Philippa Kaur

Since the bone discontinuities noted in the three vertebrae analyzed show no clear sign of bone overgrowth, it is pivotal to rule out the possibility that we are dealing with preservation damages before proposing an accurate diagnosis for the lesions. The close-up view examination of the abnormalities shows that their edges have clear signs of smoothing and rounding (Fig. 1), which represent important evidence of osteoblastic activity18,19. Additionally, the similar color of the cortical damage and normal bone can be used as secondary evidence to rule out post-mortem processes as a possible origin of the alterations, since recent destructive processes are lighter than the rest of the bone19. Therefore, as taphonomic processes can be ruled out, the pointed evidence strongly suggests that the discontinuities observed are of pathological origin. More specifically, these breaks found in all three vertebrae are indicative of bone fracture.Based on fracture analysis criteria applied here20, which consider the location and morphological pattern of the fractures, we classified the fractures noted in all vertebrae as traumas belonging to Type A (vertebral body compression), Group A2 (split fractures), and subgroup A2.1 (sagittal split fracture). This diagnosis implies that the traumatic episode was likely caused by a compressive force on the vertebral column, which split the vertebral bodies in the sagittal plane. This type of injury is considered stable—i.e., the fracture does not have a tendency to displace after reduction—and neurological deficit is uncommon20,22,23. Although stable traumas cause only moderate pain, without generating significant movement limitations20, the Eremotherium individual here analyzed died with unhealed bones, as there is no evidence of callus formation.The absence of other skeletal signs that point to the presence of another type of disease concomitantly to the fractures allows us to reject the possibility that they have been generated as a result of a pre-existing disease (e.g., infection, neoplasm). We also consider that the vertebral injuries were not caused by repetitive force (stress fractures) because this type of injury is commonly characterized as a nondisplaced line or crack in the bone, called hairline fracture3. Those refer to situations where the broken bone fragments are not visibly out of alignment and exhibit very little relative displacement21. Although the Eremotherium vertebrae fractures’ can be described as nondisplaced, they also have a noticeable gap between their edges that is mostly narrow with wider parts in the middle, something found in split fractures20 but that is not characteristic of hairline fractures. Lastly, the subgroup C1.2.1 (rotational sagittal split fracture) might be a source of confusion due to similar morphological pattern with subgroup A2.1 (sagittal split fracture). However, in subgroup C1.2.1 there are compressive and rotational forces acting simultaneously, producing total separation into two parts20, which clearly did not occur in the vertebrae analyzed here.In humans, compression fractures are most commonly caused by osteoporosis, although infection, neoplasm and trauma can also be etiological factors23,24,25. However, as aforementioned, the absence of other pathological skeletal marks is an important characteristic to take note as it serves to disregard the possibility of the fractures’ genesis to be secondary to another pathology. As such, in this case, osteoporosis, infection and neoplasm are unlikely etiologies. On the other hand, a compression fracture in a healthy individual is commonly generated after a severe traumatic event such as a fall from great height23,26. This scenario seems to better explain the origin of the vertebral fractures in the case of the Eremotherium ground sloth herein studied.The three fractured vertebrae were recovered in the Toca das Onças site (Fig. 2), a small cave considered as one of the richest paleontological sites of the Brazilian Quaternary15. Two complete skeletons of Eremotherium laurillardi and fragments belonging to at least thirteen other individuals, together with several other bones assigned to different smaller species are known to this cave14. It comprises of a single dry chamber that can only be entered through vertical entrances approximately 4.5 m high (Figs. 2b–d and 3). Two different hypotheses concerning the depositional process of Toca da Onças were previously proposed: (1) the animals climbed down into the cave in search of water14; or (2) due to the vertical character of the cave entrance, it could have functioned as a natural trap where animals accidentally fell into the cave15.Figure 2Location map of the Toca das Onças site and images of the cave. (a) Detail of the location, (b) cave entrance area view, (c) view from inside the cave, (d) Cave entrance detail. Scale bars 10 m in (b) and 5 m in (c). This figure was generated by Adobe Photoshop CS6 software (https://www.adobe.com/br/products/photoshop.html).Full size imageFigure 3Schematic representation of the Toca das Onças site. (a) Ground plan of the cave illustrating its morphology and dimension, (b) Cross-section illustrating the abyss-shaped entrance.Full size imageThe first hypothesis would indicate that the animal fell into the cave during an attempt to climb down. However, there is no report in the literature indicating that Eremotherium laurillardi could have been a climbing animal. In addition, the vertical morphology of the cave entrance would be a limiting factor for climbing behavior (see Fig. 3).Therefore, based on the type of fracture (compression sagittal split fracture) observed in the three vertebrae of Eremotherium as well as the inferred origin mechanism (fall from a great height), the presence of the individual here analyzed in the fossil accumulation of Toca das Onças is more likely explained by the second hypothesis. This idea is not particularly new as ‘entrapment due to fall’ has been described as a fossil accumulation mode to several other caves worldwide (e.g.,27,28). However, the use of bones fractures as an indicator of fossil accumulation mode is an interesting novelty. Of course, a detailed taphonomic investigation in the Toca das Onças still needs to be conducted in order to accurately interpret the formation of this important Quaternary fossil accumulation from Brazil.In sum, we suggest that the animal accidentally fell into the cave, fractured at least three sequential vertebrae (12th, 13th thoracic vertebrae and 1st lumbar vertebra) after the impact on the ground, survived for a while, but succumbed trapped inside the cave without food and water (Fig. 4). Other animals found in the cave, but without signs of bone fracture, may have fallen and not fractured their bones or not survived after the fall, especially the smaller ones. Finally, the proposal of falls to explain the unusual record of giant ground sloth fossils preserving much of its skeleton in caves, as reported for Toca das Onças site, contrasts with the better-documented pattern of skeletal accumulation via hydraulic action.Figure 4Artistic reconstruction of the suggested fall of the individual Eremotherium laurillardi into the cave. Artwork by Júlia d’Oliveira.Full size image More

