Models of Echinococcus multilocularis

The approaches used in the modelling of E. multilocularis depend to some degree on whether the intention is to capture the sylvatic cycle which exists between foxes and small mammals, or the semi-domestic cycle between domestic dogs and small mammals (no models to date have attempted to capture both simultaneously, which is an issue addressed in chapter 7 of the current thesis).

One of the first models of the E. multilocularis sylvatic cycle attempted to combine the dynamics of infection in both foxes and voles with transmission between these species in one formulation (Roberts and Aubert, 1995). This model is based upon the following structure:

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Where λ_𝑓 is the infectious contact rate for foxes, λ_𝑣 is the infectious contact rate for voles, 𝜏𝑓 is the prepatent period (time to worm maturity) in foxes (=1⁄_{𝑟𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑤𝑜𝑟𝑚𝑠}), 𝜏_𝑣 is the prepatent period (time to cyst maturity) in voles (=1⁄_{𝑟𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑐𝑦𝑠𝑡𝑠}), 𝜂 is the duration of egg production (=1⁄_{𝑙𝑖𝑓𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑎𝑛𝑐𝑦 𝑜𝑓 𝑎𝑑𝑢𝑙𝑡 𝑤𝑜𝑟𝑚}), 𝛿𝑣 is the mortality rate of voles, 𝛿𝑓 is the mortality rate of foxes, and 𝑁_𝑓 is the density of foxes.

More recent models have also attempted to incorporate the spatial and/or temporal factors which can affect the distribution and location of intermediate and definitive hosts. Temporally-explicit models have accounted for the variability in parasite and host survival at different times of the year (Ishikawa et al., 2003; Ishikawa, 2006; Nishina and Ishikawa, 2008), and spatially-explicit models have accounted for either the locations of different hosts types in relation to each other (Hansen et al., 2003, 2004), or habitat suitability from an ecological perspective (Milner-Gulland et al., 2004). One challenge faced when constructing a spatial model is deciding on which spatial scale the model should be developed: intermediate hosts often have a smaller home range than definitive hosts, and so some models have been developed based on the definitive host range. However, working at the level of the range of a single definitive host will result in the modelling of few (possibly only one) definitive hosts. As variation in susceptibility between hosts is suspected to be an important factor affecting the distribution of parasites within hosts, ideally a reasonable number of

1/η Prepatent foxes Infectious foxes Susceptible foxes

Prepatent voles Susceptible voles Infectious voles λf λ_v δf δf δf δv δv δv Fox density (Nf ) 1/τf 1/τ v

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hosts should be modelled (Morgan et al., 2004). However, increasing the scale of the model will also increase the computational load. A major challenge for spatially explicit modelling of E. multilocularis therefore lies in finding a suitable balance between these two issues.

The most commonly adopted approach to the modelling of the semi-domestic cycle of E. multilocularis is to treat it in a similar way to E. granulosus, as described above. Although no models to date have attempted to model transmission dynamics over time, the force of domestic dog infection with E. multilocularis in Tibetan and Kyrgyz communities has been estimated in the same way as that described for E. granulosus above (Budke et al., 2005b; Ziadinov et al., 2008).

1.6.5.1 Compartmental models

The models developed by Roberts and Ishikawa are forms of compartmental model, which rather than specifically modelling the numbers of worms, classify fox infection into compartments (uninfected, prepatent infection, infectious for the Roberts model (Roberts and Aubert, 1995); uninfected, prepatent infection, peak egg production, and declining egg production for the Ishikawa model (Ishikawa et al., 2003); non-infectious and infectious for the Vervaeke model (Vervaeke et al., 2006)). This approach is similar to the simple ‘SEIS’ (susceptible, exposed, infectious, recovered) compartmental model used for microparasites such as bacteria and viruses. In the case of the Ishikawa model, both juvenile foxes and adult foxes are also modelled, with different mortality rates. Similarly, voles were classified as belonging to one of two or three compartments: uninfected, infected but not yet infectious (left out in the final form of the Vervaeke model), and infectious (Roberts and Aubert, 1995; Vervaeke et al., 2006), with five different age classes (0-1 months; 1-2 months; 2-3 months; 3-4 months; > 4 months) considered in the Ishikawa model. The Vervaeke model included a compartment for abundance of eggs in the environment (Vervaeke et al., 2006), and the density of foxes and voles were varied according to season in the Ishikawa model (Ishikawa et al., 2003). More recently, stochastic models of transmission have been developed in order to account for individual variation between foxes (Nishina and

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Ishikawa, 2008). A compartmental model classifying foxes as either infected or uninfected was recently incorporated with an economic model for the evaluation of fox anthelmintic dosing schemes (Kato et al., 2010).

