生物数学最新进展学术研讨会

Workshop on Recent Advances in Mathematical Biology

2015年11月20--22日, 同济大学

为了交流生物数学领域的最新研究成果和学术发展动态，促进学术交流与合作，经协商决定在同济大学数学系举行为期三天（11月20-22，其中 20号报到）的生物数学最新进展研讨会。围绕流行病学、生态学、病毒动力学、生物医学等领域的实际问题，深入探讨动力系统最新研究成果及其在生物数学理论研究中的重要作用。研讨会的主题包括传染病的防治、生物模型的建模、模型的动力学分析及数据分析等。我们真诚地邀请您参加这次会议。

组织委员会: 舒洪英、张凤琴、宋永利、原三领、娄本东。

会议地点：同济大学数学系致远楼102-107

会议组委会、同济大学数学系

2015年11月

会议日程安排

报告摘要

Prevalence of stable periodic solutions for Duffing equations

储继峰

河海大学

jifengchu@126.com

We study the prevalence of stable periodic solutions of the Duffing equations for external force which guarantees the existence of periodic solutions. We use the anti-periodic eigenvalues of the linearized equations to characterize the conditions. Both the dissipative case and the conservative case are considered.

网络传播动力学进展与问题

傅新楚

上海大学

xcfu@shu.edu.cn

本报告先简要介绍流行病建模的几种典型方法，包括随机接触过程 (Contact process)，平均场近似 (Mean-field approximation)，对逼近 (Pair approximation)，非均匀网络 (Heterogeneous network)，淬火平均场理论 (Quenched mean-field theory)，渗流模型 (Percolation modelling)。然后简述最有代表性的三种网络传播模型，即2001年由Pastor-Satorras和Vespignani提出的平均场模型；2002年由Newman提出的渗流模型以及相关的分支过程方法和概率生成函数方法；2003年由Wang等人提出的离散概率模型，该模型的特点是面向网络的连接矩阵来分析疾病传播过程，直观而实用。最后介绍我们的相关研究进展和有待进一步研究的问题。

A Mathematical Model Verifying Potent Oncolytic Efficacy of M1 Virus

郭志明

广州大学

guozm@gzhu.edu.cn

Motivated by the latest findings in a recent medical experiment which identify a naturally occurring alphavirus (M1) as a novel selective killer targeting zinc-finger antiviral protein (ZAP)-deficient cancer cells, we propose a mathematical model to illustrate the growth of normal cells, tumor cells and the M1 virus with limited nutrient. In order to better understand biological mechanisms, we discuss two cases of the model, without competition and with competition. In the first part, the explicit threshold conditions for the persistence of normal cells and tumor cells are obtained accompanying with the biological explanations. The second part indicates that when competing with tumor cells, the normal cells will extinct if M1 virus is ignored; Whereas, when M1 virus is considered, the growth trend of normal cells is similar to the one without competition, and the minimum effective dosage of medication is explicitly found which is not reported in references. Furthermore, numerical simulations are given to support our results.

Evaluating the Impact of Test-and-Treat on the HIV Epidemic

among MSM in China Using a Mathematical Model

韩丽涛

中国人民大学

hanlitao@263.net

Background: Various studies have modeled the impact of test-and-treat policies on the HIV epidemics worldwide. However, few modeling studies have taken into account China’s context. To understand the potential effect of test-and-treat on the HIV epidemic among MSM in China, we developed a mathematical model to evaluate the impact of the strategy. Method: Based on the natural history of the CD4 count of people living with HIV (PLHIV), we constructed a dynamic compartmental model of HIV transmission among Chinese MSM to project the number of new HIV infection and prevalence over 10 years. We predicted the annual number of new HIV infections and the total number of MSM living with HIV and AIDS (based on Beijing data) between 2010 and 2022 under the following conditions: (1) current practice (testing rate of 50% and ART coverage of 39%); (2) both testing rate and ART coverage increasing to 70% in 2013; (3) both testing rate and ART coverage increasing to 90% in 2013; and (4) testing rate and ART coverage increasing by 5% each year until 90% since 2013. Results: Based on our model, if HIV test-and-treat policy was implemented among Chinese MSM, the total number of new HIV infections over 10 years (2013-2022) would be reduced by 50.6-70.9% compared with the current practice. When ART coverage for PLHIV increases to 57.5%, the turning point would occur on the curve of HIV new infections by 2015. A 25% reduction in annual number of new HIV infections by 2015 may be achieved if the testing rate increases from 50% to 70% and treatment coverage for PLHIV increases to 55%. Conclusion: Implementation of the test-and-treat strategy may significantly reduce new HIV infections among MSM in China. Great efforts need to be made to scale up HIV testing and ART coverage among Chinese MSM.

