# American Institute of Mathematical Sciences

May  2019, 24(5): 2073-2091. doi: 10.3934/dcdsb.2019085

## Delay reaction-diffusion equation for infection dynamics

 1 Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, 199178 Saint Petersburg, Russia 2 Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina Street 8, 119333 Moscow, Russia 3 Département de Mathématiques, Faculté des Sciences, Université de Tlemcen, Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, Tlemcen, BP 119, 13000, Algeria 4 Instituto de Matemática y Fisica, Universidad de Talca, Casilla 747, Talca, Chile 5 Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France 6 INRIA Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France 7 RUDN University, ul. Miklukho-Maklaya 6, Moscow, 117198, Russia

* Corresponding author: Vitaly Volpert

Received  November 2017 Revised  January 2019 Published  May 2019 Early access  March 2019

Nonlinear dynamics of a reaction-diffusion equation with delay is studied with numerical simulations in 1D and 2D cases. Homogeneous in space solutions can manifest time oscillations with period doubling bifurcations and transition to chaos. Transition between two regions with homogeneous oscillations is provided by quasi-waves, propagating solutions without regular structure and often with complex aperiodic oscillations. Dynamics of space dependent solutions is described by a combination of various waves, e.g., bistable, monostable, periodic and quasi-waves.

Citation: Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085
##### References:
 [1] N. Apreutesei, N. Bessonov, V. Volpert and V. Vougalter, Spatial structures and generalized travelling waves for an integro-differential equation, Discrete Continuous Dynam. Systems - B, 13 (2010), 537-557.  doi: 10.3934/dcdsb.2010.13.537. [2] G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Modelling the dynamics of virus infection and immune response in space and time, International Journal of Parallel, Emergent and Distributed Systems, 2017. doi: 10.1080/17445760.2017.1363203. [3] G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Spatiotemporal dynamics of virus infection spreading in tissues, PlosOne, 2016, 0168576. doi: 10.1371/journal.pone.0168576. [4] A. Ducrot and G. Nadin, Asymptotic behaviour of travelling waves for the delayed Fisher KPP equation, J. Differential Equations, 256 (2014), 3115-3140.  doi: 10.1016/j.jde.2014.01.033. [5] D. Duehring and W. Huang, Periodic Traveling Waves for Diffusion Equations with Time Delayed and Non-local Responding Reaction, Journal of Dynamics and Differential Equations, 19 (2007), 457-477.  doi: 10.1007/s10884-006-9048-8. [6] K. Hasik, J. Kopfova, P. Nabelkova and S. Trofimchuk., Traveling waves in the nonlocal KPP-Fisher equation: Different roles of the right and the left interactions, J. Differential Equations, 260 (2016), 6130-6175.  doi: 10.1016/j.jde.2015.12.035. [7] K. Hasik and S. Trofimchuk, Slowly oscillating wavefronts of the KPP-Fisher delayed equation, Discrete Continuous Dynam. Systems, 34 (2014), 3511-3533.  doi: 10.3934/dcds.2014.34.3511. [8] W. Huang, Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response, J. Differential Equations, 244 (2008), 1230-1254.  doi: 10.1016/j.jde.2007.10.001. [9] M. Jankovic, S. Petrovskii and M. Banerjee, Delay driven spatiotemporal chaos in single species population dynamics models, Theoretical Population Biology, 110 (2016), 51-62.  doi: 10.1016/j.tpb.2016.04.004. [10] E. Trofimchuk, V. Tkachenko and S. Trofimchuk, Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay, J. Differential Equations, 245 (2008), 2307-2332.  doi: 10.1016/j.jde.2008.06.023. [11] S. Trofimchuk and V. Volpert, Global continuation of monotone waves for a unimodal bistable reaction-diffusion equation with delay, Nonlinearity, to appear. doi: 10.1137/17M1115587. [12] S. Trofimchuk and V. Volpert, Travelling waves for a bistable reaction-diffusion equation with delay, SIAM J. Math. Anal., 50 (2018), 1175-1199.  doi: 10.1137/17M1115587. [13] A. Volpert, Vit. Volpert and Vl. Volpert, Traveling Wave Solutions of Parabolic Systems, Translation of Mathematical Monographs, Vol. 140, Amer. Math. Society, Providence, 1994. [14] V. Volpert, Elliptic Partial Differential Equations, Reaction-Diffusion Equations, 2, Birkhäuser, 2014. doi: 10.1007/978-3-0348-0813-2.

