In our study, we have investigated the influence of the intermediate host population density on the prevalence of Echinococcus multilocularis in the definitive host using a mathematical model of transmission. For the vole population (intermediate host) in Hokkaido, a model of population dynamics has been constructed in this paper which follows the seasonal and annual fluctuations. In the northeastern area, the vole density appears to fluctuate periodically with a 4 year cycle. The prevalence of Echinococcus multilocularis in the fox population (definitive host) can be affected by the density of vole through the fox ingesting infectious voles. Therefore we have prepared a food habit function of foxes and the logistic distribution has been proposed. The simulations which have been carried out using the mathematical model for transmission of Echinococcus multilocularis together with the vole dynamics have indicated that the prevalence in foxes is correlated and synchronized with the population dynamics of vole. In addition they have also made us recognize that it is necessary to introduce a suitable food habit function into the transmission model

Doi, Rikuo

Ishii, Hiroyuki

Ishikawa, Hirofumi

Ohga, Yukio

Okayama University Scientific Achievement Repository

Journal of the Faculty of Enviromental Science and Technology,Okayama UniversityVol. 7, No.I. pp. 1-5, March 2002Simulations on Prevalence ofEchinococcus multilocularis in Hokkaido on theBasis of Vole Population DynamicsYukio Ohgat, Hirofumi Ishikawa2, Rikuo Doe and Hiroyuki IshiP(Received November 30, 2001)In our study, we have investigated the influence of the intermediate host population density on theprevalence of Echinococcus multilocularis in the definitive host using a mathematical model oftransmission. For the vole population (intermediate host) in Hokkaido, a model of population dynamicshas been constructed in this paper which follows the seasonal and annual fluctuations, In the northeasternarea, the vole density appears to fluctuate periodically with a 4 year cycle. The prevalence ofEchinococcus multilocularis in the fox population (definitive host) can be affected by the density of volethrough the fox ingesting infectious voles. Therefore we have prepared a food habit function of foxes andthe logistic distribution has been proposed. The simulations which have been carried out using themathematical model for transmission of Echinococcus multilocularis together with the vole dynamicshave indicated that the prevalence in foxes is correlated and synchronized with the population dynamicsof vole. In addition they have also made us recognize that it is necessary to introduce a suitable foodhabit function into the transmission model.Key words: Echinococcus multilocularis, food habit offoxes, Hokkaido, population dynamics, vole1. INTRODUCTIONThe parasites Echinococcus multilocularis in foxpopulations prevail in almost all areas of Hokkaido,Japan. There are also signs of alveolar Echinococcus inthe human population. Our paper focuses on therelationship between the prevalence of Echinococcusmultilocularis in the definitive host population, fox, andthe density of the intermediate host population, vole,based on a mathematical model of transmission.There are various characteristics in the populationdynamics of vole which depend on seasonal, annual andgeographical factors (Kaneko et aI., 1998; Saitoh et al.,1998a). The infection rate in foxes depends on the densityof vole through its food habit, because a fox feeds onvoles by preference but it does also have other availableDepartment of Environmental Synthesis and Analysis, Graduate Schoolof Natural Science and Technology, Okayama University 700-8530,Japan 1, Department of Environmental and Mathematical Science,Faculty of Environmental Science and Technology, OkayamaUniversity, 700-8530, Japan', Department of Hygiene, School ofMedicine, Yokohama City University, 236-0004, Japan'food. Yoneda (1981) reported on the diversity of foodhabits for seasons and areas. The winter food habit offoxes which is influenced mainly by snowfall plays animportant role in the infection rate of foxes (Kondo et at,1986). The fluctuation in the infection rate of fox~s iseither synchronized with that of