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Design parametersA new type of Debris-flow grille dam is proposed to be built with a height of 8 m. Column section 500 mm × 700 mm, spacing 5000 mm. The cross section of the beam is 400 mm × 300 mm, and the spacing is 4000 mm. The section steel adopts I-steel 45a, the spacing is 250 mm. The counterfort wall is 300 mm thick and 6500 mm high. Pile foundation adopts manual digging pile, pile by 1000 mm, 5000 mm deep. The concrete is C30; Stressed bar is HRB335; Stirrups is HRB300; Stay Cable is 3 (emptyset) s15.2. The design size of the anchor piers is shown in Fig. 12. In the Figure where (T = 2 times 10^{5} N); (L_{l} = 8500;{text{mm}}); (E_{l} = 1.95 times 10^{5} ;{text{N/mm}}^{2}); (A_{l} = 420;{text{mm}}); (D_{e} = 1000;{text{mm}}); (L_{m} = 1200;{text{mm}}); (E_{e} = 3.0 times 10^{4} ;{text{N/mm}}^{2}); (H = 1000;{text{mm}}); (mu = 0.2); (E = 20;{text{N/mm}}^{2}). The parameter of gully bed soil is shown in Table 1.Figure 12The parameters of anchor piers.Full size imageTable 1 The parameters of gully bed soil.Full size tableAnalysis of results(1) The effect of the elastic modulus and Poisson’s ratio of the surrounding soil on the displacement deformation of the anchor-pulling system.The elastic modulus (E) and Poisson’s ratio (mu) are important parameters for calculating the displacement deformation of soil. They have something to do with both the properties of materials and the stress level. To analyze the effect of the physical parameter variation of the surrounding soil on the displacement deformation of the anchor-pulling system, we can study changing the elastic modulus and Poisson’s ratio. The variation range of the elastic modulus is 15–45 N/mm2, and the variation range of Poisson’s ratio is 0.15–0.25.Figure 13 shows the variation curve in which the displacement deformation increases with the elastic modulus of the soil around the anchor pier. We can see that as the elastic modulus of the soil around the anchor pier increases, the displacement deformation decreases gradually. When the elastic modulus is in the range of 15–35 N/mm2, the curve is steep, and the decrease in deformation is apparent. After 35 N/mm2, the curve becomes smooth, and the decrease in deformation tends to be stable.Figure 13The effect of the elastic modulus E(15–45 N/mm2) of the surrounding soil on the displacement of the anchor-pulling system.Full size imageIn Fig. 14, the displacement deformation increases linearly with Poisson’s ratio of the soil around the anchor pier. However, the total impact is not large. From calculation, the variation of elastic modulus of the soil around the anchor pier has nothing to do with elastic deformation of the stayed cable ((S_{1} )), but mainly influences relative shear displacement between anchor piers and the surrounding soil ((S_{2} )) and the compression performance of the soil on the front of anchor piers ( (S_{3} )). where ((S_{2} )) accounted for 89% and (left( {S_{3} } right)) accounted for 11%. When the Poisson ratio increases, the displacement deformation also increases. Poisson’s ratio has the greatest influence on the relative shear displacement ((S_{2} )) of the anchor pier and soil, accounting for approximately 96.4%. The design parameters should be selected correctly during design. The influence of parameters on the deformation of anchor system is analyzed by using control variable method. The influence of a single variable on the results can be intuitively obtained. However, the elastic modulus E and Poisson ‘ s ratio (mu) of rock and soil are not independent. Therefore, Matlab is used to analyze the influence of the two aspects on the deformation of the tensile anchor system, and the results are shown in Fig. 15. It can be seen from Fig. 15 that the influence of elastic modulus E on the deformation of tensile anchor system is much greater than that of Poisson’s ratio (mu). And the variation of the curve is basically the same, so the interaction between the two is weak.Figure 14The effect of Poisson’s ratio (mu)(0.15–0.26) of the surrounding soil on the displacement of the anchor-pulling system.Full size imageFigure 15Influence of elastic modulus E (15–45 N/mm2) and Poisson’s ratio (mu left( {0.15 – 0.26} right)) on deformation of anchor system.Full size image(2) The effect of the design parameters of anchor piers on the displacement deformation of the anchor-pulling system.The design parameters of anchor piers include the equivalent width (D_{e}), length (L_{m}) and height (H). Different design parameters have varying effects on the displacement deformation of the anchor-pulling system. Keep other parameters unchanged and let ( D_{e} ) vary in 0.5–1.5 m, (L_{m}) vary in 0.6–2.0 m, and (H) vary in 0.5–1.5 m. Analyzing their effect on the displacement deformation of the anchor-pulling system, the results are shown in Figs. 16 and 17.Figure 16The effect of equivalent width (D_{e})(500–1500 mm) on the displacement of the anchor-pulling system.Full size imageFigure 17The effect of equivalent length (L_{m})(600–2000 mm) on the displacement of the anchor-pulling system.Full size imageAs illustrated in Figs. 16 and 17, the effects of the design parameters of the anchor piers on the displacement deformation of the anchor-pulling system are almost the same. As the size increases, the displacement deformation gradually decreases, and the front section decreases quickly, while the rear section becomes gradually smooth. Here, the equivalent width (D_{e}) and length (L_{m}) mainly affect the compression performance of the soil on the front of anchor piers (left( {S_{3} } right)). The anchor piers can be seen as rigid bodies where horizontal displacement takes place. Increasing the size means increasing the contact area between the anchor pier and soil body. With this increase, the compression performance of the soil on the front of the anchor piers decreases. However, the effect of the height (H) on the displacement deformation of the anchor-pulling system is the contribution to the relative shear displacement between the anchor piers and the surrounding soil ((S_{2} )). When (H) grows, ((S_{2} )) grows accordingly. However, theoretically, the larger the effect of the size, the better it is. Because of the constraint of topographic conditions, construction conditions and economic benefits in practical engineering, it is necessary to choose the best size. the anchor pier provides enough anchor force and saves all kinds of resources. The best design dimensions suggested are (D_{e}) = 1.2 m–1.8 m, (L_{m}) = 1.5 m–2.5 m, and (H) = 1.0 m–1.6 m.It can be seen from Fig. 18 that the width (D_{e}) and the height (L_{m}) of anchor pier influence each other greatly. When (D_{e}) is 600 mm, with the increase of (L_{m}), the deformation of tension anchor system will first decrease and then increase. When (D_{e}) is greater than 800 mm, with the increase of (L_{m}), the deformation of tension anchor system will continue to decrease. And with the increase of (L_{m}), the decreasing trend is more obvious. When (L_{m}) is 500 mm, with the increase of the height of the anchor pier (D_{e}), the deformation of the anchor system will increase first. When (L_{m}) is greater than 800 mm, with the increase of (D_{e}), the deformation of the anchor system will continue to decrease. But the decreasing trend is not much different.Figure 18Influence of Anchor Pier Width (D_{e} left( {500 – 1500;{text{mm}}} right)) and Anchor Pier Height (L_{m} left( {600 – 2000;{text{mm}}} right)) on Deformation of Anchorage System.Full size imageThe numerical validationThe establishment of the finite element modelWhen the finite element model of the anchor-pulling system and surrounding soil is created, the constitutive model of the surrounding soil uses the Mohr–Coulomb elastoplastic model. The anchor pier and surrounding soil use eight nodes as oparametric elements, such as solid45, of which the basic grid unit is cubic units. When the grid is divided, the grid between the anchor pier and the surrounding soil contact is dense. The LINK10 unit is used to simulate cables, which have a bilinear stiffness matrix. It can simulate not only tensile bar units but also compressed bar units. For example, when the pull-up option is used alone, if the unit is under pressure, its stiffness disappears, so it can be used to simulate the relaxation of cables or chains. This feature is very significant for the static problem of wire rope, which uses a unit to simulate the entire cable. It can also be used for dynamic analysis with inertial or damping effects when the needed relaxation unit should pay attention to its performance rather than its movement. The soil is homogeneous. The soil physical parameters and structure design parameters are consistent with the theoretical calculation parameters mentioned above. The tensile force of the cable is exerted on the nodes as a force. The top surface of the model is free, and the normal displacements of the remaining faces are constrained such that the displacements are zero. The contact of the anchor pier and surrounding soils is a rigid-flexible surface-to-surface contact element to reflect the interaction. The surface of the anchor pier is regarded as the “target” surface, and the surface of the soil body is regarded as the “contact” surface. The coefficient of friction and normal penalty stiffness are 0.35 and 0.15, respectively. The scope of interaction between the anchor pier and the surrounding soil in the model is taken as 15 m × 11 m × 12 m, referring to past experience in engineering and the research data of the effect scope that the related anchors have had on the soil. The values of the model geometric parameters and physical and mechanical parameters are the same as in “Design parameters” section. The finite element model is shown in Fig. 19.Figure 19Finite element model of the anchor-pulling system and surrounding soil.Full size imageResearch on finite element model gridIn order to verify the convergence of numerical simulation, the soil was divided into three different mesh sizes. Condition 1 is fine finite element meshing. The stress nephogram of condition 1 is shown in Fig. 20. Condition 2 is medium finite element mesh. The stress nephogram of condition 1 is shown in Fig. 21. Condition 3 is coarse finite element mesh. The stress nephogram of condition 1 is shown in Fig. 22. See Table 2 for specific grid division.Figure 20Condition 1 stress cloud diagram.Full size imageFigure 21Condition 1 stress cloud diagram.Full size imageFigure 22Condition 1 stress cloud diagram.Full size imageTable 2 Mesh size of three working conditions.Full size tableIt can be seen from the stress nephogram of the three working conditions that the thicker the grid is, the greater the displacement of the anchor system is. The maximum displacement difference between condition 2 and condition 3 is 2.6%; the maximum displacement of condition 1 is 17% different from that of condition 2. The finer the mesh, the more accurate the numerical simulation results. But with the increase in computing time. It can be seen from Table 2 that the maximum iteration of condition 1 is 10 times, and the result will converge. The maximum iterations of condition 2 and 3 only need 7 times, and the results can converge.The calculation resultsFigure 23 and Fig. 24 are the displacement nephograms of the soil around the anchor piers for 100 kN and 400 kN, respectively. The soil displacement increases with increasing load, the affected area will increase and become uniform, and the area under load will also increase. The soil within the range of 1–3 m around the anchor pier is greatly affected, accounting for 80% of the total force. The soil around the anchor pier should be reinforced, and the anchoring force should be enhanced in the design.Figure 23Displacement fringe of soil around the anchor piers for 100 kN.Full size imageFigure 24Displacement fringe of soil around the anchor piers for 400 kN.Full size imageIn order to further study the influence of anchorage pier size on the displacement and deformation of anchorage system, finite element models with different sizes are established by finite element method. The stress nephogram is shown in Figs. 25, 26 and 27.Figure 25Top 800 mm, bottom 800 mm anchor pier stress nephogram.Full size imageFigure 26Top 1000 mm, bottom 1000 mm anchor pier stress nephogram.Full size imageFigure 27Top 800 mm, bottom 1000 mm anchor pier stress nephogram.Full size imageFrom Figs. 25, 26 and 27, it can be seen that when the anchor pier is rectangular, the deformation of the tensile anchor system decreases with the increase of the size of the anchor pier, but the degree is small. When the anchor pier is trapezoidal, the material is small, but the deformation is more ideal than the rectangular. It can be seen