Bifurcation and traveling wave solutions for the Fokas equation

李继彬

浙江师范大学

lijb@zjnu.cn

In this talk, we discuss bifurcation and traveling wave solutions for the Fokas equation.

Through investigating the dynamical behavior with phase space analysis, we may derive all possible exact traveling wave solutions, including compactons, periodic pekon wave solutions, and smooth solitary wave solutions.

Finding the role of time delays in dynamical systems

林伟

复旦大学

wlin@fudan.edu.cn

Time delays are omnipresently observed in many nature and artificial systems including physical, biological, and chemical systems. Naturally, two kinds of questions arise: “How to identify the time delays when a certain amount of datasets are obtained from the experiments or real world systems?” and “How to characterize the intrinsic roles of time delays that are played in coupled network systems?” In this talk, we introduce recent works that address the previous two questions, and show the significance of time delays in dealing with various systems of physical/biological significance.

Global stability of two classes of SIR-SVS epidemic models

with nonlinear incidence and vaccination

李建全

西安空军工程大学

jianq_li@263.net

Two SIR-SVS epidemic models with nonlinear incidence and vaccination are investigated, where one of the two models assumes that the rate of a vaccinated individual losing immunity depends on the vaccine age and is expressed by a function with general form, the other assumes that, before the vaccine begins to wane, there is a period during which the vaccinated individuals have the complete immunity against the infection. The two models are described by a partial differential system and a delay differential system, respectively. Their basic reproduction numbers are found, and their global stability is proved by constructing the appropriate Lyapunov functionals. The obtained results show that, when their basic reproduction numbers are not greater than one, their disease-free equilibria are globally stable; when their basic reproduction numbers are greater than one, their disease-free equilibria are unstable, and both of them have the unique endemic equilibria, which are globally stable. And it is illustrated that the basic reproduction numbers of the two models can be expressed with the same expression.

西尼罗河病毒扩散边沿及其基本再生数

林支桂

扬州大学

zglin@yzu.edu.cn

我们用反应扩散方程组描述西尼罗河病毒的空间扩散，用自由边界表示病毒扩散的边沿。为了检查空间特征对病毒扩散的影响，我们定义了四个基本再生数，分别对应于常微分方程组问题、具齐次Neumann问题，齐次Dirichlet问题和自由边界问题。结果表明，在高风险区域，如果感染区域范围大或者扩散慢，病毒将蔓延；在低风险区域，小的初始感染病例，小的感染范围和大的扩散速率有利于病毒的消退。当病毒蔓延时我们证明了其空间扩散速度接近于一个常数。

Modelling and analysis of dynamics for a 3D mixed Lorenz system

with a damped term

李先义

扬州大学

mathxyli@yzu.edu.cn

The work in this talk consists of two parts. The one is modelling. After a method of classification for three dimensional (3D) autonomous chaotic systems and a concept of mixed Lorenz system are introduced, a mixed Lorenz system with a damped term is presented. The other is the analysis for dynamical properties of this model. First, its local stability and bifurcation in its parameter space are in detail considered. Then, the existence of its homoclinic and heteroclinic orbits, and the existence of singularly degenerate heteroclinic cycles, are studied by rigorous theoretical analysis. Finally, by using the Poincare compactification for polynomial vector fields in R^3, a global analysis of this system near and at infinity is presented, including the complete description of its dynamics on the sphere near and at infinity. Simulations corroborate corresponding theoretical results. In particular, a possibly new mechanism behind the creation of chaotic attractors, consisting of the change for the dimensional number of stable manifold of the saddle at the origin as the parameter b crosses the null value, is proposed. Based on the knowledge of this mechanism, some different structure types of chaotic attractors are numerically found in the case of small b>0.

Modeling of entry-exit screening measures on spreading of infectious disease

刘胜强

哈尔滨工业大学

sqliu@hit.edu.cn

A multi-patch SEIQR epidemic model is formulated to investigate the long-term impact of entry-exit screening measures on the spread and control of infectious diseases. A threshold dynamics determined by the basic reproduction number $/Re_0$ is established: the disease can be eradicated if $/Re_0<1$, while the disease persists if $/Re_0>1$. Six different screening strategies are explored to examine how the screening impacts the control of infectious diseases. We find that it is crucial to screen travelers from and to high-risk patches and it is not necessary to implement screening in all connected patches, and the minimum number of patches that should implement screening depends critically on the dispersal rates and the success detect rate of screening measures. 报告源自论文：王新新, 刘胜强, 王林,张巍巍, An epidemic patchy model with entry-exit screening,Bulletin of Mathematical Biology, 77(2015): 1237-1255

Age-structured delay models in population growth and subject to predation

楼一均

香港理工大学

yijun.lou@polyu.edu.hk

In this talk, I am going to present two models, one describing the population growth with intra-specific competition and the other addressing the predator-prey interaction of age-structured populations. These two delay models are reduced from the PDE equation and turn to be incorporating novel structures. The models will be analyzed from the point view of dynamical systems. This talk is based on joint work with Profs. Jian Fang and Stephen Gourley.

