Differential susceptibility and infectivity epidemic models
James M. Hyman Jia Li
Mathematical Biosciences & Engineering 2006, 3(1): 89-100 doi: 10.3934/mbe.2006.3.89
We formulate differential susceptibility and differential infectivity models for disease transmission in this paper. The susceptibles are divided into n groups based on their susceptibilities, and the infectives are divided into m groups according to their infectivities. Both the standard incidence and the bilinear incidence are considered for different diseases. We obtain explicit formulas for the reproductive number. We define the reproductive number for each subgroup. Then the reproductive number for the entire population is a weighted average of those reproductive numbers for the subgroups. The formulas for the reproductive number are derived from the local stability of the infection-free equilibrium. We show that the infection-free equilibrium is globally stable as the reproductive number is less than one for the models with the bilinear incidence or with the standard incidence but no disease-induced death. We then show that if the reproductive number is greater than one, there exists a unique endemic equilibrium for these models. For the general cases of the models with the standard incidence and death, conditions are derived to ensure the uniqueness of the endemic equilibrium. We also provide numerical examples to demonstrate that the unique endemic equilibrium is asymptotically stable if it exists.
keywords: endemic equilibrium differential susceptibility reproductive number global stability. differential infectivity
An SEI infection model incorporating media impact
Xuejuan Lu Shaokai Wang Shengqiang Liu Jia Li
Mathematical Biosciences & Engineering 2017, 14(5&6): 1317-1335 doi: 10.3934/mbe.2017068

To study the impact of media coverage on spread and control of infectious diseases, we use a susceptible-exposed-infective (SEI) model, including individuals' behavior changes in their contacts due to the influences of media coverage, and fully investigate the model dynamics. We define the basic reproductive number $\Re_0$ for the model, and show that the modeled disease dies out regardless of initial infections when $\Re_0 < 1$, and becomes uniformly persistently endemic if $\Re_0>1$. When the disease is endemic and the influence of the media coverage is less than or equal to a critical number, there exists a unique endemic equilibrium which is asymptotical stable provided $\Re_0 $ is greater than and near one. However, if $\Re_0 $ is larger than a critical number, the model can undergo Hopf bifurcation such that multiple endemic equilibria are bifurcated from the unique endemic equilibrium as the influence of the media coverage is increased to a threshold value. Using numerical simulations we obtain results on the effects of media coverage on the endemic that the media coverage may decrease the peak value of the infectives or the average number of the infectives in different cases. We show, however, that given larger $\Re_0$, the influence of the media coverage may as well result in increasing the average number of the infectives, which brings challenges to the control and prevention of infectious diseases.