that reasonable selection of anchor pier size is crucial, not blindly increase the size of anchor pier.Figure 28 shows that the displacement of the soil around the anchor pier increases with increasing load, and the added value is obvious at approximately 2–3 mm. Figure 29 shows that the increase in load has a great effect on the soil in front of the anchor pier. As the load increases, the compressive deformation of the soil gradually increases. As the distance from the anchor pier increases, the displacement of the soil decreases, and the scope of influence gradually decreases. The displacement of the soil tends to be stable beyond 4–5 m from the anchor pier.Figure 28The displacement of soil around anchor pier.Full size imageFigure 29The horizontal displacement of soil along cable axis.Full size imageComparison of theoretical calculation and numerical simulation results at the time of load variationTo verify the correctness of the theoretical calculation, we compare the theoretical calculation with numerical simulation results of displacement deformation of anchor-pulling system under different pulling force of stayed cable. The results are shown in Fig. 30, see Table 3 for data.Figure 30Comparison of theoretical calculation and numerical simulation results.Full size imageTable 3 Comparison between theoretical calculation and numerical simulation.Full size tableAs seen from Fig. 30, the theoretical and numerical simulation results are consistent, showing a linear growth trend. The slope difference of the two straight lines is approximately 5%, which meets the accuracy requirements of geotechnical engineering. As the restraint effect of the surrounding soil on the anchor pier is not fully considered, the theoretical calculation result is too large. The deformation of anchor (left( {S_{1} } right)) in displacement deformation is the same, and the relative shear displacement (left( {S_{2} } right)) of the anchor pier and the soil and the compressive deformation ((S_{3} )) of the soil at the front end of the anchor pier are 1.25 times and 1.08 times the numerical simulation results, respectively. The change in (left( {S_{2} } right)) in the calculation results is large and should be taken into account in the design. More

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Following the work in36, we construct an epidemiological model which tracks the disease dynamics and population of two species of hosts following the introduction of a pathogen. The native host (hereafter simply referred to as “type 1”) is vulnerable to the disease, but due to being well adapted to the native habitat has high fecundity when uninfected. The invasive host (hereafter referred to as “type 2”), has coevolved defenses to the pathogen that increase both its tolerance of and resistance to the disease, but is not inherently as well-adapted to the habitat in the absence of infection (i.e., its intrinsic rate of growth in the new habitat is lower than that of the native).Our initial conditions correspond to a population of uninfected type 1 hosts with a small number of both uninfected and infected type 2 hosts, representing an invasion by a novel competitor carrying a novel pathogen into the type 1 population. We consider a vector-borne pathogen, and make the simplifying assumption that there is an already abundant competent vector species in the habitat. (For this initial formulation, we considered a scenario of mosquito-borne infections in birds, such as avian malaria37 or West Nile virus38, to motivate concrete choices.)The model couples two biological dynamics: the daily vector-borne spread of the disease among hosts, and a yearly host breeding cycle. We simulate in discrete time-steps that represent days using an SIR model taking into account the interactions between the disease, the two species of host, and the vectors. The model also includes a passive death rate for hosts of vectors, which increases for hosts while infected. While the vectors are assumed to breed daily, the hosts reproduce as part of an assumed annual breeding season, every (t_c) time-steps (typically equal to 365). These dynamics were informed by considering an annually breeding bird population in a tropical environment, however, they are not meant to reflect the realism of any one biological system. They are chosen here merely to allow a clean interpretation of modeled scenarios. Future models should explore the impact of greater variety in the dynamics of possible vector and host reproductive patterns.Epidemiological modelThe model tracks eight variables corresponding to combinations of host species and vectors with their infection status. Hosts may be of type 1 or 2, and are either susceptible to the disease ((S_1, S_2)), currently infected ((I_1, I_2)), or recovered ((R_1, R_2)). We assume that recovery is complete and recovered individuals suffer no residual effects from their infection aside from a lifelong immunity to becoming reinfected. (We later set the recovery rate for host type 1 to 0, so (R_1 = 0) at all times, but leave it defined for the sake of generality.) For simplicity, we model using only one stage of infection in which individuals are both infectious and symptomatic. The model also tracks the status of the vector population, which may either be susceptible ((S_v)) or infected ((I_v)). We assume that vectors do not recover from the disease, but also suffer no negative effects from being infected, acting only as carriers.For convenience of notation, we denote the total number of hosts$$begin{aligned} H = S_1 + I_1 + R_1 + S_2 + I_2 + R_2 end{aligned}$$and the relative frequencies of infection within their respective population$$begin{aligned} F_1 = frac{I_1}{H}, F_2 = frac{I_2}{H},F_v = frac{I_v}{S_v+I_v} end{aligned}$$which allows some equations to be written more compactly. Table 1 shows a summary of these variables.Table 1 Variables.Full size tableThe model also has several constant parameters that affect the dynamics. (beta _j) determines the probability that hosts of type j become infected when bitten by a single infected vector. We typically set (beta _1 > beta _2), making type 2 hosts less likely to become infected.Likewise, (delta _j) determines the probability that a vector becomes infected when biting an infected host of type j.(b_j) determines the bite rate for vectors on host type j. We assume that each vector bites the same number of hosts per day, so each vector’s probability of becoming infected depends only on the frequency of infection among hosts, while each host will be bitten more if there are more vectors.