一类方程右端相互切换的种群动力学模型的多尺度研究

倪明康

华东师范大学

xiaovikdo@163.com

众多生物种群模型往往含有多个参数，在进行无量纲变化之后有时可以化成含小参数的奇摄动动力系统。我们将讨论一类方程右端产生切换的生物种群稳态模型，其中小参数为扩散系数。小参数的出现导致了对模型的研究需要引入不同尺度，并在不同的尺度空间进行讨论。本文首先用定性分析的方法得到了该种群稳态模型具有纯边界层解、阶梯状解、脉冲状解和由它们不同组合而成的复合解。随后我们利用空间对照结构理论证明了上述解的存在性，构造了这些解的一致有效渐近解，并给出了渐近解的余项估计。

基于脉冲微分方程的进化动力学

孟新柱

山东科技大学

ty7473@sohu.com

进化动力学是国际热点研究领域之一，微分方程在进化适应动力系统中的应用得到越来越多专家学者的重视。经典的研究方法是根据自治微分系统稳定的平衡点建立进化变异体的适度函数，而基于脉冲微分系统的进化适应动力学的研究具有重要意义，我们利用稳定的长期周期平均指数增长率来建立进化变异体的适度函数，从而研究进化适应动力学的性质。利用基于脉冲微分方程的进化适应动力学理论和方法，揭示脉冲干预向不利于生物种群的方向进化，这一结论已经被实验数据证实，此外，为分析基于脉冲微分方程的进化问题提供数学理论分析方法。

The vector–host epidemic model with multiple strains

in a patchy environment

邱志鹏

南京理工大学

nustqzp@njust.edu.cn

Spatial heterogeneity plays an important role in the distribution and persistence of infectious diseases. In this article, a vector–host epidemic model is proposed to explore the effect of spatial heterogeneity on the evolution of vector-borne diseases. The model is a Ross–MacDonald type model with multiple competing strains on a number of patches connected by host migration. The multi-patch basic reproduction numbers $R_{j0}, j = 1, 2,… l are respectively derived for the model with l strains on n discrete patches. Analytical results show that if Rj0< 1, then strain j cannot invade the patchy environment and dies out. The invasion reproduction numbers R_ji, i, j = 1, 2, i /neq j are also derived for the model with two strains on n discrete patches. It is shown that the invasion reproduction numbers Rji, i, j = 1, 2, i /neq j provide threshold conditions that determine the competitive outcomes for the two strains. Under the condition that both invasion reproduction numbers are larger than one, the coexistence of two competing strains is rigorously proved. However, the two competing strains cannot coexist for the corresponding model with no host migration. This implies that host migration can lead to the coexistence of two competing strains and enhancement of pathogen genetic diversity. Global dynamics is determined for the model with two competing strains on two patches. The results are based on the theory of type-K monotone dynamical systems.

A (biased) survey on elliptic, parabolic and nonlocal eigenvalue problems

王学锋

Tulane University

xdw@tulane.edu

On the qualitative behavior of prey-taxis model with predator-prey interactions

王治安

香港理工大学

mawza@polyu.edu.hk

In this talk, we shall discuss the qualitative behavior of prey-taxis model with a variety of predator-prey interactions. In the previous limited results, various technical assumptions are imposed to ensure the boundedness of classical solutions. In our work, we shall first remove those conditions, and then establish the asymptotic behavior of solutions by which we identify the conditions under which the competition predator-prey outcomes including coexistence, exclusion and extinctions will be achieved. Then we discuss the applications of the model on the biological control. This is joint work with Haiyang Jin.

