Letter to the editors
Shigui Ruan
Mathematical Biosciences & Engineering 2009, 6(1): 207-208 doi: 10.3934/mbe.2009.6.207
Dear Editors:
I request that Mathematical Biosciences and Engineering publish this Letter of Correction to an article published in the journal for which I was the corresponding author, "The Effect of Global Travel on the Spread of SARS'' (2006;3(1):205-218). The goal of this article was to study the effect of global travel on the geographic spread of SARS. A multiregional compartmental model was proposed, mathematically analyzed, and numerically simulated to study how SARS spread out from Guangdong, China. The article consists of six sections: (1) an introduction, (2) a background section on medical geography theory, (3) the mathematical model, (4) mathematical analysis and results, (5) numerical simulations, and (6) discussion. Sections 3, 4, and 5 are the main parts of the article which are all original work.

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Interaction of diffusion and delay
Karl Peter Hadeler Shigui Ruan
Discrete & Continuous Dynamical Systems - B 2007, 8(1): 95-105 doi: 10.3934/dcdsb.2007.8.95
For reaction-diffusion equations with delay, the joint effects of diffusion and delay are studied. In particular, for two-dimensional systems where only the interaction between species is delayed, the interdependence of stability against delay and against diffusion (Turing instability) can be clearly exhibited. Turing instabilities occur largely independent of delay. But periodic oscillations, constant in space or with low spatial frequency, can be achieved via increasing the delay or changing the diffusion rates.
keywords: matrix stability. time delay Hopf bifurcation Turing instability Reaction-diffusion equations
Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting
Jicai Huang Sanhong Liu Shigui Ruan Xinan Zhang
Communications on Pure & Applied Analysis 2016, 15(3): 1041-1055 doi: 10.3934/cpaa.2016.15.1041
Recently, we (J. Huang, Y. Gong and S. Ruan, Discrete Contin. Dynam. Syst. B 18 (2013), 2101-2121) showed that a Leslie-Gower type predator-prey model with constant-yield predator harvesting has a Bogdanov-Takens singularity (cusp) of codimension 3 for some parameter values. In this paper, we prove analytically that the model undergoes Bogdanov-Takens bifurcation (cusp case) of codimension 3. To confirm the theoretical analysis and results, we also perform numerical simulations for various bifurcation scenarios, including the existence of two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1.
keywords: homoclinic bifurcation. constant-yield harvesting Bogdanov-Takens bifurcation of codimension 3 Hopf bifurcaton Predator-prey model
Preface on the special issue of Discrete and Continuous Dynamical Systems- Series B in honor of Chris Cosner on the occasion of his 60th birthday
Robert Stephen Cantrell Suzanne Lenhart Yuan Lou Shigui Ruan
Discrete & Continuous Dynamical Systems - B 2014, 19(10): i-ii doi: 10.3934/dcdsb.2014.19.1i
Chris Cosner turned 60 on June 3, 2012 and now, at age 62, continues his stellar career at the interface of mathematics and biology. He received his Ph.D. in 1977 at the University of California, Berkeley under the direction of Murray Protter, winning the Bernard Friedman prize for the best dissertation in applied mathematics. From 1977 until 1982 he was on the faculty of Texas A&M University. In 1982 he left A&M to join the faculty of the Department of Mathematics of the University of Miami as Associate Professor, rising to the rank of Professor in 1988. The academic year 2013-2014 marked his 32nd year of distinguished service to the University of Miami and its research and pedagogical missions.

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Bifurcations of an SIRS epidemic model with nonlinear incidence rate
Zhixing Hu Ping Bi Wanbiao Ma Shigui Ruan
Discrete & Continuous Dynamical Systems - B 2011, 15(1): 93-112 doi: 10.3934/dcdsb.2011.15.93
The main purpose of this paper is to explore the dynamics of an epidemic model with a general nonlinear incidence $\beta SI^p/(1+\alpha I^q)$. The existence and stability of multiple endemic equilibria of the epidemic model are analyzed. Local bifurcation theory is applied to explore the rich dynamical behavior of the model. Normal forms of the model are derived for different types of bifurcations, including Hopf and Bogdanov-Takens bifurcations. Concretely speaking, the first Lyapunov coefficient is computed to determine various types of Hopf bifurcations. Next, with the help of the Bogdanov-Takens normal form, a family of homoclinic orbits is arising when a Hopf and a saddle-node bifurcation merge. Finally, some numerical results and simulations are presented to illustrate these theoretical results.
keywords: nonlinear incidence rate stability Hopf bifurcation Bogdanov-Takens bifurcation. SIRS epidemic model
Robert Stephen Cantrell Suzanne Lenhart Yuan Lou Shigui Ruan
Discrete & Continuous Dynamical Systems - B 2015, 20(6): i-iii doi: 10.3934/dcdsb.2015.20.6i
The movement and dispersal of organisms have long been recognized as key components of ecological interactions and as such, they have figured prominently in mathematical models in ecology. More recently, dispersal has been recognized as an equally important consideration in epidemiology and in environmental science. Recognizing the increasing utility of employing mathematics to understand the role of movement and dispersal in ecology, epidemiology and environmental science, The University of Miami in December 2012 held a workshop entitled ``Everything Disperses to Miami: The Role of Movement and Dispersal in Ecology, Epidemiology and Environmental Science" (EDM).

