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March  2021, 14(3): 835-850. doi: 10.3934/dcdss.2020387

## Traveling wave solution for a diffusive simple epidemic model with a free boundary

 1 Department of Applied Mathematics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan 2 Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba 278-8510, Japan

* Corresponding author: Takeo Ushijima

Received  January 2019 Revised  March 2020 Published  March 2021 Early access  June 2020

In this paper, we proved existence and nonexistence of traveling wave solution for a diffusive simple epidemic model with a free boundary in the case where the diffusion coefficient $d$ of susceptible population is zero and the basic reproduction number is greater than 1. We obtained a curve in the parameter plane which is the boundary between the regions of existence and nonexistence of traveling wave. We numerically observed that in the region where the traveling wave exists the disease successfully propagate like traveling wave but in the region of no traveling wave disease stops to invade. We also numerically observed that as $d$ increases the speed of propagation slows down and the parameter region of propagation narrows down.

Citation: Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387
##### References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1. [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics, Lecture Notes in Math., Springer, Berlin, 446 (1975), 5-49. [3] W. W. Ding, Y. H. Du and X. Liang, Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 1: Continuous initial functions, J. Differential Equations, 262 (2017), 4988-5021.  doi: 10.1016/j.jde.2017.01.016. [4] Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic mode with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089. [5] Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary. Ⅱ, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011. [6] Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568. [7] Y. H. Du, B. D. Lou and M. L. Zhou, Nonlinear diffusion problems with free boundaries: Convergence, transition speed and zero number arguments, SIAM J. Math. Anal., 47 (2015), 3555-3584.  doi: 10.1137/140994848. [8] Y. H. Du, H. Matsuzawa and M. L. Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pure Appl. (9), 103 (2015), 741-787.  doi: 10.1016/j.matpur.2014.07.008. [9] A. Ducrot and T. Giletti, Convergence to a pulsating travelling wave for an epidemic reaction-diffusion system with non-diffusive susceptible population, J. Math. Biol., 69 (2014), 533-552.  doi: 10.1007/s00285-013-0713-3. [10] A. Ducrot, T. Giletti and H. Matano, Spreading speeds for multidimensional reaction diffusion system of the prey-predator type, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 137, 34 pp. doi: 10.1007/s00526-019-1576-2. [11] J. Ge, K. I. Kim, Z. G. Lin and H. P. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035. [12] Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Mathematical Models and Methods in Applied Sciences, 5 (1995), 935-966.  doi: 10.1142/S0218202595000504. [13] A. Källén, Thresholds and traveling waves in an epidemic model for ravies, Nonlinear Analysis, Theory & Applications, 8 (1984), 851-856.  doi: 10.1016/0362-546X(84)90107-X. [14] Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., 428 (2015), 43-76.  doi: 10.1016/j.jmaa.2015.02.051. [15] Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differential Equations, 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017. [16] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc., A115 (1927), 700-721. [17] K. I. Kim, Z. G. Lin and Q. Y. Zhang, An SIR epidemic model with free boundary, Nonlinear Analysis: Real World Applications, 14 (2013), 1992-2001.  doi: 10.1016/j.nonrwa.2013.02.003. [18] H. Murakawa, Reaction-diffusion system approximation to degenerate parabolic systems, Nonlinearity, 20 (2007), 2319-2332.  doi: 10.1088/0951-7715/20/10/003. [19] S. X. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014. [20] J. Yang and B. D. Lou, Traveling wave solutions of competitive models with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817-826.  doi: 10.3934/dcdsb.2014.19.817. [21] M. Zhu, X. F. Guo and Z. G. Lin, The risk index for an SIR epidemic model and spatial spreading of the infectious disease, Math Biosci Eng., 14 (2017), 1565-1583.  doi: 10.3934/mbe.2017081.

