Models of Echinococcus granulosus

1.6.4.1 Modelling parasite burden

The first model of definitive host infection with Echinococcus focussed on modelling the numbers of worms in individuals (Roberts et al., 1986), in a similar fashion to the first mathematical model of helminth infection (described in “Symbiose, Parasitisme et Évolution” by Kostitzin (1934) (Anderson and May, 1985)). Details of this original ‘Roberts, Lawson and Gemmell’ model will be given here (Roberts et al., 1986), as this model (or slight variations of it) is still commonly used. This model considered four general host statuses for definitive hosts: infected, noninfected, immune (𝑦) and nonimmune (𝑥). Within infected individuals, the number of worms (𝑛) was explicitly modelled:

𝜕𝑥_𝑛

𝜕𝑡 = −(𝛽 + 𝜇)𝑥𝑛+ 𝛽(1 − 𝛼𝑛−1)𝑥𝑛−1+ 𝛾𝑦𝑛 𝜕𝑦_𝑛

𝜕𝑡 = −(𝛾 + 𝜇)𝑦𝑛+ 𝛽𝛼𝑛−1𝑥𝑛−1

The differential equations for the numbers of uninfected dogs over time are as follows: 𝜕𝑥₀ 𝜕𝑡 = −𝛽𝑥0+ 𝛾𝑦0 + 𝜇 ∑ 𝑥𝑛+ 𝜇𝛿𝑛∑ 𝑦𝑛 ∞ 𝑛=1 ∞ 𝑛=1 𝜕𝑦0 𝜕𝑡 = 𝛾𝑦0+ 𝜇(1 − 𝛿) ∑ 𝑦𝑛 ∞ 𝑛=1

Where 𝑥 denotes susceptible dogs, and 𝑦 denotes immune dogs. 𝑛 is the worm burden; 𝛽 is the infection pressure; 𝜇 is the rate of complete parasite loss (including host death); 𝛼 is the probability of development of immunity upon exposure; and 𝛿 is the probability of loss of immunity upon loss of parasites (or replacement of dead

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hosts with naive individuals). 𝛾 denotes the rate of loss of immunity amongst immune animals, and is related to 𝛽 (since reinfection is likely to boost immunity). 𝑡 may either represent time (if creating an overall model of transmission) or age of dog (in a closed population, where the relationship between age and level of infection is being investigated).

In the absence of immunity, the following equations can be used: 𝜕𝑥𝑛 𝜕𝑡 = −(𝛽 + 𝜇)𝑥𝑛+ 𝛽𝑥𝑛−1 𝜕𝑥0 𝜕𝑡 = −𝛽𝑥0 + 𝜇 ∑ 𝑥𝑛 ∞ 𝑛=1

For the investigation of infection in intermediate hosts, it can be assumed that infections are permanent (i.e. 𝜇=0), meaning that the rate of change in the mean number of cysts (𝑚) over time (𝑡) can be modelled as the product of the infection pressure (in terms of the rate of acquisition of parasites in the absence of density- dependent constraints) and the proportion of susceptible individuals. If it is assumed that the infection pressure is constant, the differential equation can be formulated as:

𝜕𝑚 𝜕𝑡 = ( 𝛾ℎ 𝛾 + 𝑎ℎ) + 𝑎ℎ2 𝛾 + 𝑎ℎ(𝑒𝑥𝑝−(𝛾+𝑎ℎ)𝑡)

This model differentiates between exposures and infections, due to the clustered nature of infection. 𝑁 is the number of parasites which become established per exposure; 𝛽 is the infection pressure in terms of rate of exposure (which varies with time); ℎ is the infection pressure in terms of rate of acquisition of parasites (ℎ = 𝑁𝛽); and 𝑎 is the rate of development of immunity per parasite (_{𝑎 = 𝛼 𝑁}⁄ ).

1.6.4.2 Estimating model parameters from field data

Estimation of model parameters can be achieved by fitting age-stratified field data collected at an endemic ‘steady state’ to a model using maximum likelihood or

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Bayesian techniques. This approach is known as ‘catalytic modelling’, as movement between groups (such as the ‘uninfected’ group to the ‘infected’ group) takes at a particular rate (in this example, the ‘force of infection’), which is not dependent upon the numbers of individuals in the groups themselves (Muench, 1959). This will therefore differ from the ‘force of infection’ used in transmission models, which will depend upon the numbers of infectious and susceptible individuals in the population. For both intermediate and definitive hosts, the instantaneous rate of change in the proportion of susceptible animals over time is estimated as the balance of the rate of loss of immunity amongst immune individuals (1 − 𝑆) and that of acquisition of immunity amongst susceptible individuals (𝑆) (this equation can also be obtained by summing the equations for 𝑥𝑛 and 𝑥0 above):

𝜕𝑆

𝜕𝑡 = 𝛾(1 − 𝑆) − 𝑎ℎ𝑆

This differential equation can be solved to obtain the following formula for estimating the proportion of susceptible animals by age 𝑡 (Torgerson et al., 2003c):

𝑆(𝑡) = 1

𝛾 + 𝑎ℎ(𝛾 + 𝑒𝑥𝑝−(𝛾+𝑎ℎ)𝑡)

