Numerical solution of a spatio-temporal gender-structured model for hantavirus infection in rodents

Raimund BÜrger; Gerardo Chowell; Elvis GavilÁn; Pep Mulet; Luis M. Villada; Raimund BÜrger; Gerardo Chowell; Elvis GavilÁn; Pep Mulet; Luis M. Villada

doi:10.3934/mbe.2018004

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 1: 95-123. doi: 10.3934/mbe.2018004

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Numerical solution of a spatio-temporal gender-structured model for hantavirus infection in rodents

1.
CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
2.
School of Public Health, Georgia State University, Atlanta, Georgia, USA
3.
Simon A. Levin Mathematical and Computational Modeling Sciences Center, Arizona State University, Tempe, AZ 85287, USA
4.
Division of International Epidemiology and Population Studies, Fogarty International Center, National Institutes of Health, Bethesda, MD 20892, USA
5.
Departament de Matemàtiques, Universitat de València, Av. Dr. Moliner 50, E-46100 Burjassot, Spain
6.
GIMNAP-Departamento de Matemáticas, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile
7.
CI2MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Received: 29 September 2016 Accepted: 14 January 2017 Published: 01 February 2018
MSC : Primary: 92D30; Secondary: 97M60, 65L06, 65M20

In this article we describe the transmission dynamics of hantavirus in rodents using a spatio-temporal susceptible-exposed-infective-recovered (SEIR) compartmental model that distinguishes between male and female subpopulations [L.J.S. Allen, R.K. McCormack and C.B. Jonsson, Bull. Math. Biol. 68 (2006), 511-524]. Both subpopulations are assumed to differ in their movement with respect to local variations in the densities of their own and the opposite gender group. Three alternative models for the movement of the male individuals are examined. In some cases the movement is not only directed by the gradient of a density (as in the standard diffusive case), but also by a non-local convolution of density values as proposed, in another context, in [R.M. Colombo and E. Rossi, Commun. Math. Sci., 13 (2015), 369-400]. An efficient numerical method for the resulting convection-diffusion-reaction system of partial differential equations is proposed. This method involves techniques of weighted essentially non-oscillatory (WENO) reconstructions in combination with implicit-explicit Runge-Kutta (IMEX-RK) methods for time stepping. The numerical results demonstrate significant differences in the spatio-temporal behavior predicted by the different models, which suggest future research directions.
Citation: Raimund BÜrger, Gerardo Chowell, Elvis GavilÁn, Pep Mulet, Luis M. Villada. Numerical solution of a spatio-temporal gender-structured model for hantavirus infection in rodents[J]. Mathematical Biosciences and Engineering, 2018, 15(1): 95-123. doi: 10.3934/mbe.2018004

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Abstract

In this article we describe the transmission dynamics of hantavirus in rodents using a spatio-temporal susceptible-exposed-infective-recovered (SEIR) compartmental model that distinguishes between male and female subpopulations [L.J.S. Allen, R.K. McCormack and C.B. Jonsson, Bull. Math. Biol. 68 (2006), 511-524]. Both subpopulations are assumed to differ in their movement with respect to local variations in the densities of their own and the opposite gender group. Three alternative models for the movement of the male individuals are examined. In some cases the movement is not only directed by the gradient of a density (as in the standard diffusive case), but also by a non-local convolution of density values as proposed, in another context, in [R.M. Colombo and E. Rossi, Commun. Math. Sci., 13 (2015), 369-400]. An efficient numerical method for the resulting convection-diffusion-reaction system of partial differential equations is proposed. This method involves techniques of weighted essentially non-oscillatory (WENO) reconstructions in combination with implicit-explicit Runge-Kutta (IMEX-RK) methods for time stepping. The numerical results demonstrate significant differences in the spatio-temporal behavior predicted by the different models, which suggest future research directions.