show all references

##### References:
 [1] N. Apreutesei, N. Bessonov, V. Volpert and V. Vougalter, Spatial structures and generalized travelling waves for an integro-differential equation, Discrete Continuous Dynam. Systems - B, 13 (2010), 537-557.  doi: 10.3934/dcdsb.2010.13.537. [2] G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Modelling the dynamics of virus infection and immune response in space and time, International Journal of Parallel, Emergent and Distributed Systems, 2017. doi: 10.1080/17445760.2017.1363203. [3] G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Spatiotemporal dynamics of virus infection spreading in tissues, PlosOne, 2016, 0168576. doi: 10.1371/journal.pone.0168576. [4] A. Ducrot and G. Nadin, Asymptotic behaviour of travelling waves for the delayed Fisher KPP equation, J. Differential Equations, 256 (2014), 3115-3140.  doi: 10.1016/j.jde.2014.01.033. [5] D. Duehring and W. Huang, Periodic Traveling Waves for Diffusion Equations with Time Delayed and Non-local Responding Reaction, Journal of Dynamics and Differential Equations, 19 (2007), 457-477.  doi: 10.1007/s10884-006-9048-8. [6] K. Hasik, J. Kopfova, P. Nabelkova and S. Trofimchuk., Traveling waves in the nonlocal KPP-Fisher equation: Different roles of the right and the left interactions, J. Differential Equations, 260 (2016), 6130-6175.  doi: 10.1016/j.jde.2015.12.035. [7] K. Hasik and S. Trofimchuk, Slowly oscillating wavefronts of the KPP-Fisher delayed equation, Discrete Continuous Dynam. Systems, 34 (2014), 3511-3533.  doi: 10.3934/dcds.2014.34.3511. [8] W. Huang, Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response, J. Differential Equations, 244 (2008), 1230-1254.  doi: 10.1016/j.jde.2007.10.001. [9] M. Jankovic, S. Petrovskii and M. Banerjee, Delay driven spatiotemporal chaos in single species population dynamics models, Theoretical Population Biology, 110 (2016), 51-62.  doi: 10.1016/j.tpb.2016.04.004. [10] E. Trofimchuk, V. Tkachenko and S. Trofimchuk, Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay, J. Differential Equations, 245 (2008), 2307-2332.  doi: 10.1016/j.jde.2008.06.023. [11] S. Trofimchuk and V. Volpert, Global continuation of monotone waves for a unimodal bistable reaction-diffusion equation with delay, Nonlinearity, to appear. doi: 10.1137/17M1115587. [12] S. Trofimchuk and V. Volpert, Travelling waves for a bistable reaction-diffusion equation with delay, SIAM J. Math. Anal., 50 (2018), 1175-1199.  doi: 10.1137/17M1115587. [13] A. Volpert, Vit. Volpert and Vl. Volpert, Traveling Wave Solutions of Parabolic Systems, Translation of Mathematical Monographs, Vol. 140, Amer. Math. Society, Providence, 1994. [14] V. Volpert, Elliptic Partial Differential Equations, Reaction-Diffusion Equations, 2, Birkhäuser, 2014. doi: 10.1007/978-3-0348-0813-2.
Typical examples of the function $f(u)$. It can be monotone (left) or non-monotone (right)
Period doubling bifurcations for the non-monotone function $f(u)$. Upper row: simple oscillations ($f_3 = 2.5$) and period 2 oscillations ($f_3 = 2$). Lower row: period 4 oscillations ($f_3 = 1.899$) and period 8 oscillations ($f_3 = 1.898$). The dots show the beginning of the periods. The values of parameters: $f_1 = 0, f_2 = 0.1, f_3$ varies, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.5$, $\tau = 2$
Behavior of solutions of equation (1) with periodic boundary conditions for $D = 0.001$ (left), $D = 0.0004$ (middle), $D = 0.0001$ (right). The upper row of the graphs specifies the spatial pattern of solutions at some time $t$. The lower row shows the spatiotemporal structure of solutions with the positions of the maxima of solutions in time. The values of parameters: $f_1 = 0.1, f_2 = 6, f_3 = 6$, $u_1 = 0.1, u_2 = 0.9, u_3 = 0.95$, $\tau = 1.25$, $L = 0.5$
Behavior of solutions of equation (1) with periodic boundary conditions for $D = 10^{-3}$ (left), $D = 10^{-4}$ (middle), $D = 10^{-5}$ (right). The values of parameters: $f_1 = 0.1, f_2 = 0.1, f_3 = 1.9$, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.5$, $\tau = 2$, $L = 0.5$
