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General characteristics of study soilsTable 2 presents the descriptive statistics regarding the soil characteristics. Significant changes were observed in the distribution of sand (110–850 g kg−1), silt (50–530 g kg−1), clay (100–610 g kg−1), and soil textural class (7 texture classes) showing the diversity of natural and human processes involved in the formation and development of these soils28. Almost all soil samples were alkaline (with reaction at a range of 7.4–8.1) and calcareous (with CCE at a range of 5.5–35%). The EC of some soils was  > 4 dS/m (about 7% of the soil samples), indicating the partial salinity of the study soils. The organic carbon and total N contents of the soils were, on average, 2% (0.8–3.1%) and 0.28% (0.05–0.51%), respectively, placing them within the range of the moderate class. Likewise, the mean CEC of the soil, which is an effective indicator of soil fertility and quality, was in the moderate class of 12–25 cmol kg−129. The CEC was found to be highly correlated with clay (r = 0.76 P  Pb (58 mg kg−1)  > Ni (55.4 mg kg−1)  > Cu (38.8 mg kg−1)  > Cd (0.88 mg kg−1). In most soil samples, these ranges are comparable with data reported for other urban soils around the world—e.g. Ref.30 in Poland, Ref.31 in China, and Ref.32 in Greece. The values of Cd, Cu, and Zn were below their acceptable ranges as per the international standards4 in all soil samples. Nonetheless, the Pb and Ni contents were higher than their acceptable ranges in 13.1% and 17.4% of the samples, respectively. Furthermore, the concentrations of the five elements were higher than their background values in all urban soil samples. This difference was considerable for Cd, Pb, and Ni. The heavy metals had CV in the order of Cd (53%)  > Pb (51%)  > Ni (46%)  > Zn (21%)  > Cu (18%). This CV variation implies great variations in Cd, Pb, and Ni, which is linked to anthropogenic activities33. The background values of the metals, estimated by the median absolute deviation method10,14, were 52.3, 18.7, 0.45, 29.1, and 30.8 mg kg−1 for Zn, Cu, Cd, Pb, and Ni, respectively.We compared the concentrations of the heavy metals between urban and non-urban soils and found significant increases in the concentration of the metals in most soil types (Fig. 2). The urban soils had 17–36%, 14–21%, 41–70%, 43–69%, and 13–24% higher Zn, Cu, Cd, Pb, and Ni contents than the non-urban soils. The effluent and waste entry from multiple food processing and storage units, dying plants, metal plating facilities, and plastic production in close proximity of the study area is believed to be the reason for the high concentration of these trace elements. Research in various parts of the world, e.g., Ref.34 in India, Ref.35 in Brazil, and Ref.36 in China, has documented that the facilities have introduced significant quantities of heavy metals to soils. However, traffic and agrochemicals also play a key role in the accumulation of heavy metals in this region10.Figure 2The comparison of the mean values of Zn (a), Cu (b), Cd (c), Pb (d), and Ni (e) between urban and non-urban soils in different soil types. Different letters indicate significant differences in metal content within each soil type at P  Ni  > Cu. These findings are comparable to the results reported by37 and12. The highest EF for all five elements was observed in the Fluvisols soil type, reflecting that this soil type had been exposed to element pollution induced by urban activities to a greater extent than the other soil types. In a study on the pollution potential of four soil types in Central Greece, Ref.38 reported different ranges of element pollution across different soil types.Figure 3The comparison of the mean enrichment factor of Zn (a), Cu (b), Cd (c), Pb (d), and Ni (e) between urban and non-urban soils in different soil types. Different letters indicate significant differences in enrichment factor within each soil type at P  Pb (1.89)  > Ni (1.86)  > Cu (1.73)  > Zn (1.51). Mean PI for non-urban soils followed the order Cd (1.5)  > Zn (1.4)  > Cu (1.33)  > Pb (1.31)  > Ni (1.29). Nearly 7% and 16% of the urban soils showed moderate pollution (MP, PI = 2–3) and high pollution classes (HP, PI  > 3) of PI for Cd and 39% and 4% showed the MP and HP class of PI for Pb, respectively. However, the PI class was low pollution (PI = 1–2) for all soil samples and soil types in the non-urban soils. The results on the