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P. Magal Shigui Ruan
Discrete & Continuous Dynamical Systems - B 2007, 8(1): i-ii doi: 10.3934/dcdsb.2007.8.1i
This special issue is the proceedings of the International Workshop on Differential Equations in Mathematical Biology held in Le Havre, France, July 11-13, 2005. The workshop brought together internationals researchers in Differential Equations and Mathematical Biology to communicate with each other about their current work. The topics of the workshop included various types of differential equations and their applications to biology and other related subjects, such as, ecology, epidemiology, medicine, etc. There were more than 60 participants came from Algeria, Canada, Cameroun, Finland, France, Germany, Hungary, Italy, Japan, Lithuania, Mexico, The Netherlands, Portugal, Romania, Spain, South Africa, UK, and USA. The ple- nary speakers were Pierre Auger (IRD Bondy, France), Josef Hofbauer (University College London, UK), Michel Langlais (Bordeaux 2, France), Hal Smith (Arizona State, USA), Horst Thieme (Arizona State, USA), Glenn Webb (Vanderbilt, USA) and Jianhong Wu (York, Canada). There were also more than 40 presentations by other participants.
    The 17 articles which appear in this special issue are from the participants of the Workshop and from other leading researchers in these subjects. Topics include malaria intra-host models, stem cell dynamics, tumor invasion, reaction-diffusion systems for competition and predation, traveling waves, optimal control in age structured models, host-parasitoid models, predator-prey models, HIV infection, immune system memory, bacteria infection, innate immune response, and antibiotic treatment.

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A mathematical model for the seasonal transmission of schistosomiasis in the lake and marshland regions of China
Yingke Li Zhidong Teng Shigui Ruan Mingtao Li Xiaomei Feng
Mathematical Biosciences & Engineering 2017, 14(5&6): 1279-1299 doi: 10.3934/mbe.2017066

Schistosomiasis, a parasitic disease caused by Schistosoma Japonicum, is still one of the most serious parasitic diseases in China and remains endemic in seven provinces, including Hubei, Anhui, Hunan, Jiangsu, Jiangxi, Sichuan, and Yunnan. The monthly data of human schistosomiasis cases in Hubei, Hunan, and Anhui provinces (lake and marshland regions) released by the Chinese Center for Disease Control and Prevention (China CDC) display a periodic pattern with more cases in late summer and early autumn. Based on this observation, we construct a deterministic model with periodic transmission rates to study the seasonal transmission dynamics of schistosomiasis in these lake and marshland regions in China. We calculate the basic reproduction number $R_{0}$, discuss the dynamical behavior of solutions to the model, and use the model to fit the monthly data of human schistosomiasis cases in Hubei. We also perform some sensitivity analysis of the basic reproduction number $R_{0}$ in terms of model parameters. Our results indicate that treatment of at-risk population groups, improving sanitation, hygiene education, and snail control are effective measures in controlling human schistosomiasis in these lakes and marshland regions.

keywords: Seasonal schistosomiasis model basic reproduction number extinction uniform persistence
Chris Cosner Yuan Lou Shigui Ruan Wenxian Shen
Discrete & Continuous Dynamical Systems - B 2017, 22(3): ⅰ-ⅱ doi: 10.3934/dcdsb.201703i
Traveling wave solutions for time periodic reaction-diffusion systems
Wei-Jian Bo Guo Lin Shigui Ruan
Discrete & Continuous Dynamical Systems - A 2018, 38(9): 4329-4351 doi: 10.3934/dcds.2018189

This paper deals with traveling wave solutions for time periodic reaction-diffusion systems. The existence of traveling wave solutions is established by combining the fixed point theorem with super- and sub-solutions, which reduces the existence of traveling wave solutions to the existence of super- and sub-solutions. The asymptotic behavior is determined by the stability of periodic solutions of the corresponding initial value problems. To illustrate the abstract results, we investigate a time periodic Lotka-Volterra system with two species by presenting the existence and nonexistence of traveling wave solutions, which connect the trivial steady state to the unique positive periodic solution of the corresponding kinetic system.

keywords: Super- and sub-solutions asymptotic behavior Lotka-Volterra competitive system

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