Introduction
Pakistan is mainly an agricultural country, having rich natural resources, appropriate climatic conditions, favourable lands and water resources, thus the country has enormous potential for livestock production. Agriculture is playing an important role in the economy of Pakistan by contributing about 21% to gross domestic product, employs 45% of the total work force and 60% of its rural population depends upon this sector for its livelihood (Rehman et al., Reference Rehman, Muhammad, Ashraf, Mahmood, Ruby and Bibi2013).
Echinococcus multilocularis is one of the zoonotic tapeworms among the different species of Taeniidae family. It is mostly located in northern hemisphere especially in Asia and Europe. This parasite completes its life cycle in two types of host namely definitive hosts (fox) and intermediate hosts (rodents) while humans are the accidental hosts. It is responsible for causing disease known as alveolar echinococcosis (AE) in a metacestode stage in which small cyst is formed known as locules which spread inside the body through blood and lymph and reach to the internal organs i.e. liver, lungs, kidney and brain (Eckert and Deplazes, Reference Eckert and Deplazes2004).
AE is one of the chronic life-threatening infections, which could have endemic resource-poor settings and high economic impact. The geographic distribution is expanding and this pathogen is considering emerging and re-emerging entity in many countries. The risk of AE increases due to the presence of infective eggs present in food, water, environment and their accessibility to humans (Eckert et al., Reference Eckert, Deplazes, Kern, Palmer, Soulsby, Torgerson and Brown2011).
The high prevalence (23.9–57.3%) of E. multilocularis among red foxes is reported from Europe. In Europe, Vulpes vulpes (red fox) is responsible for AE and harbouring heavily infected animals and this parasite burden is being responsible for the parasitic environmental egg contamination. The parasite distribution is expanded due to the control of successful rabies campaigns. However, it is not exploring until now whether parasite remains undetected or range of E. multilocularis has recently extended. Biological behaviour of larval E. multilocularis in human is similar to a malignant tumour that is determined by growth of damaging tissues and metastasis to distant organs. The disease has a high mortality rate (more than 90% within 10 years and virtually 100% within 15 years of the onset of symptoms) in untreated cases (Reuter et al., Reference Reuter, Nussle and Kolokythas2001).
The exploratory study of the data showed that foxes up to 3 years' age, represented 86% of total samples, responsible for about 88% of all the infected animals and harboured up to 94% of the total parasite biomass (Fischer et al., Reference Fischer, Reperant, Weber, Hegglin and Deplazes2005). Comparing the transmission models possible changes in infection pressure or acquired immunity were compared to analyse the hypothesis that parasite induced immunity, spatial differences, host age and seasonality may be the contributing factor of the abundance parasite in the foxes. Most of the study reported showed a gradual decrease in the prevalence of parasites from rural areas and the periphery of the different cities towards the urbanized zones (Reperant et al., Reference Reperant, Hegglin, Fischer, Kohler, Weber and Deplazes2007).
Foxes in transition areas of the city and outside areas are more exposed rodents and prey are likely more exposed to parasite infection. The high density of intermediate hosts bearing a high number of parasites and are responsible for prevalence in outskirt of cities. Control of the E. multilocularis can be possible through the judicious use of praziquantel baits distributed to foxes (Heglin and Deplazes, Reference Heglin and Deplazes2008) and also represent the E. multilocularis abundance in the animal host of different age groups.
However, whatever the intervention strategy, the economic efficiency of control will depend upon the societal burden of disease. In the recent literature, networks, specifically the discrete networks have attracted the attention of researchers enormously due to their association with the biological insight and diversity, such as studies reported by many more (Heiner et al., Reference Heiner, Gilbert and Donaldson2008; Sohail, Reference Sohail2019; Wootton et al., Reference Wootton, Andrews, Lloyd, Smith, Arul, Vinod, Prasad and Garg2019). However, for epidemics and specifically a study focusing on the drug administration through Petri net (PN) modelling approach has not been reported in the literature. We, in this study present for the first time, application of a novel strategy to explore the epidemiology of fox and vole and the E. multilocularis, where the control measure is discussed with the aid of Hill function formalism and with the aid of PN modelling (Fig. 1).
Fig. 1. PN modelling fox & vole interactions.