1.6.5.2 Measuring the force of infection

A recent model used a Bayesian framework to identify the force of vulpine infection with E. multilocularis from prevalence data. Conceptually, this was an extension of the force of infection models described earlier (Budke et al., 2005b; Ziadinov et al., 2008), but estimated transmission parameters (probability of immunity upon exposure, number of insults per unit time, rate of immunity loss, parasite mortality rate) directly from the application of the system of differential equations to age-stratified prevalence data, rather than using the algebraic solutions of the differential equations. Using this technique, a number of different models were compared (with different immunity and force of infection structures). This model suggested seasonal and geographical differences in the force of infection, with periodic increases in the force of infection during the winter months and higher forces of infection for foxes in non-urban areas (Lewis et al., 2014).

1.6.5.3 Mean worm burden models

Although these can be considered a form of compartmental model, they will be described separately here as the formulation is quite different. A common approach to modelling transmission of other helminths is based upon the mean worm burden (MWB), as first described by Macdonald (1965) (despite a number of challenges incorporating overdispersion into this framework (Gurarie et al., 2010)). However, this approach has not been commonly adopted in the modelling of echinococcosis. The only examples of models for echinococcosis based upon the MWB at the population level are the models of Takumi and others (Takumi and van der Giessen, 2005; Takumi et al., 2008). These models describe the dynamics of E. multilocularis infection by modelling the parasite biomass within a 1km2 area, divided according to the stage of the parasite: total eggs, total protoscolices, and total adult worms. The latter two

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estimates are adjusted according to the number of available hosts in order to estimate the “mean worm” or “mean protoscolex” burden within these hosts. As this model therefore functions at the level of the parasite rather than the host, it is able to directly incorporate a lag period prior to patency for both intermediate and definitive host infection as well as the persistence of eggs in the environment – making it useful for evaluation of potential control schemes (Takumi and van der Giessen, 2005). However, it does not account for seasonality in infection, the age structure of the population, or density dependence in transmission. This model was developed further in chapter 7 of the current thesis in order to incorporate the E. granulosus transmission cycle, along with some seasonal effects. Another modified version of the model included a spatial spreading component of the model, allowing the spread of infection from an initial focus to be modelled (Takumi et al., 2008).

1.6.5.4 Simulation models

The ‘Echi’ model developed by Hansen is a spatially explicit simulation model combining individual- and grid- based modelling approaches, allowing it to incorporate both spatial factors and individual fox movements (Hansen et al., 2003, 2004). Despite incorporating a lot of information, the general rules governing the model remain relatively simple. This model has been useful in both the evaluation of different control measures (Hansen et al., 2003) and in the investigation of various characteristics relevant to the transmission of E. multilocularis (Hansen et al., 2004). The importance of incorporating spatial factors in modelling of the sylvatic (fox-based) cycle E. multilocularis is well-recognised (Milner-Gulland et al., 2004; Morgan et al., 2004; Pleydell et al., 2004), but the importance of this in the case of the semi-domestic cycle is less clear.

1.6.5.5 Metapopulation models

Another spatially-explicit model framework, developed by Milner-Gulland et al, used an ecological approach derived from the concept of metapopulation dynamics in order to investigate the transmission of E. multilocularis (Milner-Gulland et al., 2004).

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Metapopulations are ‘populations of populations’, which although spatially separated by unsuitable habitat types, interact on some level (Hanski, 1991, 1998). Although this concept was originally introduced from an ecological perspective, it can also be applied to infectious pathogens and parasites (Hess, 1996; Grenfell and Harwood, 1997). The Milner-Gulland model adapted this approach to E. multilocularis by modelling populations of parasites (whether in the adult, metacestode or egg form) with hosts (and the environment, in the case of eggs) treated as habitats. Therefore, like the Takumi models (Takumi and van der Giessen, 2005; Takumi et al., 2008), this model operates at the parasite level rather than the host level.