In this paper we look at a non-age-structured model for the spread of the Zika Virus and Microcephaly in Brazil. We first outline the non-seasonal differential equation model, and discuss parameter values and their estimation. Then we talk about the basic reproduction number and details of the calculation of the number of Microcephaly cases. Next we estimate how the model can be made more realistic by introducing seasonality into the mosquito population. Finally we consider sensitivity of the results to the mosquito biting rate

Greenhalgh, David

Liang, Yanfeng

University of Strathclyde Institutional Repository

Liang, Yanfeng and Greenhalgh, David (2017) Mathematical modelling the spread of Zika and Microcephaly in Brazil. In: Proceedings of the 17th International Conference on Computational and Mathematical Methods in Science and Engineering. CMMSE, [S.I.], pp. 1264-1268. ISBN 978-84-617-8694-7 , This version is available at https://strathprints.strath.ac.uk/61481/Strathprints is  designed  to  allow  users  to  access  the  research  output  of  the  University  of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may  not  engage  in  further  distribution  of  the  material  for  any  profitmaking  activities  or  any commercial gain. 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It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output.Proceedings of the 17th International Conferenceon Computational and Mathematical Methodsin Science and Engineering, CMMSE 20174–8 July, 2017.Mathematical Modelling the Spread of Zika andMicrocephaly in Brazil.Yanfeng Liang1 and David Greenhalgh11 Department of Mathematics and Statistics, University of Strathclyde, Glasgow, UKemails: yanfeng.liang@strath.ac.uk, david.greenhalgh@strath.ac.ukAbstractIn this paper we look at a non-age-structured model for the spread of the Zika Virusand Microcephaly in Brazil. We first outline the non-seasonal differential equationmodel, and discuss parameter values and their estimation. Then we talk about the basicreproduction number and details of the calculation of the number of Microcephaly cases.Next we estimate how the model can be made more realistic by introducing seasonalityinto the mosquito population. Finally we consider sensitivity of the results to themosquito biting rate.Key words: Zika, Brazil, Microcephaly, Aedes Aegypti mosquito, basic reproductionnumber, differential equation model, seasonality.MSC 2000: AMS codes 92B15, 92C60, 92D30.1 Extended Abstract1.1 Non-seasonal ModelThe Zika virus is spread by the same species of mosquito, namely the Aedes Aegypti (A.Aegypti), as Dengue. Zika is a member of the virus family Flaviviridae. The first discoveryof the Zika virus was in 1947, however despite being around for a while, Zika has notreceived much attention until recently when it has been discovered that it is associated withMicrocephaly which is a serious birth defect in newborns, caused if women are infectedwith Zika during pregnancy. Most importantly there is still no vaccine to prevent the Zikavirus. Apart from causing severe birth defects to newborn babies, infected individuals canalso experience fever, rash and joint pain. As a result, in this paper we will use an existingc©CMMSE ISBN: 978-84-617-8694-7NON-AGE-STRUCTURED MODEL FOR ZIKAParameter values Biological meanings Valuesa A. Aegypti biting rate 0.5× 7/week [1, 5]Probability of transmission of Zika when 0.10− 0.75 [1]b an infectious mosquito bites a susceptiblehumanProbability of transmission of Zika when 0.30− 0.75 [3]c a susceptible mosquito bites an infectioushumanNH Human population in Brazil in 2015 207, 848, 000 [12]µH Per capita human mortality rate in Brazil 1/(75× 52)/week [12]γ Per capita human recovery rate 7/6/week [5]µv Per capita mortality rate for A. Aegypti 0.025× 7/week [9]Nv A. Aegypti population 1.5×NH [4, 6]τ Zika extrinsic incubation period 8.2/7 weeks [5]Table 1: Parameter values given in Equation (1).time-delayed mathematical model for Dengue mentioned in [9] to analyse the