(gamma _j) determines the proportion of infected hosts of type j that recover from the disease each day. We typically set (gamma _1 = 0 < gamma _2), meaning infected hosts of type 1 do not recover, while infected type 2 recover after an average of (1/gamma _2) days.(mu _{j-}) determines the daily death rate for uninfected hosts of type j and (mu _{j+}) determines the death rate for infected host of type j. We typically set (mu _{1-} = mu _{2-}< mu _{2+} < mu _{1+}), meaning uninfected hosts have the same death rate regardless of type, infected type 2 have a higher death rate than uninfected hosts, and infected type 1 have the highest. (Both susceptible and recovered hosts are considered to be uninfected.) Table 2 shows a summary of parameters related to the SIR dynamics.Equation 1 shows continuous ordinary differential equations approximating the dynamics. Note that the actual model instantiates these in discrete time-steps using the forward Euler method with (h = 1).$$ begin{aligned}&frac{dS_1}{dt} = - S_1 beta _1 b_1 I_v /H - S_1 mu _{1-} \&frac{dI_1}{dt} = S_1 beta _1 b_1 I_v /H - gamma _1 I_1 - I_1 mu _{1+} \&frac{dR_1}{dt} = I_1 gamma _1 - R_1 mu _{1-} \&frac{dS_2}{dt} = -S_2 beta _2 b_2 I_v /H - S_2 mu _{2-} \&frac{dI_2}{dt} = S_2 beta _2 b_2 I_v /H - I_2 gamma _2 - I_2 mu _{2+} \&frac{dR_2}{dt} = I_2 gamma _2 - R_2 mu _{2-}\&frac{dS_v}{dt} = alpha _v H -S_v delta _1 b_1 F_1 -S_v delta _2 b_2 F_2 -S_v mu _v\&frac{dI_v}{dt} = S_v delta _1 b_1 F_1 + S_v delta _2 b_2 F_2 - I_v mu _v\ end{aligned} $$ (1) Table 2 Parameters for SIR dynamics.Full size tableFollowing a standard SIR model, susceptible hosts can become infected, and infected hosts become recovered, but each equation also contains a negative term corresponding to deaths. Thus, the total population of hosts is strictly decreasing in this time-frame. We assume that the vectors breed on a much shorter timescale than hosts, so we include a term for their births here, while host births are implemented by a yearly breeding event. We assume no vertical disease transmission, so all new vectors begin in the susceptible category. We assume that the daily birthrate for each vector increases with access to hosts, and decreases with competition among other vectors for hosts and breeding sites, so we set it equal to (frac{alpha _v H}{S_v + I_v}), where (alpha _v) is a constant scaling factor. Since the birthrate for each vector contains the total number of vectors in its denominator, the total number of vector births in the population will simply be (alpha _v H).A population with a larger number of hosts will be able to sustain a larger number of vectors. For a population with a constant number of hosts, the equilibrium vector population will be proportional to the number hosts: aH where (a = frac{alpha _v}{mu _v}) is the equilibrium vector density (number of vectors per host). For instance if (a = 2), then in equilibrium there will be twice as many vectors as hosts. Given a fixed number of hosts, the population of vectors will asymptotically approach the equilibrium value. In practice the total number of hosts is constantly changing, so the population of vectors will chase after this moving equilibrium, though for our standard parameters (alpha _v) and (mu _v) are sufficiently large such that this will occur on a short timescale, and the population of vectors remains close to the current equilibrium value.Breeding eventTable 3 shows a summary of parameters related to the breeding event. Every (t_c) days (typically 365), a breeding event occurs according to the following process.Table 3 Parameters for breeding event.Full size tableLet$$begin{aligned}&Delta S_1 = t_c alpha _{1-}(S_1+R_1)+t_calpha _{1+} I_1 \&Delta S_2 = t_c alpha _{2-}(S_2+R_2)+t_calpha _{2+} I_2 \ end{aligned}$$be the number of new host offspring of each type born this generation. In order to maintain consistency of temporal units among the parameters, each birthrate parameter is multiplied by (t_c). Let H be the current total number of hosts. Let$$begin{aligned} c = {left{ begin{array}{ll} 0 &{} hbox {if } H ge kappa \ 1 &{} hbox {if } H + Delta S_1 + Delta S_2 le kappa \ frac{kappa -H}{Delta S_1 + Delta S_2} &{} hbox {otherwise} \ end{array}right. } end{aligned}$$be the proportion of offspring that survive to adulthood. (None, if the population is already above carrying capacity. All, if the difference between the reproducing population size and the carrying capacity exceeds the new births. If the population is approaching carrying capacity, juvenile mortality scales proportionally so that the population will hit carrying capacity but not exceed it.)Then$$begin{aligned}&S_1 + c Delta S_1 rightarrow S_1 \&S_2 + c Delta S_2 rightarrow S_2 \ end{aligned}$$We assume there is no vertical disease transmission, so all new hosts begin in the susceptible category. We assume that the host population is iteroparous, such that the new offspring and the existing adult population both carry over to the next generation. If the new population would exceed the carrying capacity, we assume the limited space or supplies reduces the number of successful offspring so that the population exactly reaches the carry capacity by reduction in juvenile survival rather than population-wide competition that could also reduce the adult population.The carrying capacity is therefore what drives the interspecific host competition. Because births of both species are summed and then normalized by the total number of births, the higher the birthrate of one host, the larger a fraction of the available space it will capture during the breeding event. Similarly, the lower the death-rate of a host, the less space it frees up for the next breeding event. Even if one host species would be able to sustain a stable population on its own, the presence of a more fit competitor can lead to the extinction of the less fit type by driving its effective birth rate down.Immune-reproductive trade-offs and boundary conditionsWe assume that host type 1 is evolutionarily stable in the absence of the disease; an uninfected monoculture population below the carrying capacity will have at least as many births as deaths each cycle. In a continuous version of this model where births and deaths happened simultaneously, this might be defined by (alpha _{1-} ge mu _{1-}) . However in our model, the population spends many days decreasing due to deaths before the next breeding event occurs. The population exponentially decays throughout the cycle, and then jumps up during the breeding event. The number of new host births is proportional to the number of hosts at the start of the breeding event, which will be the lowest value of any other time during the cycle. Thus, the birth rate needs to be high enough that the surviving hosts can compensate despite their diminished