# American Institute of Mathematical Sciences

• Previous Article
Sex-biased prevalence in infections with heterosexual, direct, and vector-mediated transmission: a theoretical analysis
• MBE Home
• This Issue
• Next Article
Mathematical analysis of a weather-driven model for the population ecology of mosquitoes
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:

show all references

##### References:
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 -
 [1] Sihong Shao, Huazhong Tang. Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 623-640. doi: 10.3934/dcdsb.2006.6.623 [2] Jan Haskovec, Ioannis Markou. Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime. Kinetic & Related Models, 2020, 13 (4) : 795-813. doi: 10.3934/krm.2020027 [3] Xueying Wang, Drew Posny, Jin Wang. A reaction-convection-diffusion model for cholera spatial dynamics. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2785-2809. doi: 10.3934/dcdsb.2016073 [4] Wenjuan Zhai, Bingzhen Chen. A fourth order implicit symmetric and symplectic exponentially fitted Runge-Kutta-Nyström method for solving oscillatory problems. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 71-84. doi: 10.3934/naco.2019006 [5] Yoon-Sik Cho, Aram Galstyan, P. Jeffrey Brantingham, George Tita. Latent self-exciting point process model for spatial-temporal networks. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1335-1354. doi: 10.3934/dcdsb.2014.19.1335 [6] Cédric Wolf. A mathematical model for the propagation of a hantavirus in structured populations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1065-1089. doi: 10.3934/dcdsb.2004.4.1065 [7] Daniil Kazantsev, William M. Thompson, William R. B. Lionheart, Geert Van Eyndhoven, Anders P. Kaestner, Katherine J. Dobson, Philip J. Withers, Peter D. Lee. 4D-CT reconstruction with unified spatial-temporal patch-based regularization. Inverse Problems & Imaging, 2015, 9 (2) : 447-467. doi: 10.3934/ipi.2015.9.447 [8] Da Xu. Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2389-2416. doi: 10.3934/dcdsb.2017122 [9] Aniello Raffaele Patrone, Otmar Scherzer. On a spatial-temporal decomposition of optical flow. Inverse Problems & Imaging, 2017, 11 (4) : 761-781. doi: 10.3934/ipi.2017036 [10] Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124 [11] Igor Pažanin, Marcone C. Pereira. On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption. Communications on Pure & Applied Analysis, 2018, 17 (2) : 579-592. doi: 10.3934/cpaa.2018031 [12] ShinJa Jeong, Mi-Young Kim. Computational aspects of the multiscale discontinuous Galerkin method for convection-diffusion-reaction problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020101 [13] Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679 [14] Angelamaria Cardone, Zdzisław Jackiewicz, Adrian Sandu, Hong Zhang. Construction of highly stable implicit-explicit general linear methods. Conference Publications, 2015, 2015 (special) : 185-194. doi: 10.3934/proc.2015.0185 [15] Yachun Tong, Inkyung Ahn, Zhigui Lin. Effect of diffusion in a spatial SIS epidemic model with spontaneous infection. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020273 [16] Benedetto Bozzini, Deborah Lacitignola, Ivonne Sgura. Morphological spatial patterns in a reaction diffusion model for metal growth. Mathematical Biosciences & Engineering, 2010, 7 (2) : 237-258. doi: 10.3934/mbe.2010.7.237 [17] Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173 [18] Sebastian Aniţa, William Edward Fitzgibbon, Michel Langlais. Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 805-822. doi: 10.3934/dcdsb.2009.11.805 [19] Wen Tan, Chunyou Sun. Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6035-6067. doi: 10.3934/dcds.2017260 [20] Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297

2018 Impact Factor: 1.313