the vole density orfollows it with a one year leg (Saitoh et aI., 1998b). Inthis paper, we formulate a mathematical model for thepopulation dynamics of the vole as well as a functionwhich describes the food habit of foxes, and then combinethem with a model for transmission of Echinococcusmultilocularis (Ishikawa et aI., in preparation).We have carried out the simulations on the prevalenceof Echinococcus multilocularis in two typical areas: thenortheastern area in Hokkaido and the southwestern areausing the field data of Nemuro and Oshima, respectively.The comparative study on the food habit of foxes showsthat it is necessary to introduce a suitable food habitfunction into the transmission model.2 J. Fac. Environ. Sci. and Tech., Okayama Univ. 7 (l) 20022. MATERIALS AND METHODSSeasonal Population Dynamics of VoleThe population dynamics of the vole is under theinfluence of the breeding season, a breeding coefficientand a survival rate which is seasonally swayed bymeteorological conditions. It is well known (Kaneko etaI., 1998) that the breeding time covers three seasons, thatis, from April to October, that the pregnancy rate is higherin spring and fall than in sum!Uer, and that the survivalrate is higher in winter than in the other seasons, so thatthe population density curve has two peaks a year.We need several assumptions to deal with a model forthe population dynamics of vole. We assume that thebreeding season begins on the first day when the move inthe average temperature with a band width of 20 days isin excess of SoC (Stenseth et aI., 1998). We may assumethat spring runs for two months from this day, and this isfollowed by summer and fall each for two months. In ourstudy, we adopted the temperatures in the northeasternpart of Hokkaido from Nemuro the data of which wasmeasured by the Meteorological Agency, Japan. Thebreeding rate R of the vole is written by the formR = LS . SR . Prepwhere LS, SR and Prep denote litter size, sex ratio andpercentage of breeding females, respectively. We adoptthat LS =5.3 and SR =0.5. The other parameters, such asPrep and the survival rate are discussed in next section.Annual Population Dynamics of Vole in HokkaidoTo investigate annual population dynamics of vole, weused the census data on vole populations in Hokkaidocarried out by the Forestry Agency of the JapaneseGovernment. Saitoh et al. (1998a) reported that a periodicfluctuation in the vole population with a 3.5 - 4.5 yearcycle was found in the northern and eastern area inHokkaido whereas an aperiodic fluctuation was found inthe southernmost area. The proportion of breeding voleswas higher during the increase phase than during thedecline phase (Nakata, 1989). It is assumed that bothpercentage P"p of breeding females and survival ratedepend on age and season, therefore we arranged somesets of data which consisted of parameters concerned withage (0, 1, 2, 3 months old and more than 4 months old)and each season as sho~n in Table 3 of Yoccoz et al.(1998). We make suitable combinations of theseparameter sets in a model of vole population to realize aperiodic or aperiodic situation.Food Habit ofFoxes Depended on Vole PopulationVoles can transmit Echinococcus multilocularis to afox when a susceptible fox ingests an infected vole.Therefore, it is important to know the food habit of foxes,that is, the number of voles which a fox ingests a day,hereafter referred to as NVF. A fox feeds on voles bypreference but it does have other available food (Yoneda,1981). NVF depends on the population density of voleand on meteorological conditions (Abe, 1975; Yoneda,1981). Thus as NVF we introduce the food habit functionF(x) of vole population density x into our transmissionmodel.We assume that the food habit function F(x) increasesexponentially in low density, while the increasing degreeof F(x) is reduced in high density, and that F(x) becomessaturated with the maximum of NVF in a fairly highdensity. We use the logistic distribution as the food habitfunction F(x) conveniently, whose value is the number ofvole ingested by fox a day:F( ) _ VmaxX - x-f-l1 + exp(--(J-)where x and Vmax denote population density of vole per0.5 ha and the maximum number of vole which areingested by fox. The fl, and (1 denote the