dynamicalbehaviour for the Zika virus, in Brazil, as well as estimating the future expected number ofcases of Microcephaly due to Zika.The definitions of the parameter values used in the differential equation model and theircorresponding values are given in Table 1. The model that we are working with is given asfollows:dSH(t)dt= −abIv(t)SH(t)NH− µHSH(t) + µHNH ,dIH(t)dt= abIv(t)SH(t)NH− (µH + γ)IH(t),dRH(t)dt= γIH(t)− µHRH(t), (1)dSv(t)dt= −acSv(t)IH(t)NH− µvSv(t) + µvNv,dLv(t)dt= acSv(t)IH(t)NH− µvLv(t)− acSv(t− τ)IH(t− τ)NHe−µvτ ,dIv(t)dt= acSv(t− τ)IH(t− τ)NHe−µvτ − µvIv(t),with initial conditions SH(0), IH(0), RH(0), Sv(0), Lv(0) and Iv(0), where SH(t), IH(t) andRH(t) respectively represent the susceptible, infected and recovered individuals for hu-mans, while Sv(t), Lv(t) and Iv(t) respectively represent the susceptible, latent and infectedmosquitoes. Note that NH = SH + IH + RH denotes the total human population size andNv = Sv + Lv + Iv represents the total A. Aegypti population size where both populationsc©CMMSE ISBN: 978-84-617-8694-7Yanfeng Liang and David Greenhalghare constant. Although the Zika virus and Dengue are spread by the same transmissionroute and thus some parameter values would remain the same, parameters such as thetransmission probabilities between humans and A. Aegypti mosquitoes which are defined asb and c in Equation (1) may vary. Therefore one of the aims in this project is to use theleast squares estimation technique and the real Zika virus data from Brazil given in [5] toestimate these two values.We assume that a single Zika infected human enters the disease free population at sometime t0, where t0 < t1 and t1 is the first time when we have available Zika data valuesin Brazil obtained from [5] as the first week in 2015. We have estimated t0 and henceSH(t1), IH(t1), RH(t1), Sv(t1), Lv(t1) and Iv(t1), used as simulation starting values, by leastsquares. The basic reproduction number for our delayed Zika model given in [10] is definedasR0 =ma2bce−µvτµv(µH + γ), (2)where all the parameter values are defined as in Table 1. For a in the range 0.7 – 3.5/weekwe get R0 in the range 1.27 – 11.01/week.1.2 Numerical SolutionsOnce all the parameter values are obtained and estimated, we use R to solve the differentialequations given in Equation (1) and produced simulations which illustrate the number ofsusceptible, infected and recovered individuals over both a short time period, to representthe immediate future, and over a long time period, to represent what happens when theendemic equilibrium has been reached.We focus on analysing the effect of pregnant women infected with Zika virus duringtheir first trimester as various reports (e.g. [2, 8]) suggest that pregnant women who areinfected with the Zika virus during the first trimester have a much higher risk of theirbabies developing Microcephaly as opposed to those who are infected with Zika in theirsecond or third trimesters. We have obtained an estimated expected future number of casesof Microcephaly due to pregnant women infected with Zika during their first trimester bothin the short and in the long term.1.3 Model With SeasonalityIt is well-known that the life cycle of A. Aegypti is influenced by many environmental factorssuch as rainfall and temperature (e.g. [7, 11, 13]). As a result, in order to fully capturethe behaviour of the A. Aegypti mosquitoes under the influence of environmental factorsand its effect on the number of Microcephaly cases, later on we decided to improve on ourmodel by adding seasonality into the birth function of A. Aegypti mosquitoes. Similarly,with this seasonality model, we use the least squares estimation technique to parameterc©CMMSE ISBN: 978-84-617-8694-7NON-AGE-STRUCTURED MODEL FOR ZIKAestimate new values of b and c, and thus calculate the future expected number of cases ofMicrocephaly both in the short and long term due to pregnant women being infected intheir first trimester.1.4 ResultsFor both models, numerical simulations are produced to illustrate the spread