pollution index indicate a widespread intensification of soil pollution in urban soils across all studied heavy metals.Table 3 The level and terminology of PI and Ei of the analyzed heavy metals in urban and non-urban soils.Full size tableEcological risk, Ei was similarly found to be significantly higher in the urban soils than in the non-urban soils, even though the concentration of all elements except Cd fell within the low-risk class (Ei ≤ 40) in both urban and non-urban soils (Table 3). The mean Ei for Cd was 58.7 (moderate-risk class) and 39.2 (low-risk class) in the urban and non-urban soils, respectively. This means that urban activities have enhanced the ecological risk class of Cd by one grade. Overall, Cd had the highest EF, PI, and Ei among all heavy metals and in all soil samples, indicating a greater risk potential by Cd than Zn, Cu, Pb, and Ni across the water-soil–plant-human domain. Elevated Cd pollution by anthropogenic activities has been widely reported in the literature10,12,39. Cadmium as a Group 1 carcinogen element40 can accumulate in plant tissue without exhibiting visual symptoms. Therefore, Cd generally transfers from soil to the food chain covertly. Cadmium pollution can also influence soil quality and reduce crop yields and grain quality3.Similar to EF, PI, and Ei, the mean ER was significantly elevated in all urban soil types than the non-urban soils (Fig. 4). Among different soil types, the ER magnitude was in the order of Fluvisols (66.6%)  > Regosols (66.1%)  > Cambisols (59.8%)  > Calcisols (47%). These results indicate that Fluvisols carry a higher ecological risk potential for heavy metal accumulations than other soil types. In the study region, Fluvisols due to higher fertility and productivity are subject to more intense and extensive agronomic operations than other soil types13. Heavy application of agrochemicals (e.g., pesticides, herbicides, insecticides, and chemical fertilizers), accelerate the heavy metal input to the Fluvisols. Widespread application of nitrogen fertilizers and subsequent reduction in average soil pH markedly increases the solubility of certain heavy metals (e.g., Zn, Cu, Cd) which can be another factor increasing the ecological risk of heavy metal contamination in Fluvisols41. In addition, these Fluvisols are located on the margin of open urban wastewater channels, which are sometimes used for irrigation. A combination of mentioned processes can be implicated for higher ER of Fluvisols than that of other soil types as for BF, PI, and Ei.Figure 4The comparison of the mean ecological risk of selected heavy metals between urban and non-urban soils in different soil types. Different letters indicate significant differences in ecological risk within each soil type at P  Cu  > Ni  > Cd  > Pb in the roots, partially differing from that of the grain—Zn  > Cu  > Pb  > Ni  > Cd. Heavy metals concentrations observed in the corn roots and grains are almost comparable with those reported by42 in China and43 in Peru.Table 4 Summary statistical attributes of the concentration of heavy metals in corn root (R) and grain (G) along with their BCF and TF.Full size tableThe accumulation of heavy metals in the edible parts of corn is of higher importance. In the present study, the concentrations of these metals were lower than the acceptable level in the corn grains based on international references44. So, the consumption of corns grown in the regions should not threaten human and animal health in the short term, but caution should be exercised in their long-term consumption because some of these elements, especially Cd and Pb, which have long decomposition half-lives, gradually accumulate in body organs, especially in kidneys and livers45. Besides, the ratio of Zn, Cu, Cd, Pb, and Ni of the corn grain to their acceptable standard concentration, known as the pollution index of crop heavy metals, Ref.12 was lower than 0.7 for most corn samples, indicating the unpolluted risk class.The mean concentrations of Cd, Pb, and Ni were 5, 3.1, and 9.2 times as great in the corn roots as in their grains. This observation exhibits a notable phytoremediatory function of corn roots through restriction of radial translocation of heavy metals to the xylems and eventually into the grains. A similar trend of heavy metal accumulation in different plant organs has been reported in previous observations46,47. Based on Kabata-Pendias4 and Adriano22, plant cells can use