Materials and methods
The mathematical model, highlighting the impact of intra and inter fox & vole interactions, initially provided by Roberts & Aubert (Brochier et al., Reference Brochier, Kieny, Costy, Coppens, Bauduin, Lecocq, Languet, Chappuis, Desmettre and Aademanyo1991) is utilized as a benchmark in this research. We have used the Hill function formulation, which has been used a successful tool in the field of computational biology to describe the dynamic solver a period, in a more realistic manner (Reperant et al., Reference Reperant, Hegglin, Fischer, Kohler, Weber and Deplazes2007).
Control measures via mathematical modelling of fox & vole interactions
We have considered the following mathematical framework to elaborate the interactions and the control measures:
$$\displaystyle{{dF_{\rm I}} \over {dt}} = H_1\lpar {F_{\rm S}V_{\rm J}} \rpar -mF_{\rm I}-k_{\rm F}F_{\rm I}-cF_{\rm I}$$
$$\displaystyle{{dF_{\rm J}} \over {dt}} = mF_{\rm I}-aF_{\rm J}-k_{\rm F}F_{\rm J}-cF_{\rm J}$$
$$\displaystyle{{dV_{\rm I}} \over {dt}} = H_2\lpar {F_{\rm J}V_{\rm S}} \rpar -pV_{\rm I}-k_{\rm V}V_{\rm I}$$
$$\displaystyle{{dV_{\rm J}} \over {dt}} = pV_{\rm I}-k_{\rm V}V_{\rm J}$$In Table 1, H 1(F sV J) represents the Hill function which is used in this study to forecast the interaction of infected vole with the susceptible fox and H 2(F sV J) represents the Hill function which is used in this study to forecast the interaction of susceptible vole with the infected fox. M presents the inverse of the average time to maturity of the worm in the fox and a is the inverse of adult worm life expectancy in fox and p is the inverse of the average maturity of cyst in vole (Table 1). In equation (1)H 1(F sV J) represents the Hill function which is used in this study to forecast the interaction of infected vole with the susceptible fox F s. It becomes infected at the rate of m that actually represents the inverse of the average time to maturity of the worm in the foxes. Next, k FF I is the death rate of infected foxes. c is used to present the effect of the control measure against the infection in the foxes. In equation (2)m is the maturity of the worm in the infected fox i.e. F I. It became infectious at the rate of a, k FF J is the death rate of infectious foxes. In equation (3)H 2(F sV J) represents the Hill function which is used in this study to forecast the interaction of susceptible vole with the infected fox. And a is the inverse of adult worm life expectancy in fox. Infected vole becomes infectious at the rate of P, and k V is the death rate of infected vole V I. In equation (4) infected fox become infectious at the rate of P. k VV J is the death rate of the infectious vole. Tables 1 and 2 provide the model parametric values and their respective definitions.
Table 1. Description of each variable along with transition rates
Table 2. Description of each variable along with transition rates
Equation (5) represents the number of susceptible parasites, and is used to represent the rate of loss of immunity in susceptible parasite. a is the rate of acquisition of immunity and h is the infection pressure in parasite. In equation (6)M is the parasite abundance and ah use to represent the infection pressure in susceptible parasites. γ is the death rate of parasite (Table 2).
$$\displaystyle{{dS} \over {dT}} = \gamma -\lpar {\;\gamma + ah} \rpar S\;$$
$$\displaystyle{{dM} \over {dt}} = hS-\mu M$$PN model to depict the transmission model for E. multilocularis in an animal host (fox) is shown in Fig. 2.
Fig. 2. PN model represents the transmission model for E. multilocularis in an animal host.
Transition invariants
The most important property of PN in biological modelling is the transition invariants (T-invariants) (Tables 3 and 4). Using PN models, our qualitative analysis was focused on identifying two properties (place invariants and transition invariants). Place invariants are for characterizing relationships among variables, while transition invariants are for identifying a set of sub-networks in the overall network. In this paper we are using only transition invariant in the quantitative analysis, we obtained the PNs and compared them with the results obtained from Ordinary Differential Equations (ODE's). Transition invariants are a set of transitions where their sequences of firings can be reproduced in the specific states.
Table 3. Description of each variable along with transition invariants PN fox & vole interactions
Table 4. Description of each variable along with the transition invariants PN model of E. multilocularis abundance in foxes
During this research, we have derived the transition invariants for both models using technical programming language. We have presented the details of the transitions, which can help in analysing the real transition from one compartment to other in both models. The purpose of this study is to provide a network analysis that can help to forecast such thresholds. In this paper we presented the transition invariants via a quantitative approach of PNs. This approach is recommended for future control measurements.