of the Zikavirus over a period of time and the future expected number of cases of Microcephaly as aresult of the Zika virus are calculated. The suggested value of a= 3.5/week for the parametera, the A. Aegypti biting rate, is high compared to the other values in the literature. Sowe discuss the sensitivity of our results to different values of this parameter. We will laterextend the results to an age-structured model.AcknowledgementsThe authors are grateful to the EPSRC and the University of Strathclyde for support for thiswork under the EPSRC Global Challenges Research Fund Institutional Award 2016 (EPSRCgrant reference number EP/P511055/1) and the British Council, Malaysia for funding fromthe Dengue Tech Challenge (Application Reference DTC 16022). DG is grateful to theScience Without Borders Program, Brazil, for a Special Visiting Fellowship (CNPq grant30098/2014-7), with Professor E. Massad, Department of Legal Medicine, University ofSao Paulo, Sao Paulo, Brazil, and to the Leverhulme Trust for support from a LeverhulmeResearch Fellowship (RF-2015-88).References[1] M. Andraud, N. Hens, C. Marais and P. Beutels, Dynamic epidemiologicalmodels for Dengue transmission: a systematic review of structural approaches, PLoSone 7(11) (2012) e49085.[2] S. Cauchemez, M. Besnard, P. Bompard, T. Dub, P. Guillemette-Artur,D. Eyrolle-Guignot, H. Salje, M. D. V. Kerkhove, V. Abadie, C. Garel,A. Fontanet and H. Mallet, Association between Zika Virus and Microcephaly inFrench Polynesia, 2013-2015: A retrospective study, The Lancet 387(10033) (2016)2125–2132.[3] E. Chikak and H. Ishikawa, A Dengue transmission model in Thailand consideringsequential infections with all four serotypes, J. Inf. Develop. Countries 3(09) (2009)711–722.c©CMMSE ISBN: 978-84-617-8694-7Yanfeng Liang and David Greenhalgh[4] Centers for Disease Control and Prevention, Surveillance and control ofAedes Aegypti and Aedes Albopictus in the United States, 16 pages. Vancouver, 2017.Retrieved from https://www.cdc.gov/chikungunya/resources/vector-control.html, lastaccessed on 20th March 2017.[5] N.M. Ferguson, Z.M. Cucunub, I. Dorigatti, G. L. Nedjati-Gilan, C.A.Donnelly, M.G. Basez, P. Nouvellet and J. Lessler, Countering the Zikaepidemic in Latin America, Science 353(6297) (2016) 353–354.[6] D. Focks, N. Alexander and E. Villegas, Multicountry study ofAedes Aegypti pupal productivity survey methodology: findings and recom-mendations, UNICEF/UNDP/World Bank/WHO Special Programme forResearch and Training in Tropical Disease (TDR), (2006), retrieved fromhttp://www.who.int/tdr/publications/documents/aedes aegypti.pdf, last accessedon 20th March 2017.[7] J. Liu-Helmersson, H. Stenlund, A. Wilder-Smith and J. Rocklv, Vectorialcapacity of Aedes Aegypti: effects of temperature and implications for global Dengueepidemic potential, PLoS one 9(3) (2014) e89783.[8] M.A. Johansson, L. Mier-y-Teran-Romero, J. Reefhuis, S.M. Gilboa andS. L. Hills, Zika and the risk of of Microcephaly, New Eng. J. Med. 375(1) (2016)1–3.[9] E. Massad, F.A.B. Coutinho, M.N. Burattini and M. Amaku, Estimation ofR0 from the initial phase of an outbreak of a vector-borne infection, Trop. Med. Int.Health 15(1) (2010) 120–126.[10] E. Massad, F.A.B. Coutinho, M.N. Burattini and L.R. Lopez, The risk ofYellow Fever in a Dengue-infested area, Trans. Roy. Soc. Trop. Med. Hyg. 95 (2001)370–374.[11] H. S. Rodrigues, M.T. Monteiro and D.M. Torres, Seasonality effects onDengue: basic reproduction number, sensitivity analysis and optimal control, Math.Meth. Appl. Sci. 39(16) (2016) 4671–4679.[12] World Health Organization, Countries: Brazil, 2017 Retrieved fromhttp://www.who.int/countries/bra/en/, last accessed on 20th March 2017.[13] S. Wiwanitkit and V. Wiwanitkit, Predicted pattern of Zika Virus infection dis-tribution with reference to rainfall in Thailand, Asian Pacif. J. Trop. Med. 9(7) (2016)719–720.c©CMMSE ISBN: 978-84-617-8694-7

Mathematical modelling the spread of Zika and Microcephaly in Brazil

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