the defensive tools of the roots to cope with heavy metals, especially Cd and Pb—highly toxic metals to plant cytosols. Accordingly, plant cells can fix these elements in the root system by such approaches as precipitating on cell walls, storing in vacuoles, and/or chelating by phytochelatins, thereby alleviating their toxic effects and inhibiting their translocation to plant shoots. For Zn, Cu, and Cd metals, a significant correlation was observed between their concentration in corn roots and grains. But, a less significant correlation (P  Cu (0.17)  > Zn (0.12)  > Ni (0.02)  > Pb (0.01). This implies that Cd, and to a smaller extent Cu is taken up by corn roots from the soil more readily, but Pb and Ni are less absorbable. These results are consistent with the reports of48 and46. The greater value of BCF-Cd may be related to a combination of the specific factors e,g., Cd concentration and chemistry, as well as soil characteristics (e.g., soil texture, pH, and calcium carbonate content)4. As was already discussed, the examined soils were characterized by high alkaline (pH = 7.4–8.1) and calcareous properties (CCE = 5.5–35%) with a high concentration of Soluble salts (EC = 0.7–6.6 dS m−1). These characteristics can result in the formation of complex Cd ions, especially CdOH+, CdCl20, CdCl+, CdSO40, and CdHCO3+4,22. These ions are plant-available, resulting in a further increase in Cd BCF. Regarding Ni and Pb, the alkaline and calcareous properties of the soils may have motivated insoluble compounds such as NiHCO3+ and NiCO30 (for Ni) and Pb(OH)2, PbCO3, PbSO4, and PbO (for Pb)4,22. These compounds cannot be uptake by plant roots, which may have resulted in a significant decrease in the BCF of these metals versus the other analyzed elements.Like BCF, the heavy metals had TF of  Pb (0.21)  > Cd (0.2)  > Ni (0.15). This implies that Zn and Cu are translocated from roots to grains readily, about four times as great as the other metals, while Ni, Cd, and Pb are translocated in smaller concentrations.The comparison of BCF and TF of Cd showed that less than 30% of Cd, on average, accumulated in the corn roots were translocated to the grains. This states that Cd is immobilized by various mechanisms before it can find its way into the grains. Some of the important mechanisms include (i) the antagonistic effects of Cd with other equivalent elements, especially Zn, Fe, and Ca, in the vascular system of corn, which reduces its mobility in the corn root-stem-grain system22, (ii) Cd sequestration in active exchange sites on the cell wall in the corn root-stem pathway10, and (iii) the binding of Cd with some specific compounds, e.g., phytochelatins of root vacuoles, which immobilizes it before its translocation to grains4,22. Lin and Aarts52 remarked that Cd mostly tends to be trapped in root vacuoles, which reduces its translocation to the upper parts of the plants. In general, it was found that corn plants have a high potential to absorb and accumulate Cd in their roots and Zn in their grains, which is consistent with previous studies41. For the majority of heavy metals, the values of BCF and TF in different soil types were in the order of Fluvisols  > Regosols  > Cambisols  > Calcisols, indicating that the great variety of soil types for the uptake and translocation of heavy metals in the soil-root-grain of the corn (Fig. 5).Figure 5Effect of soil type on the mean bioconcentration factor (a) and translocation factor (b) of selected heavy metals in urban soils. Different letters indicate significant differences in bioconcentration and translocation factors among soil types for each metal at P  Zn  > Cu  > Pb  > Ni for children, differing from that for adults (Cu  > Cd  > Pb  > Zn  > Ni). The values of HQ was  1 in over 87% of the samples, implying the low non-carcinogenic risk of this metal for corn-consuming children in the study region53. Rapidly developing children’s nervous system are highly sensitive to environmental factors, including heavy metals, so even a relatively low concentration of Cd in children’s blood may irreversibly affect their mental growth and functioning54.The highest HI was observed in children (min = 1.16, max = 2.31, mean = 1.63) followed by women and men which was similar to the found pattern of HQ (Table 7). These data show a moderate non-carcinogenic health risk (1 ≤ HI  More

