    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  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 & Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085
##### References:
  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.  Google Scholar  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. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  A. Volpert, Vit. Volpert and Vl. Volpert, Traveling Wave Solutions of Parabolic Systems, Translation of Mathematical Monographs, Vol. 140, Amer. Math. Society, Providence, 1994. Google Scholar  V. Volpert, Elliptic Partial Differential Equations, Reaction-Diffusion Equations, 2, Birkhäuser, 2014. doi: 10.1007/978-3-0348-0813-2.  Google Scholar

show all references

##### References:
  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.  Google Scholar  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. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  A. Volpert, Vit. Volpert and Vl. Volpert, Traveling Wave Solutions of Parabolic Systems, Translation of Mathematical Monographs, Vol. 140, Amer. Math. Society, Providence, 1994. Google Scholar  V. Volpert, Elliptic Partial Differential Equations, Reaction-Diffusion Equations, 2, Birkhäuser, 2014. doi: 10.1007/978-3-0348-0813-2.  Google Scholar 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: $f_1 = 0.2, f_2 = 0.1, f_3 = 1.5$, $u_1 = 0.1, u_2 = 0.3, u_3 = 0.6$, $\tau = 4$, $D = 0.0002$
  Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316  Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321  Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319  H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433  Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242  Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116  José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376  Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118  Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074  Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243  Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466  S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435  Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079  Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448  Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247  Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276  Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264  Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273  Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048  Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

2019 Impact Factor: 1.27