As the rate of change of parasite abundance (𝑀) over time will be related to the balance of the basic infection pressure (ℎ) and the parasite death rate (𝜇), this can be represented by the following differential equation:

𝜕𝑀

𝜕𝑡 = ℎ𝑆 − 𝜇𝑀

These two equations can be combined and solved to give an equation for the expected variation in the number of parasites over time (Torgerson et al., 2003c):

𝑀(𝑡) = 𝑎ℎ

2

(𝛾 + 𝑎ℎ)(𝜇 − 𝛾 − 𝑎ℎ)(𝑒𝑥𝑝−(𝛾+𝑎ℎ)𝑡− 𝑒𝑥𝑝−𝜇𝑡) +

𝛾ℎ

𝜇(𝛾 + 𝑎ℎ)(1 − 𝑒𝑥𝑝−𝜇𝑡) In the absence of immunity, 𝑎=0 and the equation becomes:

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𝑀(𝑡) = ℎ

𝜇(1 − 𝑒𝑥𝑝−𝜇𝑡)

This approach was used in a Bayesian framework to parameterise a transmission model of E. granulosus in Kazakhstan, which suggested that farm dogs were developing immunity, whereas village dogs were not (Torgerson et al., 2003c). A similar strategy was used in Kyrgyzstan, incorporating the results of a number of imperfect tests in order to evaluate test characteristics and estimate the force of infection for E. granulosus (and E. multilocularis) in dogs (Ziadinov et al., 2008).

1.6.4.3 Modelling prevalence of infection

Due to the highly overdispersed nature of infection in both intermediate and definitive hosts, attempting to model the prevalence of infection rather than the parasite burden is not ideal. However, in some cases it is not possible to obtain suitable estimates of worm burden and so only prevalence data may be available. A model based upon the burden model described above has been developed, and has been applied to field data in China and Kyrgyzstan (Budke et al., 2005b; Ziadinov et al., 2008). Infected dogs are classified in one of two groups:

𝜕𝑌

𝜕𝑡 = −(𝛾 + 𝜇)𝑌 + 𝛼𝛽𝑆 𝜕𝑋

𝜕𝑡 = −(𝛽 + 𝜇)𝑋 + 𝛽(1 − 𝛼)𝑆 + 𝛾𝑌

Where 𝑌 represents the proportion of dogs which are infected but immune, and 𝑋 represents those which are infected but susceptible to further infection. The total proportion of dogs which are susceptible to further infection (whether already infected or not) is represented as 𝑆 and is modelled in the same fashion as described earlier:

𝜕𝑆

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The prevalence of infection, 𝑃(𝑡), can be either estimated as 𝑋(𝑡) + 𝑌(𝑡), or estimated directly in the absence of immunity from the following equation:

𝑃(𝑡) = 𝛽

𝛽 + 𝜇(1 − 𝑒𝑥𝑝−(𝜇+𝛽)𝑡)

The equation used to estimate susceptibility above can also be adjusted in order to be used for prevalence data as follows:

𝑆(𝑡) = 1

𝛾 + 𝛼𝛽(𝛾 + 𝛼𝛽𝑒𝑥𝑝−(𝛾+𝛼𝛽)𝑡)

1.6.4.4 Simulation modelling

Individual-based simulation models which operate at the level of the individual animals involved in the transmission cycle have also been developed in an attempt to explicitly account for stochasticity in individual infection with E. granulosus (Heinzmann et al., 2011b; Huang et al., 2011). These model individual animals as autonomous units and aim to reproduce the complex behaviour of these units using simple rules, and so differ from the population-based differential equation methods described above. The first of these models is based on the compound process models described earlier (Heinzmann et al., 2009, 2011a; b), which were fitted to data from Kazakhstan (Torgerson et al., 2003b; c). Independent models of the infection of sheep by eggs and of the infection of dogs by protoscolices were linked by simulating a contact pattern between the two hosts in order to create a complete simulation model. Dogs become exposed through ingesting sheep offal, with the infection risk (and therefore the proportion of dogs becoming infected/reinfected in each time step) being related to the number of fertile cysts, modelled as one minus the probability that none of the cysts are fertile. Sheep become exposed through contact with dog faeces, with the probability of contact with faeces selected for each individual sheep at birth/start of simulation from a gamma distribution. Infectivity of canine faeces is not assumed to be related to the worm burden. Deaths of sheep and dogs result in removal from the population and replacement with a new uninfected animal of age zero.

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Model fit and parameter estimation was based on the component models, and the model was used to evaluate the possible effects of praziquantel dosing of dogs and of seasonality on transmission of infection (Heinzmann et al., 2011b).

The model by Huang (Huang et al., 2011) was a largely theoretical model using parameters (often point estimates) from other studies and surveys. Dogs, intermediate hosts, parasites and egg contaminations were specified as the agents of interest, each of which functions according to defined characteristics (including survival, ageing and acquisition of infection). The model also includes ‘objects’ (in this case, the number of egg contaminations and the number of deaths of infectious intermediate hosts), which are not autonomous but are dependent upon the status of the agents, and the environment (a community in western Sichuan province). The model was used to investigate the possible effects of a wide range of control strategies.