Characteristics of heavy metal concentrationsOn the basis of the soil sample collection and chemical analysis, the concentration data for heavy metals in the urban soils of Wuxi were obtained. Through the statistical analysis of the soil heavy metal concentration data (Table 1), on the whole, the concentration of each heavy metal is as follows: Mn  > Zn  > Cr  > Ni  > Pb  > Cu  > Co  > Be  > Cd. Among these, the concentration range of Cr was 64.5–99 mg kg-1, and the average concentration was 72.9 mg kg−1. The concentration range of Ni was 31.4–67.5 mg kg−1, and the average concentration was 38.2 mg kg−1. The concentration range of Cu was 19.8–37.2 mg kg−1, and the average concentration was 25.5 mg kg−1. The concentration range of Zn was 72.4–1146 mg kg−1, and the average concentration was 90.2 mg kg−1. The concentration range of Cd was 0.34–1.06 mg kg−1, and the average concentration was 0.51 mg kg−1. The concentration range of Pb was 25.6–66.4 mg kg−1, and the average concentration was 37.6 mg kg−1. The variation coefficients of urban soil heavy metal concentration in Wuxi is between 0.09 and 0.33, which is less than 1. The spatial fluctuation of urban soil heavy metal concentration in Wuxi is small, indicating that the sources may be the same or similar.Table 1 Statistics of the heavy metal concentrations and Pb isotope ratios in the urban soils of Wuxi city (unit of heavy metal: mg kg−1; CV: coefficient of variation).Full size tableBy analysing the spatial distributions of the urban soil heavy metal concentrations in Wuxi, several obvious spatial distribution characteristics are found (Fig. 2). First, the heavy metals have high values in the central area of Wuxi, due to where has a high population density and various industries. The central aggregation of Pb is more obvious. Due to the dense roads in the city centre, vehicle traffic, bus stop signs and gas stations are mostly concentrated here, which will lead to Pb contents in this area that are significantly higher than those in other areas. In addition to the heavy metal concentrations, such as those for Cu, Zn and Cr in the downtown area, there are also areas with high values in western Wuxi and low values in eastern Wuxi. This phenomenon may be related to the land use types in Wuxi. In the western area of Wuxi, most land use types are urban and construction land, and the soils in this area are greatly disturbed by human activities. In the eastern region of Wuxi, woodland and grassland account for a large proportion of the land use types, which are less disturbed by human activities.Figure 2Spatial distribution characteristics of heavy metals in the urban soils of Wuxi city (unit: mg kg−1) [the figure was generated by Yan Li using the ArcGIS 10.2 (http:// https://developers.arcgis.com/)].Full size imageSource analysis of heavy metalsExploring for heavy metal pollution from emission sources is an important prerequisite for the study of urban soil pollution. By analysing the sources of heavy metals in soil environments, we can accurately determine which industries are major sources28,29,30 and whether there is homologous pollution. This is not only a theoretical basis for the study of lake sediment pollution and to clarify the risks brought by different pollution sources to the urban soil environment but also provides important guides for local government control of specific polluting industries and pollutant emissions. Based on this, the correlations and significance of heavy metals in the urban soils of Wuxi were analysed (Table 2). Generally, a heavy metal pollution source will emit multiple heavy metals at the same time. If the pollution source has a large emission, the concentration of these heavy metals in the environment will show a high level; on the contrary, if the emission of this pollution source is small, the concentration of these heavy metals in the environment will show a low level10. The correlations between the heavy metals Zn, Cr, Ni, Pb, Cu and Cd are between 0.655–0.907 and show strong correlations and significance at a level of 0.01. The strong significant correlations between different heavy metals indicate that these heavy metals have similar emission sources and transmission routes, which also means that they have consistent sources.Table 2 Correlations of Heavy Metals in the Urban Soils of Wuxi City.Full size tableTo further determine which industries are the sources of the heavy metals found in the urban soil of Wuxi, we analysed the Pb isotope data. The variation range of 208Pb/206Pb in soil is 