Unravelling Transmission in Epidemiological Models and its Role in the Disease-Diversity Relationship.

1 There are a multitude of pathways a pathogen can invade and spread through a host population. The assumptions 2 of the transmission model used to capture disease propagation determines the outbreak potential, the pathogen’s 3 net reproductive success ( R 0 ). This review offers an insight into the assumptions and motivation behind common 4 transmission mechanisms, and introduces a general framework with which we unify all, where contact rate, the 5 most important parameter in disease dynamics, determines the type of transmission model. This general trans- 6 mission model framework helps bridge the gap between mathematical disease modelling and the much debated 7 disease-diversity relationship, by expanding common disease models to multiple hosts systems, and in considering 8 the role of host diversity in disease transmission. By describing the mechanisms of transmission as a stepwise 9 process, we provide a guide for modelling pathogens in multi-host systems and elaborate how these transmis- 10 sion mechanisms can affect host communities, but also how host communities affect the pathogen’s success. We 11 further expand on these models and introduce an approach for including host species’ evolutionary history into 12 transmission dynamics to hereafter aid the debate on the effect of biodiversity and community composition on 13 disease outbreak potential.


Introduction
Box 1: The Biology of Transmission Transmission is the transfer of infectious particles, or propagules, from an infected (donor) host, to a naive, receiving host. Transmission potential is then dependent on a threshold propagule load within the donor host (Wilber et al., 2016;Blaser et al., 2014). As pathogen particles often have to travel through space between leaving the donor host and establishing in the recipient host, the environment and the time a particle is free-roaming can play a role in transmission success (Tien and Earn, 2010). McCallum et al. (2017) deconstructed transmission into five discrete stages: • Stage 1: Dynamics of propagules within donor host.
• Stage 2: Production of pathogen-infective stages in donor host.
• Stage 3: Pathogen survival and growth in the environment (including the environment of an intermediate host).
• Stage 4: Dose acquired by recipient host at exposure.
• Stage 5: Pathogen load in the recipient host.
Between these stages of transmission, heterogeneities can arise and threshold-like behaviours may emerge. Conventional, linear frameworks of transmission may overlook such heterogeneity, for example, super-spreaders resulting from high infection loads (McCallum et al., 2017). For instance, nematode crowding within hosts causes an increased immune response, and saturates the relationship between transmission Stages 1 and 2, such that at high pathogen levels the production of the dispersal pathogen (nematode) life-stage plateaus (McCallum et al., 2017). Age of infection, host immunosuppressive capabilities and pathogen competition (in case of co-infection) can also have an effect on the production of pathogen dispersal stages (Stage 2). An example is Malaria, where physiological traits affect the susceptibility of individuals: People with sickle cell anaemia are more resistant to the disease (Elguero et al., 2015). In diseases such as cholera, high pathogen loads cause diarrhoea and vomiting and therefore increased shedding of infectious particles.
For Stage 3, the environment determines the subsequent received load, but the donor/recipients host behaviour is of importance too, determining the exposure to the shed pathogen. Some dispersal stages can linger outside hosts longer than others. For example, the decay rate of Mycobacteria, causing tuberculosis, is dependent on the temperature of the medium and likely also to light-exposure, therefore differing among seasons (Fine et al., 2011). However, some pathogens (e.g. HIV) are never exposed to the environment, and are directly transmitted, these pathogens skip Stage 3.
The load the recipient host acquires depends on its exposure (Stage 4), which is impacted by multiple factors including: the relative density of the donor and recipient hosts, the survival rate of the free-living pathogen stage, and, for trophically transmitted pathogens, the Holling type II function response.
The final load that establishes in the recipient host (Stage 5) depends on the quality of the received particle, the host immune response (which may depend on its genotype), and the presence of other pathogens (co-infection). These stages can differ between each pathogen, complicating their simplification into standard models. Figure 1 shows these transmission stages in the spillover of the bacterium bovine Tuberculosis (bTB) from the reservoir host (African buffalo, Syncerus caffer ), to a novel host (elephant, Loxodonta africana) (Miller et al., 2021). This example emphasises the complex dynamics underlying transmission.
β, which has been described in numerous formulations (see Table 1). The transmission term can also be described where β is the transmission rate which determines the spread of infection from infected (I) to susceptible (S) hosts.