keywords: Media impact Hopf bifurcation stability infectious disease
A malaria model with partial immunity in humans
Jia Li
Mathematical Biosciences & Engineering 2008, 5(4): 789-801 doi: 10.3934/mbe.2008.5.789
In this paper, we formulate a mathematical model for malaria transmission that includes incubation periods for both infected human hosts and mosquitoes. We assume humans gain partial immunity after infection and divide the infected human population into subgroups based on their infection history. We derive an explicit formula for the reproductive number of infection, $R_0$, to determine threshold conditions whether the disease spreads or dies out. We show that there exists an endemic equilibrium if $R_0>1$. Using an numerical example, we demonstrate that models having the same reproductive number but different numbers of progression stages can exhibit different transient transmission dynamics.
keywords: malaria endemic equilibrium compartmental models reproductive number
Modeling of mosquitoes with dominant or recessive Transgenes and Allee effects
Jia Li
Mathematical Biosciences & Engineering 2010, 7(1): 99-121 doi: 10.3934/mbe.2010.7.99
To study the impact of releasing transgenic mosquitoes on malaria transmission, we formulate discrete-time models for interacting wild and transgenic mosquitoes populations, based on systems of difference equations. We start with models including all homozygous and heterozygous mosquitoes. We then consider either dominant or recessive transgenes to reduce the 3-dimensional model systems to 2-dimensional systems. We include density-dependent vital rates and incorporate Allee effects in the functional mating rates. Dynamics of these models are explored by investigating the existence and stability of boundary and positive fixed points. Numerical simulations are provided and brief discussions are given.
keywords: period-doubling bifurcation. stability discrete-time population model malaria transmission transgenic mosquito
A mathematical model of HTLV-I infection with two time delays
Xuejuan Lu Lulu Hui Shengqiang Liu Jia Li
Mathematical Biosciences & Engineering 2015, 12(3): 431-449 doi: 10.3934/mbe.2015.12.431
In this paper, we include two time delays in a mathematical model for the CD8$^+$ cytotoxic T lymphocytes (CTLs) response to the Human T-cell leukaemia virus type I (HTLV-I) infection, where one is the intracellular infection delay and the other is the immune delay to account for a series of immunological events leading to the CTL response. We show that the global dynamics of the model system are determined by two threshold values $R_0$, the corresponding reproductive number of a viral infection, and $R_1$, the corresponding reproductive number of a CTL response, respectively. If $R_0<1$, the infection-free equilibrium is globally asymptotically stable, and the HTLV-I viruses are cleared. If $R_1 < 1 < R_0$, the immune-free equilibrium is globally asymptotically stable, and the HTLV-I infection is chronic but with no persistent CTL response. If $1 < R_1$, a unique HAM/TSP equilibrium exists, and the HTLV-I infection becomes chronic with a persistent CTL response. Moreover, we show that the immune delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations. Our numerical simulations suggest that if $1 < R_1$, an increase of the intracellular delay may stabilize the HAM/TSP equilibrium while the immune delay can destabilize it. If both delays increase, the stability of the HAM/TSP equilibrium may generate rich dynamics combining the ``stabilizing" effects from the intracellular delay with those ``destabilizing" influences from immune delay.
keywords: Hopf bifurcation. Epidemic threshold HTLV-I infection Lyapunov functional time delay
Malaria model with stage-structured mosquitoes
Jia Li
Mathematical Biosciences & Engineering 2011, 8(3): 753-768 doi: 10.3934/mbe.2011.8.753
A simple SEIR model for malaria transmission dynamics is formulated as our baseline model. The metamorphic stages in the mosquito population are then included and a simple stage-structured mosquito population model is introduced, where the mosquito population is divided into two classes, with all three aquatic stages in one class and all adults in the other class, to keep the model tractable in mathematical analysis. After a brief investigation of this simple stage-structured mosquito model, it is incorporated into the baseline model to formulate a stage-structured malaria model. A basic analysis for the stage-structured malaria model is provided and it is shown that a theoretical framework can be built up for further studies on the impact of environmental or climate change on the malaria transmission. It is also shown that both the baseline and the stage-structured malaria models undergo backward bifurcations.
keywords: backward bifurcation. mosquito metamorphic stages reproductive number climate change Malaria transmission stage-structured model
Epidemic models with differential susceptibility and staged progression and their dynamics
James M. Hyman Jia Li
Mathematical Biosciences & Engineering 2009, 6(2): 321-332 doi: 10.3934/mbe.2009.6.321
We formulate and study epidemic models with differential susceptibilities and staged-progressions, based on systems of ordinary differential equations, for disease transmission where the susceptibility of susceptible individuals vary and the infective individuals progress the disease gradually through stages with different infectiousness in each stage. We consider the contact rates to be proportional to the total population or constant such that the infection rates have a bilinear or standard form, respectively. We derive explicit formulas for the reproductive number $R_0$, and show that the infection-free equilibrium is globally asymptotically stable if $R_0<1$ when the infection rate has a bilinear form. We investigate existence of the endemic equilibrium for the two cases and show that there exists a unique endemic equilibrium for the bilinear incidence, and at least one endemic equilibrium for the standard incidence when $R_0>1$.
keywords: reproductive number staged progression global stability. endemic equilibrium differential susceptibility
Azmy S. Ackleh Linda J. S. Allen Graciela Canziani Shandelle M. Henson Jia Li Zhien Ma
Mathematical Biosciences & Engineering 2008, 5(4): i-iii doi: 10.3934/mbe.2008.5.4i
Thomas Guy Hallam began his career as a faculty member in the Department of Mathematics at Florida State University, working in the area of comparison theorems for ordinary differential equations. While at Florida State he organized a mathematical modeling course and thus became interested in mathematical biology. He began to wonder how he, as a mathematician, might address the mounting environmental problems. He took courses in oceanography and ecology and delved deeply into the literature. During the summer of 1974, he gave a full series of lectures on mathematical biology at the University of São Carlos in São Paulo, Brazil. In 1976, he took a year's leave at the University of Georgia, Athens, in the Departments of Mathematics, Zoology, and the Institute of Ecology, where he met Tom Gard and Ray Lassiter, with whom he has had career-long interactions. In Athens, he became interested in ecotoxicology and partial differential equation models of physiologically structured populations.

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Wavelet frame based color image demosaicing
Jingwei Liang Jia Li Zuowei Shen Xiaoqun Zhang
Inverse Problems & Imaging 2013, 7(3): 777-794 doi: 10.3934/ipi.2013.7.777
Color image demosaicing consists in recovering full resolution color information from color-filter-array (CFA) samples with 66.7% amount of missing data. Most of the existing color demosaicing methods [14, 25, 16, 2, 26] are based on interpolation from inter-channel correlation and local geometry, which are not robust to highly saturated color images with small geometric features. In this paper, we introduce wavelet frame based methods by using a sparse wavelet [8, 22, 9, 23] approximation of individual color channels and color differences that recovers both geometric features and color information. The proposed models can be efficiently solved by Bregmanized operator splitting algorithm [27]. Numerical simulations of two datasets: McM and Kodak PhotoCD, show that our method outperforms other existing methods in terms of PSNR and visual quality.
keywords: operator splitting method inpainting. Wavelet frame color image demosaicing
From the Guest Editors
Fred Brauer Carlos Castillo-Chavez Thomas G. Hallam Jia Li Jianhong Wu Yicang Zhou
Mathematical Biosciences & Engineering 2006, 3(1): i-ix doi: 10.3934/mbe.2006.3.1i
Zhien Ma's love for mathematics has strongly shaped his educational pursuits. He received formal training from the strong Chinese School of Dynamical System over a period of two years at Peking University and during his later visit to Nanjing University where the internationally renowned professor Yanqian Ye mentored him. Yet, it is well known that Zhien's curiosity and love of challenges have made him his own best teacher. Hence, it is not surprising to see his shift from an outstanding contributor to the field of dynamical systems to a pioneer in the field of mathematical biology. Zhien's vision and courage became evident when he abandoned a promising career in pure mathematics and enthusiastically embraced a career in the field of mathematical biology soon after his first visit to the United States in 1985. His rapid rise to his current role as an international leader and premier mentor to 11 Ph.D. students has facilitated the placement of Chinese scientists and scholars at the forefront of research in the fields of mathematical, theoretical and computational biology.

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