2.09–2.12, and the average value is 2.10. The variation range of 206Pb/207Pb in soil is 1.17–1.18, and the average value is 1.177 (Table 1). After consulting relevant literature and materials, the main pollution sources of heavy metals in cities in eastern China include coal combustion, oil combustion, factory emissions, municipal wastes and so on3. Therefore, we collected the corresponding Pb isotope data in the emissions of heavy metal pollution sources. By collecting and comparatively analysing the Pb isotope data of known pollution sources (Fig. 3), it was determined that the Pb isotopes of the urban soil heavy metals in the soils of Wuxi have distinct characteristics. First, the Pb isotope distributions in the soils of Wuxi are relatively concentrated, and the ranges of variation are relatively small, which indicate that these heavy metals may have the same source or similar sets of sources. Second, the Pb isotopes in the urban soils of Wuxi city have few similarities with those of the uncontaminated soils and granites in eastern China; in contrast, the Pb isotopes in the urban soils of Wuxi are distributed in areas that are associated with coal combustion, automobile exhaust and urban waste (supplementary materials). The urban soil heavy metals in Wuxi generally have similar pollution sources and are greatly affected by human activities such as coal combustion and automobile exhaust emissions. Wuxi has a developed industrial economy and large numbers of factories. In the production and processing activities, the combustion of energy and fuel and the incomplete utilization of raw materials will lead to the enrichment of pollutants in the surrounding environment. By comparing other studies30,31, the Pb isotope analysis results in this study well indicate the source of soil heavy metals in Wuxi and make up for the Pb isotope data in this area. In the process of urban development, we should develop and apply clean energy, reduce the utilization of petroleum fossil fuels, and control the enrichment of heavy metals and other pollutants in the soil from the source.Figure 3Comparison of the Pb isotope compositions in the urban soils of Wuxi city with known sources.Full size imageEcological risk analysisBy calculating the potential ecological risk index for the heavy metals in the urban soils of Wuxi, the risks of heavy metals in the Wuxi soils were evaluated (Table 3). According to previous studies21, an Ei value lower than 40 indicates that a heavy metal is in a low-risk state at this location, and Ei values greater than or equal to 40 indicate that a heavy metal represents a high-risk state at this location. The average value of the potential ecological risk index of soil heavy metal Cd in Wuxi is 80.3, which represents a high-risk state. The average distributions of the potential ecological risk indexes of the heavy metals Cr, Cu, Zn, Pb and Ni are 1.8, 4.3, 1.1, 5.5 and 4.8, respectively, which all indicate a low-risk state. The risk statuses of different heavy metals may show certain correlations in space, which may be mutually complementary or antagonistic. Examining the spatial interactions of different heavy metal compound pollutants in urban soils plays an important role in the prevention and control of urban heavy metal pollution. Based on this, we used the Lisa analysis method to explore the spatial correlations of the different heavy metal risks in the urban soils of Wuxi (Fig. 4). The Moran scatter diagram can be divided into four quadrants that correspond to four different spatial patterns. High means that the variable value is higher than the average value, and Low means that the variable value is lower than the average value. In the upper right quadrant (High–High), a high-value area is surrounded by high-value neighbours; in the upper left quadrant (Low–High), a low-value area is surrounded by high-value neighbours; in the lower left quadrant (LL), a low-value area is surrounded by low-value neighbours; and in the lower right quadrant (High–Low), a high-value area is surrounded by low-value neighbours. High-High and Low-Low indicate that the differences between the region and its surrounding areas are small; that is, the regions with higher or lower values are concentrated, while the Low–High and High–Low quadrants indicate that the variable values between a region and its surrounding areas are different to a certain extent.Table 3 Ecological risk and health risk analysis of