other types of 590 encounters as well. For example, when consumers are forced to make long treks 591 to scarce resources, they may be exposed to unfamiliar individuals. Distinguishing   In what follows (and in the main text) when we write ϕ(x) ∼ x α as x → a, we 665 mean that there exists some constant C ∈ (0, ∞) such that 666 lim x→a ϕ(x) x α = C.

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For example, a result we will use below is that if Y ∼ Pois(λ) for some λ > 0, 668 then P {Y > 1} ∼ λ 2 as λ → 0. This is because and using the Taylor series expansion for the exponential (or simply L'Hôpital's 671 rule), we have For higher order terms we will use Big-Oh notation: we say that f (x) = O(g(x)) 674 near x = a if there exist constants C > 0 and L > 0 such that if |x − a| < L, then As in the main text, κ and denote the resource intensity and maximum dis-677 tance of detection respectively. In the presentation of our results we will assume the results when ρ = 1.

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We take the domain O to be a circle of radius R > 3 centered at the origin.

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There is a focal consumer located exactly at the origin. Resources are distributed 682 throughout O as a Poisson spatial process with intensity κ. Other, non-focal con- We say that O i is a basin of attraction for resource i: all consumers located in O i 699 will choose resource i as their resource to visit if it is within their detection radius. 700 We define B( x; r) to be the circle of radius r centered at the location x. Then the 701 distribution of the encounter variable β conditioned on a given resource landscape where we recall that η is the index of the resource chosen by the focal consumer 705 and 1 A = 1 if the event A occurs, and is zero otherwise.
consumers and resources within a distance r of the focal consumer. We proceed 712 by conditioning on the number of resources that are near the focal consumer. We 713 partition the sample space Ω as follows: 714 Naturally, it follows that E(β) = ∑ i E(β | Ω i ) P {Ω i } and we will find that the 716 dominant term is the one associated with Ω 10 . Looking at the other terms, first 717 observe that E(β | Ω 0 ) = 0 since, if there are no resources to consume, the focal 718 consumer will not have any encounters.

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For the event Ω 11 we again exploit that, when the detection distance or re-729 source density is small, it is unlikely that there will be more than one resource by the focal consumer, we have E(β | Ω 11 ) ≤ π 2 . Therefore 737 E(β | Ω 11 ) P {Ω 11 } = O(κ 2 6 ).
Turning our attention to the event Ω 10 , if there is only one resource in the focal consumer's detection radius, and the resource is the only one in the larger focal resource will choose the same resource as the focal consumer. In other words, 742 the number of encounters conditioned on Ω 10 is β| Ω 10 ∼ Pois(π 2 ). What was an 743 upper bound in previous cases is now equality. It follows that E(β | Ω 10 ) = π 2 .

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To compute the event's probability we argue as before, In the high resource density and large distance of detection we are unable to get 753 exact results. This is due to a fundamental barrier in the analysis that we will 754 describe below. In the high density regime we can provide what appears to be a 755 lower bound on E that, from the numerics, seems to scale with E as κ → ∞. Proof.
where, in the last line, we have used the Cauchy-Schwarz inequality. The exponential integral function can be written in terms of the series [36].

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Because this holds for all L κ , the proposition follows. meaning thatκ = κ 2 /˜ 2 = κ/ρ. 829 We note that this reduction of the problem to two parameters amounts to a The infectious population process is then a Galton-Watson branching process.

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The only two outcomes for such a process are extinction or explosion to infin-847 ity. The analysis reduces to recasting the CTMC as a discrete time generation-by-848 generation branching process that is defined in terms of the offspring distribution,