show all references

##### References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1. [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics, Lecture Notes in Math., Springer, Berlin, 446 (1975), 5-49. [3] W. W. Ding, Y. H. Du and X. Liang, Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 1: Continuous initial functions, J. Differential Equations, 262 (2017), 4988-5021.  doi: 10.1016/j.jde.2017.01.016. [4] Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic mode with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089. [5] Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary. Ⅱ, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011. [6] Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568. [7] Y. H. Du, B. D. Lou and M. L. Zhou, Nonlinear diffusion problems with free boundaries: Convergence, transition speed and zero number arguments, SIAM J. Math. Anal., 47 (2015), 3555-3584.  doi: 10.1137/140994848. [8] Y. H. Du, H. Matsuzawa and M. L. Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pure Appl. (9), 103 (2015), 741-787.  doi: 10.1016/j.matpur.2014.07.008. [9] A. Ducrot and T. Giletti, Convergence to a pulsating travelling wave for an epidemic reaction-diffusion system with non-diffusive susceptible population, J. Math. Biol., 69 (2014), 533-552.  doi: 10.1007/s00285-013-0713-3. [10] A. Ducrot, T. Giletti and H. Matano, Spreading speeds for multidimensional reaction diffusion system of the prey-predator type, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 137, 34 pp. doi: 10.1007/s00526-019-1576-2. [11] J. Ge, K. I. Kim, Z. G. Lin and H. P. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035. [12] Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Mathematical Models and Methods in Applied Sciences, 5 (1995), 935-966.  doi: 10.1142/S0218202595000504. [13] A. Källén, Thresholds and traveling waves in an epidemic model for ravies, Nonlinear Analysis, Theory & Applications, 8 (1984), 851-856.  doi: 10.1016/0362-546X(84)90107-X. [14] Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., 428 (2015), 43-76.  doi: 10.1016/j.jmaa.2015.02.051. [15] Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differential Equations, 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017. [16] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc., A115 (1927), 700-721. [17] K. I. Kim, Z. G. Lin and Q. Y. Zhang, An SIR epidemic model with free boundary, Nonlinear Analysis: Real World Applications, 14 (2013), 1992-2001.  doi: 10.1016/j.nonrwa.2013.02.003. [18] H. Murakawa, Reaction-diffusion system approximation to degenerate parabolic systems, Nonlinearity, 20 (2007), 2319-2332.  doi: 10.1088/0951-7715/20/10/003. [19] S. X. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014. [20] J. Yang and B. D. Lou, Traveling wave solutions of competitive models with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817-826.  doi: 10.3934/dcdsb.2014.19.817. [21] M. Zhu, X. F. Guo and Z. G. Lin, The risk index for an SIR epidemic model and spatial spreading of the infectious disease, Math Biosci Eng., 14 (2017), 1565-1583.  doi: 10.3934/mbe.2017081.
Parameter regions for existence and non existence of traveling wave ($d = 0$): (left) $\gamma \lambda$ plane, (right) $\gamma a$ plane
$uv$ phase plane for (2.5), $a>a^*$ (left), $a < a^*$ (right)
Numerical shooting in $uv$ phase plane (left), Profile of the traveling wave (right) $d = 0, \gamma = 0.3, a = 0.05 (\lambda \sim 0.05128)$
Evolution of the solutions in spreading case ($\gamma = 0.3$): Left column; $\lambda = 0.054159$, $T = 100$, ① $t = T$, ② $t = 2T$, ③ $t = 3T$, ④ $t = 4T$, ⑤ $t = 5T$, (top left) $d = 0$, (middle left) $d = 0.1$, (bottom left) $d = 1$. Right column; (top right) Profile of traveling wave $\lambda = 0.054159$, $d = 0$, (middle right) $\lambda = 0.254297$, $d = 0$, support of $I_0$ is the origin
Evolution of the solutions in vanishing case ($\gamma = 0.3$, $d = 0$): (left) $\lambda = 0.027$, (right) $\lambda = 0.281066$, support of $I_0$ is fairly large
Estimated speed of propagation: $\Box\ d = 0, \bigcirc\ 0.1, \ast\ 1$, (left) speed of propagation versus $\lambda$, $\gamma = 0.5$, (right) speed of propagation versus $\gamma$, $\lambda = 0.4$
Vanishing versus spreading diagram: $\times$ vanishing, $+$ spreading, horizontal axis $\gamma = 1/R_0$, vertical axis $\lambda = 1/\mu$, (top left) $d = 0$, (top right) $d = 0.1$, (bottom) $d = 1$
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