numbers. Taking this into account, we get the condition$$begin{aligned}&alpha _{1-} ge frac{1-(1- mu _{1-})^{t_c}}{(1-mu _{1-})^{t_c}} \ end{aligned}$$Which is a higher bound on (alpha _{1-}) than the simpler one above, but will be close to it if (mu _{1-}) and (t_c) are small.To implement the scenario in which type 2 has increased resistance and tolerance to the disease at the expense of overall fecundity, we implement the following boundary conditions:$$begin{aligned}&beta _1 > beta _2 \&0 = gamma _1< gamma _2 \&mu _{1-} = mu _{2-}< mu _{2+} < mu _{1+} \&alpha _{1-} > alpha _{2-} > alpha _{2+} > alpha _{1+} end{aligned}$$Type 2 hosts are less likely to contract the disease, and are able to recover from it, while type 1 lack the immunological strength to eradicate it completely. Additionally, while both types of host are weakened by the disease, type 2 suffer fewer negative effects. However, this stronger immune response comes at the cost of reducing their birth rate when compared to healthy type 1 hosts.Due to the heterogeneous population, there is ambiguity in defining (R_0) for the disease. The two types of host have different transmission rates and durations of infection, and will therefore be responsible for different amounts of disease spread. To resolve this, we define several related values. Let (R_0^j) be the (R_0) of the disease in a homogeneous population of type j hosts: the average number of hosts infected (indirectly, through vectors) from a single infected host in a population consisting entirely of type j hosts.$$begin{aligned}&R_0^1 = frac{delta _1 beta _1 a b_1^2}{mu _v mu _{1+}} \&R_0^2 = frac{delta _2 beta _2 a b_2^2}{mu _v (mu _{2+}+gamma _2)} end{aligned}$$We simplify the equation for (R_0^1) since (gamma _1 = 0). We define w to be the frequency of host type 1: (w := (S_1 + I_1)/H). Then (R_0) for the vectors is$$begin{aligned} R_0^v = R_0^1 w + R_0^2 (1-w) end{aligned}$$which will also be the effective (R_0) of the disease for the hosts in the mixed population.For simplicity of results, we restrict to the case where type 1 is more infectious overall than type 2, in particular (R_0^1 > R_0^2). This allows us to avoid edge cases in simulation outcomes which are beyond the scope of this paper. We intend to lift this restriction and study these outcomes in future work.NoteAlthough usual epidemiological model formulations can rely on the value 1 as the boundary condition for (R_0) to determine the epidemic potential of an outbreak, in this case we are calculating effective (R_0) in a dynamic host population, such that the decrease in disease spread due to saturation from recovered hosts and already infected hosts increases the actual thresholds. More accurate criteria require a technical and somewhat cumbersome analysis, which we leave for a future paper. More

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Present fire-climate classificationTo identify the different regions of the planet with suitable climatic conditions for fire activity, we compare the global distribution of climate indicators based on temperature and precipitation, with satellite-derived GFED4 burned area data21 (Fig. 1). Starting from four general climates (Tr-tropical, Ar-arid, Te-temperate and Bo-boreal) based on the Köppen–Geiger climate classification main categories22, we create four fire-prone classes using climate thresholds to define the patterns observed in Fig. 1. Each category is characterised by the element that boosts fire activity during the FS: low precipitation, high temperatures or a combination of both. The classification is made by contrasting the probability distribution of the climatic variables at data points associated with high fire activity vs. points with low fire activity within the main Köppen-Geiger categories (see Threshold Selection in Methods section for a detailed explanation).Fig. 1: Burned area observations and climate drivers.a 1996–2016 maximum annual burned area (BAmax) and monthly burned area time series for selected regions. b Average monthly precipitation percentage from the annual total for the fire season (PPFS). c Average monthly temperature anomaly from the annual mean for the fire season (TAFS).Full size imageThe environmental conditions associated with fire occurrence emerge more clearly in this comparison, yielding the different threshold sets in Table 1 that determine the fire-prone months at any location (the selection method is detailed in the Methods section). We define those years with at least 1-month meeting the thresholds, as fire-prone years (FPY). Depending on the number of FPY at each location, the categories of Table 1 are sub-divided into recurrent (r), occasional (o) and infrequent (i) (see Methods). The average number of fire-prone months during the FPY is defined as the potential FS length (PFSL), i.e., the season with climatic characteristics prone to fire activity.Table 1 Fire classification defining criteria.Full size tableFigure 2a depicts the global map of the burned areas classified according to the selected thresholds (Table 1). Savanna fires are responsible for the largest proportion of burned area on the global scale21. The FS in these areas is longer than in other climates (see Supplementary Fig. 1) and, despite savanna fires being also dependent on ignition patterns and human policies and practices, the FS is tied to a pronounced seasonal cycle of precipitation23,24,25, with fire occurring mainly during the dry part of the cycle. Because of this, the Tropical – dry season fire class (Tr-ds) coincides with the distribution of the tropical savanna climate. In Fig. 2, boreal fires are represented as hot season fires (Bo-hs) due to the large positive temperature anomaly existing in those locations during the FS (Fig. 1c). In fact, temperature variations explain much of the variability in boreal burned area26,27. Temperate fires are classified as dry and hot season (Te-dhs) because they affect regions where the dry season coincides with the warm season (Fig. 1b, c). Here, high temperatures and precipitation seasonality determine fire activity and inter-annual burned area variability, e.g., in Western North America28,29,30,31 and Southern Europe32,33. Fire activity in arid regions occurs during warm months, but the relation with precipitation is more complex. The FS is associated with a hot season in cooler (MAT  27.5 °C), the FS starts right at the beginning of the dry season (e.g., the Sahel, Supplementary Fig. 12) while where MATs are more moderate, between 18.5 and 27.5 °C, it takes longer to develop (e.g., Central Australia and the Kalahari desert, Supplementary Figs. 12 and 13). Due to the dependency between fires and the existence of fuel