Evolution of the transition zone in time. Upper row: time oscillations at the center of the interval. Middle row: $u(x, t)$ as a function of $x$ for some fixed $t$. Lower row: positions of the maxima of solutions. The values of parameters: $f_1 = 0.1, f_2 = 3, f_3 = 3$, $u_1 = 0.1, u_2 = 0.9, u_3 = 0.95$, $\tau = 1.5$ (left); $f_1 = 0.1, f_2 = 3, f_3 = 3$, $u_1 = 0.1, u_2 = 0.9, u_3 = 0.95$, $\tau = 2$ (middle); $f_1 = 0.1, f_2 = 6, f_3 = 6$, $u_1 = 0.1, u_2 = 0.9, u_3 = 0.95$, $\tau = 1.5$ (right). $u_0 = 0.2, u_1^0 = 0.001$, $D = 10^{-5}, L = 1$
Quasi-waves for the non-monotone function $f(u)$, $f_3 = 3$ (left); $f_3 = 2.3$ (middle); $f_3 = 1.9$ (right). The figure shows snapshots of solutions (upper row) and location of the maxima of solutions on the $(x, t)$-plane (lower row). The values of parameters: $f_1 = 0.1, f_2 = 0.1, f_3$ varied, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.5$, $\tau = 2$, $u_0 = 0.2, u_1^0 = 0.001$
Quasi-waves for the non-monotone function $f(u)$, $\tau = 5$ (left); $\tau = 7$ (right). Middle column shows the corresponding homogeneous oscillations for $\tau = 5$ (upper) and $\tau = 7$ (lower). The values of parameters: $f_1 = 0.1, f_2 = 0.1, f_3 = 2$, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.5$, $\tau$ varied, $u_0 = 0.2, u_1^0 = 0.001$
Three types of regimes: wave with the minimal speed $c_0$ and a constant value $v_0$ behind the wave (left); the same wave is followed periodic in time and constant in space oscillations (middle); the wave converges to a periodic space structure instead of the constant value (right). The wave and the interval of time periodic oscillations are separated by a transition zone growing linearly in time. The values of parameters: $u_1 = 0.1, u_2 = 0.9, u_3 = 0.95$, $\tau = 2$, $u_0 = 0.2, u_1^0 = 0$; $f_1 = 0.1, f_2 = 0.9, f_3 = 0.9$ (left); $f_1 = 0.1, f_2 = 1.6, f_3 = 1.6$ (middle), $f_1 = 0.1, f_2 = 3, f_3 = 3$ (right)
Different regimes in the case of non-monotone function $f(u)$. In all three cases, the most left is the bistable wave, next to it a periodic wave, the most right is time periodic homogeneous in space solution. The speed of the bistable wave is negative in the left image while the speed of the periodic wave is positive. The speed of the bistable wave is larger than the speed of the periodic wave in the middle image. As a consequence, they merge forming a single periodic bistable wave. The periodic wave is separated from the homogeneous solutions by a quasi-wave in the right image. The values of parameters: $f_1 = 0, f_2 = 0.1, f_3 = 3$, $u_1 = 0.1, u_2 = 0.5, u_3 = 0.9$, $\tau = 2$, $u_0 = 0.8, u_1^0 = 0.001$ (left); $f_1 = 0, f_2 = 0.1, f_3 = 1.2$, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.5$, $\tau = 1.2$, $u_0 = 0.8, u_1^0 = 0.001$ (middle); $f_1 = 0, f_2 = 0.1, f_3 = 3$, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.5$, $\tau = 2$, $u_0 = 0.8, u_1^0 = 0.001$ (right)
Different regimes in the case of non-monotone function $f(u)$. In all three cases, bistable waves are separated from the homogeneous solutions by quasi-waves with different structures. The values of parameters: $f_1 = 0, f_2 = 0.1, f_3 = 3$, $u_1 = 0.1, u_2 = 0.5, u_3 = 0.9$, $\tau = 4$, $u_0 = 0.8, u_1^0 = 0.001$ (left); $f_1 = 0, f_2 = 0.1, f_3 = 3$, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.5$, $\tau = 6$, $u_0 = 0.8, u_1^0 = 0.001$ (middle); $f_1 = 0.1, f_2 = 0.1, f_3 = 3$, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.5$, $\tau = 1$, $u_0 = 0.8, u_1^0 = 0.001$ (right)
The values of parameters: $f_1 = 0.1, f_2 = 0.1, f_3 = 1.6$, $u_1 = 0.1, u_2 = 0.5, u_3 = 0.9$, $\tau = 1$, $u_0 = 0.8, u_1^0 = 0$ (left); $f_1 = 0, f_2 = 0.1, f_3 = 3$, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.5$, $\tau = 6$, $u_0 = 0.8, u_1^0 = 0$ (middle); $f_1 = 0.1, f_2 = 0.1, f_3 = 3$, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.5$, $\tau = 1$, $u_0 = 0.8, u_1^0 = 0$ (right)