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There are several equally valid formulations of F , but we will focus on the one presented in Equation (2) as it is  Here we decompose commonly used transmission functions, F (S, I), of compartmental SIR models (Table 1). We 100 begin with the different relationships between susceptible and infected individuals for homogeneously and heteroge-101 neously mixing populations (not including explicit contact networks) (Table 1) and finish with explaining how the 102 contact rate (κ) determines the density or frequency-dependent nature of transmission, together with the probability of successful transmission (c). We provide a schematic illustrating our framework in Figure 2, starting with standard 104 homogeneity (left column) and finishing with some examples of heterogeneity (right column). In the model of System 1, the criterion for a disease outbreak is proportional to the transmission rate. Namely, the 108 conditions under which a disease can spread is determined by the basic reproductive rate, R 0 , which describes the 109 number of secondary infections arising from a single infected host in a fully susceptible host population. This quantity 110 is also known as the intrinsic lifetime reproductive success or fitness of the pathogen. It tells us the pathogen's lifetime 111 reproductive success early on in an infection, as the quantity growth rate tells us the growth rate in the absence of 112 competition in a model of logistic growth. As shown in Box 3, in an SIR model with density-dependent transmission 113 R 0 can be simply defined as: Emergence of disease, or pathogen persistence in a community, will occur when R 0 > 1. If R 0 < 1, the pathogen will 116 be lost from the host community, whereas an R 0 equal to 1 represents an endemic disease. In addition to determining 117 disease emergence, Equation (3) describes the intrinsic rate of increase of the pathogen. then introduce a novel general contact rate function κ(N ) from which we then derive a range of transmission rates 125 commonly used in the literature, see Table 1 for a summary.

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In a homogeneous population, the rate at which new infections occur is given by: see Appendix for a formal derivation. Using this linear relationship between κ and c, we can then determine the R 0 136 of a disease as: where τ is the duration of the infection. This formulation can be used to approximate R 0 from empirical data  Table 1: The factor ξ is a constant (contacts per unit time), which determines the metric of transmission that will be held 148 constant across population of different size. ω portrays the biotic or abiotic asymptotic effects of the transmission.

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If one wants to create heterogeneity in classes, F (N ) (and therefore also κ(N )) can be extended to be F (S, I, R) Substituting this into Equation (2), we return the familiar FD transmission function (Row 1 of Table 1): Here, infections rise with the probability I N , the density of infected hosts. Note that ξ, and hence κ is O(N ).  Figure 4A.  Table 1). This I. This dynamic, here referred to as Carrying Capacity Dependent transmission, is shown in Figure 4D.
This brings us to the second and third equations in Table 1. Note that here β D is an order of magnitude higher 192 than β F . It is of importance to notice that here, β has to be of O( 1 N ) to satisfy the F (S, I) to be of O(N ). This 193 is represented in the units for β D (see Table 1). Additionally, β A includes area in its units. This can be useful parameter q (0 < q < 1) and setting ω = 0 in Equation (8) such that: where the dimensions for ξ are individuals (−q) and A was set to 1. Combining Equation (14) with Equation (2), 220 brings us: When q = 0, N drops out and contact rate is again independent of N , thus q = 0 is FD, and q = 1 is DD. We can 228 interpret q as the relative importance of a single added host within a population to the average contact rate. Now, the , therefore neither fully DD nor FD, which is represented in the units of β q of ind −q t −1 (See 230   Table 1). Thus, it that at values of 0< q <0.5, β q has FD units (t −1 ), and at values 0.5< q <1, DD units (ind −1 t −1 ).

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Likewise, ξ changes in units due to this switch from FD to DD with increasing q. This formulation was presented transmission is FD, so shifting from DD to FD with increasing N . The transmission function is then: Note, here β is identical of the case of frequency dependence, β F . We can thus define X = N * (with 1 < X < ∞), 253 as the half saturation constant in Michealis-Menten kinetics, which is the critical level after which contacts start to 254 saturate. The lower X, the higher the affinity of individual contacts and the quicker saturation of contacts is reached.

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The F (S, I) is again of O(N ). Figure 4E shows the asymptotic transmission dynamics. The assumption in SIR models of spatial homogeneous populations is obviously an oversimplification of real pop-   Figure 5B. can average out, providing an opportunity to simplify otherwise complex models.

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Box 3: Derivation of the pathogen's net reproductive success, R 0 To calculate Eqn (3), we want to know when the pathogen can grow in the population, using Eqns 1: Using the the example of density dependent transmission, of Eqn (13), we get: Then, dividing by I and rearranging we get: So, when S ∼ N , the disease will spread when β Γ is greater than unity, which is the definition of R 0 :  The dynamics of multi-host diseases can vary greatly with the species composition of the host community. One 362 important relationship in wildlife disease dynamics is that between disease prevalence and host community diversity.