heavy metals in the urban soils of Wuxi (Cr-E represents the ecological risk of metal element Cr; Ni-E represents the ecological risk of metal element Ni; Cu-E represents the ecological risk of metal element Cu; Zn-E represents the ecological risk of metal element Zn; Cd-E represents the ecological risk of metal element Cd; Pb-E represents the ecological risk of metal element Pb; ADDderm-C is the average exposure to skin contact pathways for child; ADDderm-A is the average exposure to skin contact pathways for adult; ADDing-C is the average daily exposure to intake pathway for child; ADDing-A is the average daily exposure to intake pathway for adult; HI-C is the total health risk caused by accumulation of heavy metals in multiple ways in the same environmental medium for child; HI-A is the total health risk caused by accumulation of heavy metals in multiple ways in the same environmental medium for adult).Full size tableFigure 4LISA analysis of the ecological risks from different heavy metals [the figure was generated by Yan Li using the ArcGIS 10.2 (http:// https://developers.arcgis.com/)].Full size imageIn this study, two main results were obtained from spatial correlation Lisa analysis between different heavy metals. One is a High-High area, which is mainly distributed in the central and western regions of Wuxi city, which is consistent with the spatial distribution of the urban soil heavy metal concentrations in Wuxi city and is strongly disturbed by human activities. The other is the insignificant area, in which there are also large numbers of factories and enterprises and in which the forestland and grassland are distributed at intervals, which leads to an insignificant spatial correlation of soil heavy metal contents. Based on the above analysis, the high-risk areas for heavy metals in the urban soils of Wuxi are mainly concentrated in the central and western regions, and the relevant management activities need to be given great attention. In the eastern region, sporadic high-risk areas are also present, which should also receive due attention. Moran’s I is a method to measure the interdependence and degree of objects or phenomena by constructing statistics on certain characteristics or attributes for a certain spatial unit in the study area and the surrounding spatial units. It can be used to describe the spatial characteristics of spatial units such as aggregation or outliers in the distribution of certain attributes and is a very important technology in spatial data analysis33,34. However, few studies have applied it to the spatial relationship analysis of different heavy metals in urban soil.Health risk analysisBy using the health risk assessment model that is recommended by the U.S. EPA, this study calculated the health risks of soil heavy metals to adults and children through skin contact and ingestion. For both adults and children, the risk of soil heavy metals through ingestion was much higher than that caused by skin exposure (Table 3). For children, the total health risk that was caused by soil heavy metals is 0.078, which is four times that of adults. This may be related to children’s habits. Most children like to play with sand and climb around on the ground. These behaviours greatly increase the frequency of children contacting the soil, which thus increases the health risk caused by heavy metals in the soil. To further explore the spatial characteristics of the health risks of heavy metals in the soils of Wuxi, this study provides spatial predictions of the health risk values of soil heavy metals (Fig. 5). The total health risk values of soil heavy metals for children and adults have similar spatial distribution characteristics. High health risk values appear in the central area of Wuxi and decrease in a ring-shaped pattern. This is similar to the development degree of the city. The downtown area of Wuxi is densely populated, the pedestrian flow is very large, and the health risk of soil heavy metals in this area is very high, which poses a very serious potential threat. The health risk values for the western region of Wuxi are high, and there is also a potential threat. When compared with western Wuxi, eastern Wuxi has a lower risk.Figure 5Health risk analysis of heavy metals in the urban soils of Wuxi [the figure was generated by Yan Li using the ArcGIS 10.2 (http:// https://developers.arcgis.com/)].Full size image 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