median, andvariance of the distribution.3. RESULTSimulation ofMathematical Model for Vole PopulationDynamicsTo simulate the vole population for the situation witha 4 year cycle of periodic fluctuation in the northeasternarea of Hokkaido, we sorted the years into two groups:the years of increasing vole population and those ofdecreasing vole population. Next, we applied a suitablecombination of the 7 parameter sets chosen from Table 3of Yoccoz et al. (1998) to each group, while avoiding theeradication or explosion of the vole population. Fig. 1Y. OHGA et al. / Simulations on prevalence ofEchinococcus multilocularis 330,----------------------,shows the density curve for the vole population per 0.5 haobtained by our model, which includes seasonalfluctuations, the annual periodic ones and a range offluctuations from 6.0 to 24.0. For the purpose of acomparative study, we also prepared a model for theaperiodic situation observed in the southwestern area inHokkaido, and these results are shown in Fig. 1.1 25v;6D 20oSf:il 15'"'"~ 10""~ 54YEAR- Northeastern.- Southwestern10associates the vole population dynamics with the foodhabit function as mentioned above.Simulation of Mathematical Model for transmission ofEchinococcus multilocularisWe have carried out simulations using a model for thetransmission of Echinococcus multilocularis (Ishikawa etaI., in preparation), our model of vole populationdynamics and a simple model of the fox populationdynamics of foxes which considers only seasonalfluctuations. The prevalence curves of Echinococcusmultilocularis in the fox population obtained by thesesimulations are shown in Fig. 3 for periodic and aperiodicvole density situations. According to our simulations, theprevalence rate ranges from 20% to 80% for the periodicsituation while it only ranges limited from 40% to 60%for the aperiodic situation.YEAR10- NortheasternSouthwestern1009080~ 70" 60~"505 40 n~ /~30 . / Ii" i It /0.20 ~ V i104Fig. 3 Variation in the prevalence rate of the fox population bysimulation. The black line shows the prevalence rate inthe northeastern area of Hokkaido, and the gray line isthat in the southwestern area of Hokkaido.Fig. 1 The transition of vole population density by our model.The black line shows the density in the northeastern areaof Hokkaido, and the gray line, that in the southwesternarea of Hokkaiclo.In the light of the census data on the vole population inHokkaido (the Forestry Agency of the JapaneseGovernment), we assigned the remaining parameters j.J.,and a of the food habit function F(x) as JI :: 8.0, and (T::2.5 (x ~ 8.0), or (T:: 1.0 (x > 8.0). The maximum Vmax ofNVF was chosen as 7.0 (Yoneda, 1981). Fig. 2 shows thetransition of NVF derived by the simulation whichFood Habit Function for the Number of Vole Ingested byFox101009080~ 70"~ 60"5 50~ 400.3020104 10YEAR YEARFig. 2 The transition of the number of voles which the foxingest (NVF) using the logistic distribution as the foodhabit function.Fig. 4 Variation in the prevalence rate of the fox populationusing a constant as the food habit function. The blackline shows the prevalence rate in the case with a valueof 3.0, and gray line, with that of 6.0.4 J. Fac. Environ. Sci. and Tech., Okayama Univ. 7 0) 2002To investigate the effect of the food habit function onthe infection rate of the fox population, we havecompared the logistic distribution adopted in our modelwith the constant function (the two constant values werechosen 3.0 and 6.0) as the food habit function. The resultof the simulations indicate that it has a small effect on theconstant function case (Fig. 4).4. DISCUSSIONIn the population dynamics of vole, the annualpopulation growth rate is given theoretically by thedominant eigenvalue of an annual matrix (Yoc,coz et aL,1998). The dominant eigenvalues of the annual matricescorresponding to the parameter sets in our volepopulation model range from 0.90 to 5.07, although theactual growth rate can suffer a change according to theage distribution at the beginning of the study. The curveof population density in Fig. 1 satisfied the average rangeof the annual population densities at 8 observation points(from 3.0 to 23.0 per 0.5 ha) in the northeastern area ofHokkaido.On using the constant function with the high