Numerical simulation of equation (1) in a square domain with no-flux boundary condition and radially symmetric initial condition. The circular waves propagate from the lower left corner and gradually fill the whole domain. The left image shows a snapshot of solution, the right image the level lines of the same solution in a color code. The middle image shows the maxima of solution in time. Each green surface corresponds to the line of maxima (wave contour) propagating in space. The values of parameters: $f_1 = 0.1, f_2 = 0.1, f_3 = 2$, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.6$, $\tau = 2$, $u_0 = 0.2, u_1^0 = 0.001$, $D = 0.001$
Snapshot of solutions with quasi-wave propagation for $\tau = 2$ (left) and $\tau = 4$ (right). The values of parameters: $f_1 = 0.1, f_2 = 0.1, f_3 = 1.8$, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.5$, $u_0 = 0.2, u_1^0 = 0.001$, $D = 0.0001$
Propagation of a monostable-bistable wave. The initial condition (upper left) has a 2D perturbation. The wave remain 1D (lower left). When a quasi-wave develops between the bistable and the periodic monostable waves, the oscillations become two-dimensional (upper and lower right). The values of parameters: $f_1 = 0.2, f_2 = 0.1, f_3 = 1.5$, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.6$, $\tau = 4$, $D = 0.0002$
 [1] Ning Wang, Zhi-Cheng Wang. Propagation dynamics of a nonlocal time-space periodic reaction-diffusion model with delay. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1599-1646. doi: 10.3934/dcds.2021166 [2] Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure and Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319 [3] Maria do Carmo Pacheco de Toledo, Sergio Muniz Oliva. A discretization scheme for an one-dimensional reaction-diffusion equation with delay and its dynamics. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1041-1060. doi: 10.3934/dcds.2009.23.1041 [4] Shangbing Ai, Wenzhang Huang, Zhi-An Wang. Reaction, diffusion and chemotaxis in wave propagation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 1-21. doi: 10.3934/dcdsb.2015.20.1 [5] Wenjing Wu, Tianli Jiang, Weiwei Liu, Jinliang Wang. Threshold dynamics of a reaction-diffusion cholera model with seasonality and nonlocal delay. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022099 [6] Anouar El Harrak, Hatim Tayeq, Amal Bergam. A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062 [7] Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407 [8] Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246 [9] Tarik Mohammed Touaoula, Mohammed Nor Frioui, Nikolay Bessonov, Vitaly Volpert. Dynamics of solutions of a reaction-diffusion equation with delayed inhibition. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2425-2442. doi: 10.3934/dcdss.2020193 [10] Zhi-Xian Yu, Rong Yuan. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 709-728. doi: 10.3934/dcdsb.2010.13.709 [11] M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079 [12] Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media. Networks and Heterogeneous Media, 2016, 11 (3) : 369-393. doi: 10.3934/nhm.2016001 [13] Luisa Malaguti, Cristina Marcelli, Serena Matucci. Continuous dependence in front propagation of convective reaction-diffusion equations. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083 [14] Tarik Mohammed Touaoula. Global dynamics for a class of reaction-diffusion equations with distributed delay and neumann condition. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2473-2490. doi: 10.3934/cpaa.2020108 [15] Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242 [16] Yacheng Liu, Runzhang Xu. Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 171-189. doi: 10.3934/dcdsb.2007.7.171 [17] Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681 [18] Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach. Mathematical Biosciences & Engineering, 2015, 12 (1) : 41-69. doi: 10.3934/mbe.2015.12.41 [19] Narcisa Apreutesei, Vitaly Volpert. Reaction-diffusion waves with nonlinear boundary conditions. Networks and Heterogeneous Media, 2013, 8 (1) : 23-35. doi: 10.3934/nhm.2013.8.23 [20] Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382

2020 Impact Factor: 1.327