When host populations move north, but disease moves south: counter-intuitive impacts of climate warming on disease spread

Empirical observations and mathematical models show that climate warming can lead to the northern (or, more generally, poleward) spread of host species ranges and their corresponding diseases. Here, we explore an unexpected possibility whereby climate warming induces disease spread in the opposite direction to the directional shift in the host species range. To test our hypothesis, we formulate a reaction-diffusion equation model with a Susceptible-Infected (SI) epidemiological structure for two host species, both susceptible to a disease, but spatially isolated due to distinct thermal niches, and where prior to climate warming the disease is endemic in the northern species only. Previous theoretical results show that species’ distributions can lag behind species’ thermal niches when climate warming occurs. As such, we find that climate warming, by shifting both species’ niches forward, may increase the overlap between northern and southern host species ranges, due to the northern species lagging behind its thermal tolerance limit, thus facilitating a southern disease spread. As our model is general, our findings may apply to viral, bacterial, and prion diseases that do not have thermal tolerance limits and are inextricably linked to their hosts’ distributions, such as the spread of rabies from arctic to red foxes.

to climate warming, there exists an infected northern population, and an uninfected, but susceptible, southern population. The arctic rabies system, for example, lends itself to this 83 formulation of our pre-and post-climate warming scenarios. Indeed, historically, rabies has 84 been endemic in Arctic foxes (Vulpes lagopus) (i.e., the "northern population"), while red 85 foxes (Vulpes vulpes) (i.e., the "southern population") have remained disease-free with only 86 sporadic outbreaks (Mørk and Prestrud, 2004;Tabel et al., 1974). The movement of red foxes 87 northward, facilitated by climate change and anthropogenic disturbance, has already led to 88 an increase in overlap among the two species which can be observed in most arctic areas 89 (Gallant et al., 2012(Gallant et al., , 2020Savory et al., 2014), and might constitute a threat for potential 90 fast spread of rabies to the south, given the vast distribution of red foxes across Eurasia, 91 North America, part of North Africa and in most of Australia (Hoffmann and Sillero-Zubiri, 92 2021), with major consequences for human and animal health. Additionally, if rabies is spread 93 southward, rabies' disease range may overlap with more host species, specifically skunks and 94 raccoons (Finnegan et al., 2002), opening up new transmission pathways. It is therefore 95 imperative to understand how climate warming can contribute to the risk of the southern 96 spread of diseases, for the prevention and management of rabies, as well as other prion and 97 viral diseases.

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We formulate a temperature-driven moving habitat model based on a reaction-diffusion 100 framework (Cantrell and Cosner, 2004) to understand disease dynamics for directly trans-101 mitted pathogens in a warming climate, and in spatially structured host populations. Our 102 model combines disease dynamics with the reproduction, survival, and dispersal of two host 103 populations (i.e., the northern population, characterized by the sub index "n", and the south-104 ern population, with sub index "s") in a landscape consisting of a thermal gradient, such that 105 each population occupies a distinct region in the North or in the South. We assume that the 106 dynamics characterizing the northern and the southern host populations are identical, except 107 for the thermal tolerance limits of the two populations.

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Spatio-temporal dynamics: Susceptible and infected individuals disperse by random mo-109 tion, where the dispersal ability is quantified by the diffusion coefficient D n (for the northern 110 species) and D s (for the southern species). We assume that susceptible populations exhibit 111 logistic growth, with a temperature-dependent reproductive rate r n (T (x, t)) or r s (T (x, t)) 112 (described below, see Eq. (2)), and density-dependent mortality rate µ n or µ s . We assume 113 that infectious individuals do not reproduce, and die with a density-dependent mortality rate 114 ν n or ν s . Susceptible individuals can become infected by contacting infected individuals in 115 northern or in southern populations alike, where disease transmission occurs at rate β nn , β ss , β ns or β sn , depending on whether the contact has been between two individuals of the same 117 population (northern or southern) or of different populations.

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The system of equation describing the spatio-temporal dynamics of the northern and southern populations is given by: where S n (x, t), I n (x, t), S s (x, t) and I s (x, t) represent the densities of susceptible and infected 119 individuals in the northern and southern populations respectively, at time t and at location 120 x. Although, for application to a specific host-parasite system, the modelling of population 121 growth, disease dynamics, and dispersal may require a more complex framework than that 122 provided in Eq.
(1), in order to emphasize the broad validity of our findings we aimed for 123 the simplest possible formulation of the population dynamics, which relies on very minimal 124 assumptions. Possible extensions of the model will be discussed later in this manuscript.