Modeling of HIV-1 infection

王开发

第三军医大学

kfwang72@163.com

A novel dynamic model covering five types of cells and three connected compartments, peripheral blood (PB), lymph nodes (LNs), and the central nervous system (CNS), is here proposed. It is based on assessment of the biological principles underlying the interactions between the human immunodeficiency virus type I (HIV-1) and the human immune system. The simulated results of this model matched the three well-documented phases of HIV-1 infection very closely and successfully described the three stages of LN destruction that occur during HIV-1 infection. The model also showed that LNs are the major location of viral replication, creating a pool of latently infected T4 cells during the latency period. A detailed discussion of the role of monocytes/macrophages is made, and the results indicated that infected monocytes/macrophages could determine the progression of HIV-1 infection. The effects of typical highly active antiretroviral therapy (HAART) drugs on HIV-1 infection were analyzed and the results showed that efficiency of each drug but not the time of the treatment start contributed to the change of the turnover of the disease greatly. An incremental count of latently infected T4 cells was made under therapeutic simulation, and patients were found to fail to respond to HAART therapy in the presence of certain stimuli, such as opportunistic infections. In general, the dynamics of the model qualitatively matched clinical observations very closely, indicating that the model may have benefits in evaluating the efficacy of different drug therapy regimens and in the discovery of new monitoring markers and therapeutic schemes for the treatment of HIV-1 infection. In addition, though Pre-Exposure Prophylaxis (PrEP) has been approved by the US Food and Drug Administration, the results of several clinical trials until now are complex and inconclusive. we further investigated a theoretical trial study using single drug intervene and 100% participants’ adherence for the prevention of HIV infection, and to examine the effectiveness of different drug responses.

Modeling climate variation on range expansion of Ixodes scapularis tick population

吴孝钿

上海海事大学

xtwu@shmtu.edu.cn

Ixodes scapularis is the primary vector of Borrelia burgdorferi, the bacterial agent of Lyme disease. Both the range of Ixodes scapularis and the risk of Lyme disease are changing rapidly in Canada. Thus, a deterministic model of Ixodes scapularis was developed to measure the establishement of the tick population under the condition of climate change. The threshold temperature conditions for tick population survival (R0<1) were shown to be the same as those identified using the mechanistic model, and a map of R0 for I. scapularis, the first such map for an arthropod vector, was drawn for Canada east of the Rocky Mountains. Based on the deterministic model, we quantified potential effects of future climate change on the basic reproduction number (R0) of the tick vector of Lyme disease, and explored their importance for Lyme disease risk, and for vector-borne diseases in general.

A delay-dependent model with HIV drug resistance during therapy

王艳

中国石油大学（华东）

wangyan@upc.edu.cn

The use of combination antiretrovial therapy has proven remarkably effective in controlling HIV disease progression and prolonging survival. However, the emergence of drug resistance can occur. It is necessary that we gain a greater understanding of the evolution of drug resistance. Here, we consider an HIV viral dynamical model with general form of target cell density, drug resistance and intracellular delay incorporating antiretroviral therapy. The model includes two strains: wild-type and drug-resistant. The basic reproductive ratio for each strain is obtained for the existence of steady states. Qualitative analysis of the model such as the well-posedness of the solutions and the equilibrium stability is provided. Global asymptotic stability of the disease-free and drug-resistant steady states is shown by constructing Lyapunov functions. Furthermore, sufficient conditions related to the properties of the target cell density are obtained for the local asymptotic stability of the positive steady state. Numerical simulations are conducted to study the impact of target cell density and intracellular delay focusing on the stability of the positive steady state. The occurrence of Hopf bifurcation of periodic solutions is shown to depend on the target cell density.

Global dynamics of a delayed SEIS infectious disease model

with logistic growth and saturation incidence

徐瑞

军械工程学院

rxu88@163.com

In this work, a delayed SEIS infectious disease model with logistic growth and saturation incidence is investigated, where the time delay describes the latent period of the disease. By analyzing corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is discussed. The existence of Hopf bifurcations at the endemic equilibrium is established. By using the persistence theory for infinite dimensional dynamic systems, it is proved that if the basic reproduction number is greater than unity, the system is permanent. By means of suitable Lyapunov functionals and LaSalle’s invariance principle, sufficient conditions are obtained for the global stability of the disease-free equilibrium and the endemic equilibrium, respectively. Numerical simulations are carried out to illustrate the theoretical results.

生物钟振子：实验与模型的一些交叉研究

杨凌

苏州大学

lyang@suda.edu.cn

生物钟是一个非常合适模型与实验交叉合作研究的课题，因为它具有一些非常复杂有趣的动力学行为。由于传统的生物学手段不足以厘清其深刻的机制，因此必须有数学模型的参与。生物钟表现为以24小时为周期的动物活动/睡眠切换，以及植物叶片开/合等现象，而在细胞内的本质则是蛋白质浓度振荡。其中有一些非常有意思的生物学现象，1）例如对刺激的周期性记忆现象；2）活动睡眠的反相位切换；3）特殊刺激下振荡消失的奇异性现象等。我们与生物学家密切合作，通过生物问题-建模-实验验证的交叉研究流程，研究了这些生物钟相关问题。事实上，我们模型的一些预测已获得了实验验证，这里就上述合作研究的结果和过程作一些介绍。