in arid climates, we named this class Arid fuel limited (Ar-fl). A more in-depth discussion about the definition of this fire-climate class can be found in the section entitled Threshold selection for each climate of the Supplementary Information.Fig. 2: Fire-prone region classification.a With observed burned area data as a reference: not classified (NC, white) and misclassified (C, black) areas with BAmax = 0 ha, unclassified (NC, grey) and classified (Tr-ds, Ar-fl, Te-dhs and Bo-hs) areas with BAmax  > 0 ha. Each class is subdivided into three subcategories depending on the recurrence of the fire-prone conditions: recurrent (r), occasional (o) and infrequent (i). b Present (1996–2016) fire-prone climatic regions. c Future (2070–2099) fire-prone climatic regions with shaded grey representing a  0 ha) or fireless (BA = 0 ha). This reveals a two-way relation between fires and climate: fires take place under specific climatic conditions, and most places with these climatic conditions are indeed fire-prone, which supports our earlier hypothesis. Fire activity is controlled by weather, resources to burn and ignitions, as represented through the fire regime triangle12,20. On broad temporal scales and large spatial scales, temperature and precipitation have an important impact on fire because these climate variables influence vegetation type and the abundance, composition, moisture content, and structure of fuels34. Although ignitions may be driving fires to a greater extent than temperature or precipitation at specific locations or events35, they do not seem to limit fire activity at coarse spatial and temporal resolutions, implying that where fuels are sufficient and atmospheric conditions are conducive to combustion, the potential for ignition exists, either by lightning or human causes13,20. For all these reasons, we can identify specific climates that are prone to fires.The areas classified as fire-prone in Fig. 2b comprise 99.26% of the observed global mean annual burned area in Supplementary Fig. 2. This percentage is above 85% for all four general climates (Supplementary Fig. 20). The percentage of land area with non-zero burned area data classified as fire-prone is 91.22%. Considering for each location only the obtained FPY, the percentage of the observed burned area classified is 90.36%. Furthermore, the PFS obtained in the fire-climate classification (Fig. 3b) also correlates well with the timing of observed fire incidence, as globally 87.91% of the observed mean burned area occurs during the identified months of PFS at classified fire-prone locations.Fig. 3: Potential fire season.a Future minus present potential fire season length (PFSL) difference in months (ΔPFSL). b Present potential fire season. c Future potential fire season.Full size imageUnclassified regions (in grey in Fig. 2a) correspond for the most part to those with the least burned area or those where agricultural practices modify the climatic seasonality of fires. In addition, as the classification is conceived from a climatic point of view, locations with fire activity associated with specific meteorological conditions that are not appreciable at the monthly temporal resolution, are probably not well identified. For example, a week of extremely high temperatures could be almost unnoticeable in the monthly mean temperature, but not in fire activity. Similarly, months with the same total precipitation may have different fire activity if the precipitation falls concentrated in a few days or is distributed throughout the month. Furthermore, the short temporal sampling period of the burned area data could also be influencing our results. Locations with long fire cycles may not be well represented in the data.Future fire-climate classificationA future fire-climate classification map is derived by applying the thresholds obtained in the present fire-climate classification to future climatology variables from multiple coupled model intercomparison project phase 5 (CMIP5) global circulation model (GCM) outputs, considering the RCP8.5 scenario (the worst-case climate change scenario of the CMIP5). Two contrasting approaches can be taken for analysing future fire activity, one that considers quick vegetation adaptation to the new climatic conditions, and another that does not. These two approaches clearly diverge in the boreal regions, where the biome (mainly taiga) is strongly conditioned by the low temperatures and where future temperature changes at the end of the 21st century will have a greater amplitude. It is expected that the boreal forest of these areas will not be immediately replaced by a temperate mixed forest where the average annual temperature exceeds the range of values typical of the taiga biome. Terrestrial vegetation compositional and structural change could occur during the 21st century where vegetation disturbance is accelerated or amplified by human activity, but equilibrium states may not be reached until the 22nd century or beyond36.Based on the assumption that during the future period (2070–2099) the vegetation will not be fully adapted to the new climatic conditions, and since the present Köppen–Geiger climate classification (on which we base our Tr, Ar, Te and Bo categories) closely corresponds to the different existent biomes22, we analyse only the projected changes in the specific fire-climate classification variables, maintaining the general division of Tropical, Arid, Temperate and Boreal regions as is in present climate conditions. The future fire-climate classification is shown in Fig. 2c.We note that we determine future fire activity from relationships of the latter with the present climate; however, these relationships might not be stationary. Our approach does not contemplate possible future changes in precipitation frequency if they are not noticeable in monthly precipitation amounts. Areas with the rising incidence of extreme precipitation events due to global warming37 may experience an increase in monthly precipitation but a decrease in rainy days, which may lead us to consider the conditions there less favourable for fire activity than they actually will be.Future changes in global fire activityModelled future fire-prone regions experience significant variations with respect to the present (Fig. 2b, c). Due to global warming, the Bo-hs fire class pertaining to boreal forests would spread over a larger area, reaching most of Northern Scandinavia and undergoing a southward and northward expansion in Canada, Alaska and Russia. This category may experience a percentual expansion of 47.0% according to our results. This expansion is more accentuated for the combination of the highest recurrence subcategories Bo-hs-r