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Temperature, species niches and climate warming: In Eqs. (1a) and (1c), the birth 126 rates r n (T ) and r s (T ) are represented as functions of temperature T = T (x, t), which de-127 pends on the location x and on time t. Specifically, we assume that birth rates are constant 128 and greater than zero within the species' thermal tolerance range, identified as the species 129 "thermal" or "fundamental niche", and zero outside of the thermal tolerance range. We write: thus in the fundamental niche) of each species northwards, at a constant rate, and equally 154 at all points in space. Figure 1: Hypothesized south-to-north temperature gradient as a function of location x prior to climate warming (solid black line, t = 0) and after 25, 50 and 75 years of climate warming (dashed black lines, t = 25, 50, 75). The temperature is assumed to increase 0.1 • C per year, which corresponds to a yearly 1 km shift to the north. The location of the fundamental thermal niches of the northern (in blue, color online) and southern (red, color online) populations prior and after climate warming are represented as horizontal solid lines (for t = 0) and dashed lines (for t = 25, 50, 75). The tolerated temperature ranges of each species prior to climate change are indicated on the vertical axis, where the northern species tolerates lower temperatures (ranging from -15 • to -1 • ), and the southern species tolerates higher temperatures (ranging from 1 • to 15 • ). The impact of varying the Euclidean distance between niches, and thus the thermal tolerance limits of each species, is discussed in Fig. 3b.
Simulations: We will focus on the situation where, prior to climate change, species dis-156 tributions has reached endemic equilibrium, where the disease is present in the northern 157 population only. Although the southern population is also susceptible, it is spatially isolated 158 due to the distinct thermal niche, and thus disease-free. We assume a numerical cutoff value 159 of 0.001, below which population densities are considered to be zero.

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As climate change occurs, the temperature gradient is uniformly increased, which results

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Our numerical simulations show that climate warming may induce the southward spread of 169 disease when host species' ranges shift northwards (Fig. 2). Prior to climate change, the 170 disease reaches endemic equilibrium in the northern population and, because of the spatial 171 isolation arising from the distinct thermal niches, the disease does not spread into the southern 172 population (Fig. 2a). After 25 years of continuous climate warming, the thermal niches of both 173 populations have moved northwards, as have their population densities, but these densities 174 now lag behind their thermal tolerance limits (Fig. 2b). The infected northern individuals 175 (b; blue dashed line) shown south of x = 35 km occupy habitat that is too warm, and will 176 ultimately go extinct even if no further climate warming occurs; however, extinction takes 177 time and disease spread to the southern population is enabled via this transient persistence.

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Indeed, the lag of the northern infected population behind its southern thermal tolerance limit 179 is sufficient to "bridge the gap" to the northern limit of the southern susceptible population (b; 180 right-most red dashed line), allowing the disease to be transmitted to the previously isolated 181 and uninfected southern species. Once disease establishes in the southern population, we 182 observe a wave of infection, which moves southward in space ( Fig. 2c and d).

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A climate-induced southern spread of the disease is observed only if the thermal tolerance 184 limits, and thus fundamental niches, of the two host species are far enough to be spatially 185 separated before climate warming occurs, but close enough to allow disease spread after 186 climate warming begins. When a southern spread is observed, the Euclidean distance between 187 niches greatly affects the number of years of climate warming needed before disease spread 188 between populations is observed (Fig. 3a). Additionally, southern disease spread requires 189 a high dispersal ability and birth rate of the southern species ( Fig. 3b and c), and it is 190 more likely to be observed when the mortality rate of northern infected individuals is low 191 (Fig. 3d). Other model parameters, such as the disease transmission rates and the dispersal

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We note that the conditions required for the southern spread of disease may be restrictive: 210 1) there must exist a spatially isolated susceptible, but uninfected population in the south;

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and 2) the southern population must not be so isolated that individuals cannot disperse into change and emerging infectious diseases. Jama, 275 (3):217-223, 1996. Figure S.1: Years of climate warming elapsing before the spread of the disease in a southern population is observed, as a function of (a) the dispersal ability of the northern species (D s ), (b) the birth rate of the northern species (r n ), (c) the mortality rate of the northern species (µ n ), (d) the mortality rate of the southern species (µ s ), (e) the interspecific transmission rate (β sn and β ns ), (f) the intraspecific transmission rate (β nn or β ss ). Other parameter values are given in Fig. 2.