Physica A 403 (2014) 100–109 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Pattern formation of an epidemic model with time delay Jing Li a , Gui-Quan Sun b,c,∗ , Zhen Jin b,c a Department of Mathematics, North University of China, Taiyuan, Shan’xi 030051, People’s Republic of China b Complex Systems Research Center, Shanxi University, Taiyuan, Shan’xi 030006, People’s Republic of China c School of Mathematical Sciences, Shanxi University, Taiyuan, Shan’xi 030006, People’s Republic of China highlights • We presented an epidemic model with spatial diffusion and time delay. • We find two different types of instability. • Delay can induce regular patterns. • The interaction of diffusion and time delay may give rise to rich dynamic. article info Article history: Received 17 December 2013 Received in revised form 21 January 2014 Available online 22 February 2014 Keywords: Epidemic model Spatial diffusion Time delay Pattern formation abstract One of the central issues in epidemiology is the study of the distribution of disease. And time delay widely exists in the process of disease spread. Thus, in this paper, we presented an epidemic model with spatial diffusion and time delay. By mathematical analysis, we find two different types of instability. One is the diffusion induced instability, and the other one is delay induced instability. Moreover, we derive the corresponding patterns by performing a series of numerical simulations. The obtained results show that the interaction of diffu- sion and time delay may give rise to rich dynamics in epidemic systems. © 2014 Elsevier B.V. All rights reserved. 1. Introduction In recent years, with environmental pollution, ecological destruction and frequent population flow, the whole world major infectious diseases often breakout. Like the H7N9 avian flu, which take place in the year 2013 from February to July and have some bad effect [1–5]. Therefore it is rather important for us to research the infectious disease, in order to predict the development trend of the infectious disease and further control the epidemic. There are some works on epidemic models without spatial factor [6–9]. However, many infectious diseases are affected by the spatial distribution. In the real world, human, animal, plant and other species are to survive in the spatial environment. In order to be more close to the reality, in the study of infectious disease, it is necessary for us to consider the evolution and distribution of the infected in the space. And this influence can be reflected in the determination of reaction–diffusion models. On the other hand, in the study of infectious disease, one of the most important goals is to predict the trends of disease spreading in space. In other words, the spatial distribution of infectious diseases is very important [10–15]. From the clinical and observational experiment we found, most infectious diseases are infected, and then appear some symptoms needing a period of time (namely incubation period), such as atypical pneumonia SIRS, H1N1 flu, smallpox, which requires us to consider joining the time delay in the model [16–19]. Time delay means that the changes of t moment depend ∗ Corresponding author at: Complex Systems Research Center, Shanxi University, Taiyuan, Shan’xi 030006, People’s Republic of China. E-mail address: gquansun@126.com (G.-Q. Sun). http://dx.doi.org/10.1016/j.physa.2014.02.025 0378-4371/© 2014 Elsevier B.V. All rights reserved.