Results and discussion
During this research, we have documented the interplay between the terms involved in the system of differential equations and the networks associated with them. The results obtained after sketching the networks and the corresponding transition invariants are listed below step by step.
Firstly, the left panel of Fig. 3 was discussed. Here, the dynamics associated with mF I, i.e. the density of infected fox, with the Hill function of infectious fox and susceptible vole was explained. This image presents the major transition invariant of the PN, which were not that clear from equations (1) and (2). Similarly, Figs 2–4 depict the interesting features of the mathematical model in a novel way. These transitions are explained in detail in Table 3. Figures 4–6 presents the dynamics of infected fox & vole and infectious fox &vole with the passage of time, over a period of 50 weeks (nearly a year). The parametric values were selected from Eckert et al. (Reference Eckert, Deplazes, Kern, Palmer, Soulsby, Torgerson and Brown2011) and Fischer et al. (Reference Fischer, Reperant, Weber, Hegglin and Deplazes2005).
When no control measures were applied, the frequency of cases reported for the infected fox, accumulated after a period of 10 weeks at a higher rate. After introducing fewer control measures (Lucius and Bilger, Reference Lucius and Bilger1995), this frequency was controlled, and it remained almost equivalent to the number of the infected fox cases reported initially. However, for the higher control measures, the response was quite more than normal.
There is only a published report on E. multilocularis from Pakistan. It was reported in human and cattle from KPK province of Pakistan by using Polymerase chain reaction-restriction fragment length polymorphism (PCR-RFLP). Out of 30 cattle samples, 13 (43.3%) were found to be positive for E. multilocularis. On the other hand, among 10 human samples, 3 (30%) were found positive for E. multilocularis (Ali et al., Reference Ali, Panni, Iqbal, Munir, Ahmad and Ali2015). However, without DNA sequence analysis the results have some doubts.
We can see from the results that the number of infected fox cases decreased at a rate of three-fold to the initial cases reported. Thus, the control measures, when applied in a strategic manner, can help to in fact eradicate the disease spread. Based on the graphical interpretation, we can demonstrate the dynamics of the other three variables (infected vole and infectious fox & vole), in a similar manner. Two transition invariants (in Fig. 5 left panel) represent PN modelling of the fox and parasite interaction. This is mathematically depicted with the aid of equation (2), on the other hand, PN modelling of the fox and parasite interaction is presented in Fig. 6 right panel and is mathematically depicted by the last two terms of equation. (1). In a similar fashion, one can depict the correspondence between the dynamics interpreted in Fig. 7 and equations (1) and (2). Figure 8 and the first two terms of equation (1) are in fact linked and actually provide the network for the fox and parasite interactions.
Fig. 8. PN modelling of the fox and parasite interaction. Presented the first two terms of equation (1).
We emphasize that the disease modelling via PNs makes it easier to understand the interactions. Although in the recent literature, evidence is available that such techniques are used at cellular and molecular scales (Liu et al., Reference Liu, Heiner and Gilbert2017; Wootton et al., Reference Wootton, Andrews, Lloyd, Smith, Arul, Vinod, Prasad and Garg2019), but no attempts have been made for the parasitology research. We therefore present here a novel approach. The system of differential equations and the PNs, together, work as a useful tool to explore the dynamical analysis in a more critical manner. From this study it is very obvious that the more variables are involved in computational framework (equations (1)–(4)) the better the results are in terms of forecasting, whereas, when fewer variables are involved (equations (5) and (6)), it is more challenging to forecast the infection spread and the impact of the control measures. The major advantage of this study is that both models can be visualized with the aid of networks. These networks and the corresponding invariants work as useful interpretation and forecasting tool. Such discrete tools can prove to be fruitful in future to design and plan the control measures, which will surely help to reduce the economic burden by controlling the spread of Echinococcosis.
Conclusions
These models present the interaction of two animals and prevalence of E. multilocularis in different regions and the E. multilocularis abundance in an animal host of different age groups. For prevention of the E. multilocularis we use control measures in this model, we conclude that for better administration of the disease, clear knowledge of the interactions between the two animals, as well as the respective densities, is required. There is a threshold, for which, the infectious fox density remains stable, in a control group. The purpose of this study is to provide a network analysis that can help to forecast such thresholds. In this paper we presented the transition invariants via a quantitative approach of PNs. This approach is recommended for future control measurements.
Financial support
The research is funded by NRPU 4275.
Conflict of interest
The authors declare that there is no conflict of interest or financial disclosure about this publication.