constantvalue of 6.0 as the food habit function, for the situation ofa high density of vole population, NVF is always assureda given quantity, that differs from the actual case, whereasfor the situation of low density, the NVF should followthe result of the logistic distribution case. Its simulationyields an infection rate which is maintained with a narrowamplitude. On the other hand, using the low constantvalue of 3.0, it seems that the prevalence may only beinfluenced by the seasonal fluctuations of the foxpopulation and may be independent of the annualfluctuation in vole density. If the prevalence in foxes wasnot reflected by the state of the intermediate host, then itwould not realize the actual situation for transmission ofEchinococcus multilocularis. The food habit of foxesshould be affected by environmental conditions in theirhabitat, especially snowfall, but it is very difficult toincorporate them into the food habit function.The simulations on the prevalence of Echinococcusmultilocularis in fox carried out for the two typicalfluctuations of vole densities show that, from thestandpoint of the seasonal fluctuation, both of theprevalence curves have similar figures and reach theirpeaks in winter, while, from the standpoint of the annualfluctuation, the prevalence curve in the northeastern areahas a higher peak and wider amplitude than that in thesouthwestern area. Therefore it follows from thesesimulations that there is a positive correlation and asynchronism between the population dynamics of voleand the infection rate in foxes.ACKNOWLEDGMENTS: We sincerely thank Prof. T.Saitoh, Prof. M. Kamiya in Hokkaido University, Dr. K.Takahashi in Hokkaido Institute for Public Health, andProf. A. Ishii in Jichi Medical School for their helpfulcomments. We are indebted to the Forestry Agency ofJapanese Government for providing the census data onthe vole. This work was supported in part by Grant-in-Aidfor Science Research from Japan Society for Promotionof Science (grant no. 13640116, H. Ishikawa) and bygrants from the Ministry of Health, Labor and Welfare,Japan for "The control of emerging and reemergingdiseases in Japan".REFERENCESAbe, H. (1975): Winter food of the red fox, Vulpes vulpesschrencki Kishida (Carnivora:Canidae), in Hokkaido, withspecial reference to vole populations. Applied Entomologyand Zoology 10, 40-51.Ishikawa, H., Ohga, Y., Doi, R. and Ishii, H. (in preparation): Amodel for the transmission of Echinococcus multilocularis inHokkaidoKaneko, Y., Nakata, K., Saitoh, T., Stenseth, N. C. andBj0rnstad, O. N. (1998): The biology of the voleClethrionomys rufocanus: a review. Researches onPopulation Ecology 40,21-37.Kondo, N., Takahashi, K., and Yagi, K. (1986): Winter food ofthe red fox, Vulpes vulpes schrencki Kishida, in the endemicarea of multilocular echinococcusis. Bulletin of PreparativeOffice of Nemuro Municipal Museum 1,23-31 (in Japanesewith English summary).Nakata, K. (1989): Regulation of reproductive rate in a cyclicpopulation of the red-backed vole, Clethrionpmys rufocanusbedfordiae. Researches on Population Ecology 31, 185-209.Saitoh, T., Stenseth, N, C. and Bj0rnstad, O. N. (1998a): Thepopulation dynamics of the vole Clethrionomys rufocanus inHokkaido, Japan. Researches on Population Ecology 40,61-76Saitoh, T., Takahashi, K. (1998b): The role of vole populationin prevalence of the parasite (Echinococcus multi/ocularis)in faxes. Researches on Population Ecology 40, 97-105Stenseth, N. c., Bj0rnstad, O. N. and Saitoh, T. (1998)Seasonal forcing on the dynamics of Clethrionomysrufocanus: modeling geographic gradients in populationdynamics. Researches on Population Ecology 40, 85-95Yoccoz, N. G., Nakata, K., Stenseth, N. C. and Saitoh, T.Y. OHGA et al. f Simulations on prevalence ofEchinococcus multilocularis(1998) The demography of Clethrionomys rufocanus: frommathematical and statistical models to further field studies.Researches on Population Ecology 40, 107-121Yoneda, M. (1981) The number of the red fox, Vulpes vulpesschrencki Kishida and its food habits, 1, NortheasternForestry 33 (11), 5-8 (in Japanese)5

Simulations on Prevalence of Echinococcus multilocularis in Hokkaido on the Basis of Vole Population Dynamics

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