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The disease-diversity relationship is a well-known and highly-disputed concept in disease ecology.

Host community effects on pathogen prevalence
The dilution effect suggests higher diversity of hosts reduces the probability of a pathogen infecting a new host, 366 either directly, by reducing encounters, and therefore transmission, between hosts or indirectly, by changing total 367 host abundance (Keesing et al., 2006(Keesing et al., , 2010 (Table 3). A modelled example of encounter reduction (also known as 368 frequency-dependent dilution) is provided in FD systems by Rudolf and Antonovics (2005), here a host is rescued from 369 pathogen mediated-extinction (apparent mutualism) by a second host that is infected by the same disease. In this  Table 3). For example, higher amphibian diversity is thought to have increased Chytrid disease (caused by 384 Batrachochytrium dendrobatidis) prevalence in some species of frogs, as highly competent (amplifying) hosts are more 385 abundant in species-rich habitats (Ostfeld and Keesing, 2012). The amplification effect may also arise in vectored 386 diseases (often modelled as FD due to density-independent contacts) if increased diversity offers more competent 387 host species so that this multi-host pathogen can more easily persist in the system (Ostfeld and Keesing, 2000). In  The direct effect of diversity in DD transmission will always lead to disease amplification, assuming host populations 393 do not compete and the addition of a host is additive to the community. Increasing the number of hosts simply 394 increases encounter rates between S and I individuals, and will therefore always amplify the disease, regardless of the 395 competence of the hosts. Theoretically, assuming strict DD transmission, dilution through direct effects is impossible.

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In contrast, the direct effect of diversity in FD transmission, assuming a constant number of contacts per time unit 397 and varying competence, are mixed. The addition of a less competent host will always have a diluting effect, whereas 398 the addition of a highly competent host may have an amplifying effect. In Table 3 we summarise how transmission 399 type can cause dilution or amplification. Predicting the indirect effects of diversity is more challenging. Changes in 400 prevalence will reflect changes in the relative abundance of hosts and their relative disease competence, and thus how

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In all cases, it is assumed that the pathogen is a generalist and that hosts differ in competence, except for direct 407 amplification in DD systems, which assumes contacts are additive and there is no limit to contacts per time unit.

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However, these latter assumptions may be unrealistic for many empirical systems (Ostfeld et al., 2008) and, as we 409 discuss above, host contacts likely saturate over higher density Antonovics (2017). Thus, at a certain density, dis-410 ease dynamics may switch from DD to FD dynamics, and so we might predict dilution effects to be generally more 411 common at higher densities.  In this matrix, transmission can be defined as β i,j , where i is the receiving host and j the donating host (j infects 457 i). This matrix can be used to calculate the community R 0 , which determines the overall disease prevalence in the systems, and the contributions of each individual host (Dobson and Foufopoulos, 2001). One way to estimate the interspecific transmission rate is to take the average of the intraspecific transmission rates (Dobson, 2004): where α i,j is a scaling parameter to account for differences in transmission potential between species i and j, and 461 determines the magnitude of diluting and amplifying behaviour of each species in the system (Dobson, 2004). A 462 similar approach can also be applied to contact rates, κ, as defined in β = κ · c, if one wants to define c separately 463 for individual host species, rather than the community average (Anguelov et al., 2014). However, there can be 464 asymmetrical transmission between hosts, for example Blancou and Aubert (1997) suggest that for a fox with rabies 465 to infect another species, such as a dog or cat, requires a million times more virus particles than would be necessary 466 to infect another fox (Ostfeld et al., 2008). Such asymmetries may be overlooked when using the average of the 467 intraspecific transmission rates. Estimating the probability of successful transmission, c: Even when the contact structure of the population is known, for example through biological monitoring, it is still challenging to define the infectivity of a contact -the probability of successful transmission after contact, c (Craft, 2015; White et al., 2017). One approach used in wildlife disease dynamics is to quantify the Secondary Attack Rate (SAR), which is the ratio of the number of exposed hosts that developed the disease to the number of exposed hosts that did not (Childs et al., 2007). To accurately calculate the SAR, a clear distinction must be made between primary and secondary cases. For sexually transmitted diseases, which follow FD dynamics, the probability of becoming infected after contact can be calculated using the binomial distribution (Childs et al., 2007). The maximum likelihood of this probability following a single contact is identical to the SAR. While estimating c remains a challenge, compounded by the fact that most disease models summarize the process of transmission into a single parameter, it is important to capture accurately. These methods are described in further detail in the supplementary material.