References

[1]	[ G. Abramson and V. M. Kenkre, Spatiotemporal patterns in the Hantavirus infection Phys. Rev. E 66 (2002), 011912 (5pp).
[2]	[ M. A. Aguirre, G. Abramson, A. R. Bishop and V. M. Kenkre, Simulations in the mathematical modeling of the spread of the Hantavirus Phys. Rev. E 66 (2002), 041908 (5pp).
[3]	[ L. J. S. Allen,B. M. Bolker,Y. Lou,A. L. Nevai, Asymptotic of the steady states for an SIS epidemic patch model, SIAM J. Appl. Math., 67 (2007): 1283-1309.
[4]	[ L. J. S. Allen,R. K. McCormack,C. B. Jonsson, Mathematical models for hantavirus infection in rodents, Bull. Math. Biol., 68 (2006): 511-524.
[5]	[ R. M. Anderson,R. M. May, null, Infectious Diseases of Humans: Dynamics and Control, , Oxford Science Publications, 1991.
[6]	[ J. Arino, Diseases in metapopulations. In Z. Ma, Y. Zhou and J. Wu (Eds. ), Modeling and Dynamics of Infectious Diseases, Higher Education Press, Beijing, 11 (2009), 64-122.
[7]	[ J. Arino,J. R. Davis,D. Hartley,R. Jordan,J. M. Miller,P. van den Driessche, A multi-species epidemic model with spatial dynamics, Mathematical Medicine and Biology, 22 (2005): 129-142.
[8]	[ U. Ascher,S. Ruuth,J. Spiteri, Implicit-explicit Runge-Kutta methods for time dependent partial differential equations, Appl. Numer. Math., 25 (1997): 151-167.
[9]	[ P. Bi,X. Wu,F. Zhang,K. A. Parton,S. Tong, Seasonal rainfall variability, the incidence of hemorrhagic fever with renal syndrome, and prediction of the disease in low-lying areas of China, Amer. J. Epidemiol., 148 (1998): 276-281.
[10]	[ S. Boscarino,R. Bürger,P. Mulet,G. Russo,L. M. Villada, Linearly implicit IMEX Runge-Kutta methods for a class of degenerate convection-diffusion problems, SIAM J. Sci. Comput., 37 (2015): B305-B331.
[11]	[ S. Boscarino,F. Filbet,G. Russo, High order semi-implicit schemes for time dependent partial differential equations, J. Sci. Comput., 68 (2016): 975-1001.
[12]	[ S. Boscarino,P. G. LeFloch,G. Russo, High-order asymptotic-preserving methods for fully nonlinear relaxation problems, SIAM J. Sci. Comput., 36 (2014): A377-A395.
[13]	[ S. Boscarino,G. Russo, On a class of uniformly accurate IMEX Runge-Kutta schemes and applications to hyperbolic systems with relaxation, SIAM J. Sci. Comput., 31 (2009): 1926-1945.
[14]	[ S. Boscarino,G. Russo, Flux-explicit IMEX Runge-Kutta schemes for hyperbolic to parabolic relaxation problems, SIAM J. Numer. Anal., 51 (2013): 163-190.
[15]	[ F. Brauer,C. Castillo-Chavez, null, Mathematical Models in Population Biology and Epidemiology, Second Ed., Springer, New York, 2012.
[16]	[ M. Brummer-Korvenkontio,A. Vaheri,T. Hovi,C. H. von Bonsdorff,J. Vuorimies,T. Manni,K. Penttinen,N. Oker-Blom,J. Lähdevirta, Nephropathia epidemica: Detection of antigen in bank voles and serologic diagnosis of human infection, J. Infect. Dis., 141 (1980): 131-134.
[17]	[ J. Buceta, C. Escudero, F. J. de la Rubia and K. Lindenberg, Outbreaks of Hantavirus induced by seasonality Phys. Rev. E 69 (2004), 021908 (9pp).
[18]	[ R. Bürger,G. Chowell,P. Mulet,L. M. Villada, Modelling the spatial-temporal progression of the 2009 A/H1N1 influenza pandemic in Chile, Math. Biosci. Eng., 13 (2016): 43-65.
[19]	[ R. Bürger,R. Ruiz-Baier,C. Tian, Stability analysis and finite volume element discretization for delay-driven spatio-temporal patterns in a predator-prey model, Math. Comput. Simulation, 132 (2017): 28-52.
[20]	[ R. M. Colombo,E. Rossi, Hyperbolic predators versus parabolic preys, Commun. Math. Sci., 13 (2015): 369-400.
[21]	[ M. Crouzeix, Une méthode multipas implicite-explicite pour l'approximation des équations d'évolution paraboliques, Numer. Math., 35 (1980): 257-276.
[22]	[ O. Diekmann, H. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2013.