and Bo-hs-o, reaching a value of 111.5%.The conjunction of Te-dhs-r and Te-dhs-o fire classes of midlatitudes also undergoes a considerable expansion of 24.5% in the area (Fig. 2b, c). The most remarkable changes are expected in Southern China and Southern Europe. A large part of Europe transitions from an infrequent fire category to a more frequent fire category with Csa and Csb Mediterranean climates38.The Tr-ds fire classes with frequent fire-prone conditions in the Tropics presents fewer spatial changes (Fig. 2b, c), with a spatial contraction of 6.3%. The most important differences are found in South America. Some of the climate model results considered here indicate also that some parts of the Eastern Amazon rainforest will move from a non-fire class to Tr-ds fire class, as other studies have suggested39.The Arid fire-prone classes Ar-fl-r and Ar-fl-o would increase its area by 5.0%. Projected changes in the extent of this class are very sensitive to changes in annual precipitation, conducive to vegetation and fuel reduction or increment, thus there is significant uncertainty in the proximity of desert regions (Fig. 2c).Clearer conclusions can be drawn from the FPY and PFSL calculation (Figs. 3 and 4). The number of months meeting the set of conditions in Table 1 yields the estimated PFSL (Fig. 3b), and the number of years with at least 1-month meeting the thresholds, the FPY. In the boreal regions, we obtain a general lengthening of the PFS. The PFS of these areas is conditioned by temperature, so the amplified warming of Artic zones40 is expected to make the FS longer. Notwithstanding, in certain parts of Eastern Asia, the intense warming is counterbalanced by an increase of the precipitation in certain warm months (see Supplementary Figs. 21 and 22), leading to a slight shortening of our estimated PFS. There is evidence, however, that temperature increases may lead to drier fuels in the future despite the precipitation increase, thus augmenting fire risk, as some investigations have shown for Canada41. Our results agree in general with several other studies that have previously pointed towards an increase of the FSL in boreal areas1,17,42, even when some suggest a more pronounced lengthening in more northerly latitudes1,17. In terms of the frequency of years with fire-prone conditions, the conclusions are even clearer. A general increase of the FPY is observed, especially for northerly latitudes, where the differences reach values of more than +4 years per decade (Fig. 4a). This possible increase in fire activity in boreal areas may result in significant peatland combustion and a release of the large quantities of soil carbon that they store into the atmosphere43. These greenhouse gas emissions may create a positive feedback loop, leading to a further increase in temperature, which in turn will enhance boreal wildfire incidence and more peatland burning.Fig. 4: Fire-prone years.a Future minus a present number of years with at least one month classified as fire-prone per decade (ΔFPY). b Present fire-prone years per decade. c Future fire-prone years per decade.Full size imageThe Te-dhs fire class, corresponding to temperate climates, would also experience a general lengthening of the PFS (Fig. 3). A future precipitation decline may be especially significant in Southern Europe (Supplementary Fig. 21), associated with an increased anticyclonic circulation yielding more stable conditions44, while the temperature rise would be quite homogeneous among all Te-dhs fire-climate class areas. The FS drought intensification around the Mediterranean, together with the general warming (Supplementary Fig. 21), would lead to a lengthening of the PFS of around 2 months (Fig. 3a), but summer months could also experience this precipitation decline (Supplementary Fig. 22), meaning that the FS would be more severe. The Western US, which has already experienced over the last decades the lengthening of the FS45 and the increase of large fires46 and extreme wildfire weather47,48 due to climate change, may also experience an FS lengthening by the end of the 21st century. Some authors18,48,49,50 have studied projected fire future changes from other points of view (occurrence of very large fires, wildfire potential, etc.), finding also a general increase of fire severity by the end of the century in some of these Te-dhs fire regions. The interannual recurrence of fire-prone conditions will significantly increase in countries like France, Italy or Eastern China (Fig. 4a).The PFSL of the Tropical Tr-ds fire-climate class presents slight differences between present and future values (Fig. 3). Some areas of the Northern African savanna may experience a shortening of the PFS, while Southern Africa shows a lengthening. A dipole pattern of wetting in tropical Eastern Africa and drying in Southern Africa51 could be the reason for these future changes. There is a contrasting influence of ENSO in present African fire patterns52, which suggests that the future pattern of precipitation variations in Central Africa may be associated with ENSO future changes under climate change conditions53. Although the quantification of ENSO changes in a warmer climate is still an issue that continues to be investigated, an expansion and strengthening of ENSO teleconnections is confirmed by some authors53,54,55. The general increase in precipitation along all seasons in western equatorial Africa would lead to a significant decrease in the recurrence of interannual fire-prone conditions (Fig. 4a).Our results show that fire-prone areas in Temperate and especially Boreal climates are projected to undergo the most significant expansion and lengthening of the potential FS at the end of the XXI century driven by rising temperatures. In the Tropics, little change is expected in these respects. Notwithstanding, global warming is likely to make fire risk more severe mostly everywhere, and in particular in some regions such as Mediterranean Europe and the Eastern Amazon, where an important decrease in precipitation is also predicted during the PFS. More favourable fire conditions will potentially increment fire activity and burned areas in many places. In others, especially in the Tropics, increasing suppression efforts and a cease to agricultural and pastoral practices like vegetation clearing by fire, replaced by more intensive farming, could counteract the impact of a warmer climate. A reduction of these human-caused fires in the Tropics could bring global burned area down2, despite rising trends elsewhere, given the vast contribution of Tropical fires to the burned areas at the global scale (Fig. 1). More

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