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February  2018, 15(1): 95-123. doi: 10.3934/mbe.2018004

## 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  September 29, 2016 Accepted  January 14, 2017 Published  May 2017

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. Mathematical Biosciences & Engineering, 2018, 15 (1) : 95-123. doi: 10.3934/mbe.2018004
##### References:
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##### References:
 [1] G. Abramson and V. M. Kenkre, Spatiotemporal patterns in the Hantavirus infection Phys. Rev. E 66 (2002), 011912 (5pp). doi: 10.1103/PhysRevE.66.011912. [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). doi: 10.1103/PhysRevE.66.041908. [3] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic of the steady states for an SIS epidemic patch model, SIAM J. Appl. Math., 67 (2007), 1283-1309.  doi: 10.1137/060672522. [4] L. J. S. Allen, R. K. McCormack and C. B. Jonsson, Mathematical models for hantavirus infection in rodents, Bull. Math. Biol., 68 (2006), 511-524.  doi: 10.1007/s11538-005-9034-4. [5] R. M. Anderson and R. M. May, 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 and 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 and J. Spiteri, Implicit-explicit Runge-Kutta methods for time dependent partial differential equations, Appl. Numer. Math., 25 (1997), 151-167.  doi: 10.1016/S0168-9274(97)00056-1. [9] P. Bi, X. Wu, F. Zhang, K. A. Parton and 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.  doi: 10.1093/oxfordjournals.aje.a009636. [10] S. Boscarino, R. Bürger, P. Mulet, G. Russo and 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.  doi: 10.1137/140967544. [11] S. Boscarino, F. Filbet and G. Russo, High order semi-implicit schemes for time dependent partial differential equations, J. Sci. Comput., 68 (2016), 975-1001.  doi: 10.1007/s10915-016-0168-y. [12] S. Boscarino, P. G. LeFloch and G. Russo, High-order asymptotic-preserving methods for fully nonlinear relaxation problems, SIAM J. Sci. Comput., 36 (2014), A377-A395.  doi: 10.1137/120893136. [13] S. Boscarino and 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.  doi: 10.1137/080713562. [14] S. Boscarino and G. Russo, Flux-explicit IMEX Runge-Kutta schemes for hyperbolic to parabolic relaxation problems, SIAM J. Numer. Anal., 51 (2013), 163-190.  doi: 10.1137/110850803. [15] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Second Ed., Springer, New York, 2012.  doi: 10.1007/978-1-4614-1686-9. [16] M. Brummer-Korvenkontio, A. Vaheri, T. Hovi, C. H. von Bonsdorff, J. Vuorimies, T. Manni, K. Penttinen, N. Oker-Blom and J. Lähdevirta, Nephropathia epidemica: Detection of antigen in bank voles and serologic diagnosis of human infection, J. Infect. Dis., 141 (1980), 131-134.  doi: 10.1093/infdis/141.2.131. [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). doi: 10.1103/PhysRevE.69.021908. [18] R. Bürger, G. Chowell, P. Mulet and L. M. Villada, Modelling the spatial-temporal progression of the 2009 A/H1N1 influenza pandemic in Chile, Math. Biosci. Eng., 13 (2016), 43-65.  doi: 10.3934/mbe.2016.13.43. [19] R. Bürger, R. Ruiz-Baier and 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.  doi: 10.1016/j.matcom.2016.06.002. [20] R. M. Colombo and E. Rossi, Hyperbolic predators versus parabolic preys, Commun. Math. Sci., 13 (2015), 369-400.  doi: 10.4310/CMS.2015.v13.n2.a6. [21] M. Crouzeix, Une méthode multipas implicite-explicite pour l'approximation des équations d'évolution paraboliques, Numer. Math., 35 (1980), 257-276.  doi: 10.1007/BF01396412. [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 and I. Higueras, On stability issues for IMEX schemes applied to 1D scalar hyperbolic equations with stiff reaction terms, Math. Comp., 80 (2011), 2097-2126.  doi: 10.1090/S0025-5718-2011-02463-4. [24] C. 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Numerical solution of the ODE version of (2.1), Model 0, for the initial data (4.1)
Case 1 (Model 1, Scenario 1): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times
Case 2 (Model 2 with $K=1000$, Scenario 1): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times
Case 3 (Model 3, Scenario 1): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times
Cases 1-3: integral quantities $\mathcal{I} (X)$ defined by (4.3) for each compartment obtained by evaluating numerical solutions
Case 4 (Model 1, Scenario 2): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times
Case 5 (Model 2 with $K=1000$, Scenario 2): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times
Case 6 (Model 3, Scenario 2): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times
Cases 4-6: integral quantities $\mathcal{I} (X)$ defined by (4.3) for each compartment obtained by evaluating numerical solutions
Cases 4-6: Moran's index $\mathsf{I}$ defined by (4.4), (4.5) for each compartment obtained by evaluating numerical solutions
Case 8 (Model 3, Scenario 2, periodic variation of parameters): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times
Case 8: (Model 3, Scenario 2, periodic variation of parameters): solutions of Model 0 (left column) and integral quantities $\mathcal{I} (X)$ of Model 3 (right column) defined by (4.3) for each compartment obtained by evaluating numerical solutions
Case 7 (Model 1, order test with smooth solution): approximate total $L^1$ errors $\smash{e_M^{\smash{tot}}}$ and observed convergence rates $\theta_M$
 $M$ 8 16 32 64 128 256 $e_M^{\smash{tot}}\times 10^{3}$ 368.9 383.19 379.01 153.73 34.7 9.14 $\theta_M$ -0.05 0.02 1.3 2.15 1.92 -
 $M$ 8 16 32 64 128 256 $e_M^{\smash{tot}}\times 10^{3}$ 368.9 383.19 379.01 153.73 34.7 9.14 $\theta_M$ -0.05 0.02 1.3 2.15 1.92 -
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