[23]	[ R. Donat,I. Higueras, On stability issues for IMEX schemes applied to 1D scalar hyperbolic equations with stiff reaction terms, Math. Comp., 80 (2011): 2097-2126.
[24]	[ C. Escudero, J. Buceta, F. J. de la Rubia and K. Lindenberg, Effects of internal fluctuations on the spreading of Hantavirus Phys. Rev. E 70 (2004), 061907 (7pp).
[25]	[ S. de Franciscis and A. d'Onofrio, Spatiotemporal bounded noises and transitions induced by them in solutions of the real Ginzburg-Landau model Phys. Rev. E 86 (2012), 021118 (9pp); Erratum, Phys. Rev. E 94 (2016), 0599005(E) (1p).
[26]	[ M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models Amer. Inst. Math. Sci. , Springfield, MO, USA, 2006.
[27]	[ G. S. Jiang,C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996): 202-228.
[28]	[ P. Kachroo,S. J. Al-Nasur,S. A. Wadoo,A. Shende, null, Pedestrian Dynamics, , Springer-Verlag, Berlin, 2008.
[29]	[ A. Källén, Thresholds and travelling waves in an epidemic model for rabies, Nonlin. Anal. Theor. Meth. Appl., 8 (1984): 851-856.
[30]	[ A. Källén,P. Arcuri,J. D. Murray, A simple model for the spatial spread and control of rabies, J. Theor. Biol., 116 (1985): 377-393.
[31]	[ Y. Katznelson, null, An Introduction to Harmonic Analysis, Third Ed., Cambridge University Press, Cambridge, UK, 2004.
[32]	[ C. A. Kennedy,M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44 (2003): 139-181.
[33]	[ W. O. Kermack,A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. A, 115 (1927): 700-721.
[34]	[ N. Kumar, R. R. Parmenter and V. M. Kenkre, Extinction of refugia of hantavirus infection in a spatially heterogeneous environment Phys. Rev. E 82 (2010), 011920 (8pp).
[35]	[ T. Kuniya,Y. Muroya,Y. Enatsu, Threshold dynamics of an SIR epidemic model with hybrid and multigroup of patch structures, Math. Biosci. Eng., 11 (2014): 1375-1393.
[36]	[ H. N. Liu, L. D. Gao, G. Chowell, S. X. Hu, X. L. Lin, X. J. Li, G. H. Ma, R. Huang, H. S. Yang, H. Tian and H. Xiao, Time-specific ecologic niche models forecast the risk of hemorrhagic fever with renal syndrome in Dongting Lake district, China, 2005-2010, PLoS One, 9 (2014), e106839 (8pp).
[37]	[ X.-D. Liu,S. Osher,T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994): 200-212.
[38]	[ H. Malchow,S. V. Petrovskii,E. Venturino, null, Spatial Patterns in Ecology and Epidemiology: Theory, Models, and Simulation, , Chapman & Hall/CRC, Boca Raton, FL, USA, 2008.
[39]	[ J. N. Mills,B. A. Ellis,K. T. McKee,J. I. Maiztegui,J. E. Childs, Habitat associations and relative densities of rodent populations in cultivated areas of central Argentina, J. Mammal., 72 (1991): 470-479.
[40]	[ P. A. P. Moran, Notes on continuous stochastic phenomena, Biometrika, 37 (1950): 17-23.
[41]	[ J. D. Murray, null, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications, Third Edition, Springer, New York, 2003.
[42]	[ J. D. Murray,E. A. Stanley,D. L. Brown, On the spatial spread of rabies among foxes, Proc. Roy. Soc. London B, 229 (1986): 111-150.
[43]	[ A. Okubo,S. A. Levin, null, Diffusion and Ecological Problems: Modern Perspectives, Second Edition, Springer-Verlag, New York, 2001.
[44]	[ O. Ovaskainen and E. E. Crone, Modeling animal movement with diffusion, in S. Cantrell, C. Cosner and S. Ruan (Eds. ), Spatial Ecology, Chapman & Hall/CRC, Boca Raton, FL, USA, 2009, 63-83.
[45]	[ L. Pareschi,G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25 (2005): 129-155.
[46]	[ J. A. Reinoso and F. J. de la Rubia, Stage-dependent model for the Hantavirus infection: The effect of the initial infection-free period Phys. Rev. E 87 (2013), 042706 (6pp).
[47]	[ J. A. Reinoso and F. J. de la Rubia, Spatial spread of the Hantavirus infection Phys. Rev. E 91 (2015), 032703 (5pp).
[48]	[ R. Riquelme,M. L. Rioseco,L. Bastidas,D. Trincado,M. Riquelme,H. Loyola,F. Valdivieso, Hantavirus pulmonary syndrome, southern chile, 1995-2012, Emerg. Infect. Dis., 21 (2015): 562-568.
[49]	[ C. Robertson, C. Mazzetta and A. d'Onofrio, Regional variation and spatial correlation, Chapter 5 in P. Boyle and M. Smans (Eds. ), Atlas of Cancer Mortality in the European Union and the European Economic Area 1993-1997, IARC Scientific Publication, WHO Press, Geneva, Switzerland, 159 (2008), 91-113.
[50]	[ E. Rossi,V. Schleper, Convergence of a numerical scheme for a mixed hyperbolic-parabolic system in two space dimensions, ESAIM Math. Modelling Numer. Anal., 50 (2016): 475-497.
[51]	[ S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in S. Cantrell, C. Cosner and S. Ruan (Eds. ), Spatial Ecology, Chapman & Hall/CRC, Boca Raton, FL, USA, 2010,293-316.
[52]	[ L. Sattenspiel, The Geographic Spread of Infectious Diseases: Models and Applications Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2009.
[53]	[ C.-W. Shu,S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Ⅱ, J. Comput. Phys., 83 (1988): 32-78.
[54]	[ S. W. Smith, Digital Signal Processing: A Practical Guide for Engineers and Scientists. Demystifying technology series: by engineers, for engineers. Newnes, 2003.
[55]	[ H. Y. Tian, P. B. Yu, A. D. Luis, P. Bi, B. Cazelles, M. Laine, S. Q. Huang, C. F. Ma, S. Zhou, J. Wei, S. Li, X. L. Lu, J. H. Qu, J. H. Dong, S. L. Tong, J. J. Wang, B. Grenfell and B. Xu, Changes in rodent abundance and weather conditions potentially drive hemorrhagic fever with renal syndrome outbreaks in Xi'an, China, 2005-2012, PLoS Negl. Trop. Dis. , 9 (2015), paper e0003530 (13pp).
[56]	[ M. Treiber,A. Kesting, null, Traffic Flow Dynamics, , Springer-Verlag, Berlin, 2013.
[57]	[ P. van den Driessche, Deterministic compartmental models: Extensions of basic models, In F. Brauer, P. van den Driessche and J. Wu (Eds. ), Mathematical Epidemiology, SpringerVerlag, Berlin, 1945 (2008), 147-157.
[58]	[ P. van den Driessche, Spatial structure: Patch models, In F. Brauer, P. van den Driessche and J. Wu (Eds. ), Mathematical Epidemiology, Springer-Verlag, Berlin, 1945 (2008), 179-189.
[59]	[ P. van den Driessche,J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002): 29-48.
[60]	[ E. Vynnycky,R. E. White, null, An Introduction to Infectious Disease Modelling, , Oxford University Press, 2010.
[61]	[ J. Wu, Spatial structure: Partial differential equations models, In F. Brauer, P. van den Driessche and J. Wu (Eds. ), Mathematical Epidemiology, Springer-Verlag, Berlin, 2008,191-203.
[62]	[ H. Xiao,X. L. Lin,L. D. Gao,X. Y. Dai,X. G. He,B. Y. Chen, Environmental factors contributing to the spread of hemorrhagic fever with renal syndrome and potential risk areas prediction in midstream and downstream of the Xiangjiang River [in Chinese], Scientia Geographica Sinica, 33 (2013): 123-128.
[63]	[ C. J. Yahnke,P. L. Meserve,T. G. Ksiazek,J. N. Mills, Patterns of infection with Laguna Negra virus in wild populations of Calomys laucha in the central Paraguayan chaco, Am. J. Trop. Med. Hyg., 65 (2001): 768-776.
[64]	[ W. Y. Zhang,L. Q. Fang,J. F. Jiang,F. M. Hui,G. E. Glass,L. Yan,Y. F. Xu,W. J. Zhao,H. Yang,W. Liu, Predicting the risk of hantavirus infection in Beijing, People's Republic of China, Am. J. Trop. Med. Hyg., 80 (2010): 678-683.
[65]	[ W. Y. Zhang,W. D. Guo,L. Q. Fang,C. P. Li,P. Bi,G. E. Glass,J. F. Jiang,S. H. Sun,Q. Qian,W. Liu,L. Yan,H. Yang,S. L. Tong,W. C. Cao, Climate variability and hemorrhagic fever with renal syndrome transmission in Northeastern China, Environ. Health Perspect, 118 (2010): 915-920.
[66]	[ X. Zhong, Additive semi-implicit Runge-Kutta methods for computing high-speed nonequilibrium reactive flows, J. Comput. Phys., 128 (1996): 19-31.

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