scholarly journals A Mathematical Model of Malaria Transmission with Structured Vector Population and Seasonality

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Bakary Traoré ◽  
Boureima Sangaré ◽  
Sado Traoré
Keyword(s):  
Mathematical Model ◽  
Malaria Transmission ◽  
Climatic Factors ◽  
Positive Periodic Solution ◽  
Uniform Persistence ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Vector Population ◽  
Biting Rate

In this paper, we formulate a mathematical model of nonautonomous ordinary differential equations describing the dynamics of malaria transmission with age structure for the vector population. The biting rate of mosquitoes is considered as a positive periodic function which depends on climatic factors. The basic reproduction ratio of the model is obtained and we show that it is the threshold parameter between the extinction and the persistence of the disease. Thus, by applying the theorem of comparison and the theory of uniform persistence, we prove that if the basic reproduction ratio is less than 1, then the disease-free equilibrium is globally asymptotically stable and if it is greater than 1, then there exists at least one positive periodic solution. Finally, numerical simulations are carried out to illustrate our analytical results.

scholarly journals The Global Behavior of a Periodic Epidemic Model with Travel between Patches

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Luosheng Wen ◽  
Bin Long ◽  
Xin Liang ◽  
Fengling Zeng
Keyword(s):  
Epidemic Model ◽  
Transmission Rate ◽  
Global Behavior ◽  
Uniform Persistence ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Periodic Transmission ◽  
Theoretical Results ◽  
Disease Free

We establish an SIS (susceptible-infected-susceptible) epidemic model, in which the travel between patches and the periodic transmission rate are considered. As an example, the global behavior of the model with two patches is investigated. We present the expression of basic reproduction ratioR0and two theorems on the global behavior: ifR0< 1 the disease-free periodic solution is globally asymptotically stable and ifR0> 1, then it is unstable; ifR0> 1, the disease is uniform persistence. Finally, two numerical examples are given to clarify the theoretical results.

scholarly journals Seasonality Impact on the Transmission Dynamics of Tuberculosis

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Yali Yang ◽  
Chenping Guo ◽  
Luju Liu ◽  
Tianhua Zhang ◽  
Weiping Liu
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Autonomous Systems ◽  
Positive Periodic Solution ◽  
Transmission Dynamics ◽  
Uniform Persistence ◽  
Globally Asymptotically Stable ◽  
The Impact ◽  
Disease Free ◽  
The Basic Reproduction Number

The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show the seasonality fluctuations in Shaanxi of China. A seasonality TB epidemic model with periodic varying contact rate, reactivation rate, and disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB. Simulations show that the basic reproduction number of time-averaged autonomous systems may underestimate or overestimate infection risks in some cases, which may be up to the value of period. The basic reproduction number of the seasonality model is appropriately given, which determines the extinction and uniform persistence of TB disease. If it is less than one, then the disease-free equilibrium is globally asymptotically stable; if it is greater than one, the system at least has a positive periodic solution and the disease will persist. Moreover, numerical simulations demonstrate these theorem results.

GLOBAL DYNAMICS OF A DELAYED HIV-1 INFECTION MODEL WITH ABSORPTION AND SATURATION INFECTION

2012 ◽  
Vol 05 (03) ◽  
pp. 1260012 ◽  
Author(s):  
RUI XU
Keyword(s):  
Viral Entry ◽  
Chronic Infection ◽  
Infection Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Virus Particles ◽  
Hiv 1

In this paper, an HIV-1 infection model with absorption, saturation infection and an intracellular delay accounting for the time between viral entry into a target cell and the production of new virus particles is investigated. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and LaSalle's invariance principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable; and if the basic reproduction ratio is greater than unity, sufficient condition is derived for the global stability of the chronic-infection equilibrium.

scholarly journals Global Dynamics of a Host-Vector-Predator Mathematical Model

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Fengyan Zhou ◽  
Hongxing Yao
Keyword(s):  
Mathematical Model ◽  
Predation Rate ◽  
Disease Suppression ◽  
Uniform Persistence ◽  
Global Dynamics ◽  
Vector Population ◽  
Threshold Conditions ◽  
Vector Theory ◽  
The Impact ◽  
Disease Free

A mathematical model which links predator-vector(prey) and host-vector theory is proposed to examine the indirect effect of predators on vector-host dynamics. The equilibria and the basic reproduction numberR0are obtained. By constructing Lyapunov functional and using LaSalle’s invariance principle, global stability of both the disease-free and disease equilibria are obtained. Analytical results show thatR0provides threshold conditions on determining the uniform persistence and extinction of the disease, and predator density at any time should keep larger or equal to its equilibrium level for successful disease eradication. Finally, taking the predation rate as parameter, we provide numerical simulations for the impact of predators on vector-host disease control. It is illustrated that predators have a considerable influence on disease suppression by reducing the density of the vector population.

scholarly journals Global Dynamics of a Delayed HIV-1 Infection Model with CTL Immune Response

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Yunfei Li ◽  
Rui Xu ◽  
Zhe Li ◽  
Shuxue Mao
Keyword(s):  
Immune Response ◽  
Viral Infection ◽  
Infection Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Ctl Immune Response ◽  
Hiv 1

A delayed HIV-1 infection model with CTL immune response is investigated. By using suitable Lyapunov functionals, it is proved that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio for viral infection is less than or equal to unity; if the basic reproduction ratio for CTL immune response is less than or equal to unity and the basic reproduction ratio for viral infection is greater than unity, the CTL-inactivated infection equilibrium is globally asymptotically stable; if the basic reproduction ratio for CTL immune response is greater than unity, the CTL-activated infection equilibrium is globally asymptotically stable.

scholarly journals Mathematical analysis of an HTLV-I infection model with the mitosis of CD4+ T cells and delayed CTL immune response

2021 ◽  
Vol 26 (1) ◽  
pp. 1-20
Author(s):  
Chenwei Song ◽  
Rui Xu
Keyword(s):  
Immune Response ◽  
T Cells ◽  
Cd4 T Cells ◽  
Infection Model ◽  
Uniform Persistence ◽  
Type I ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Ctl Immune Response

In this paper, we consider an improved Human T-lymphotropic virus type I (HTLV-I) infection model with the mitosis of CD4+ T cells and delayed cytotoxic T-lymphocyte (CTL) immune response by analyzing the distributions of roots of the corresponding characteristic equations, the local stability of the infection-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium when the CTL immune delay is zero is established. And we discuss the existence of Hopf bifurcation at the immunity-activated equilibrium. We define the immune-inactivated reproduction ratio R0 and the immune-activated reproduction ratio R1. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that if R0 < 1, the infection-free equilibrium is globally asymptotically stable; if R1 < 1 < R0, the immunity-inactivated equilibrium is globally asymptotically stable; if R1 > 1, the immunity-activated equilibrium is globally asymptotically stable when the CTL immune delay is zero. Besides, uniform persistence is obtained when R1 > 1. Numerical simulations are carried out to illustrate the theoretical results.

scholarly journals A transmission model of Bilharzia : A mathematical analysis of an heterogeneous model

2011 ◽  
Vol Volume 14 - 2011 - Special... ◽  
Author(s):  
Riveau Gilles ◽  
Sallet Gauthier ◽  
Tendeng Lena
Keyword(s):  
Real Data ◽  
Transmission Model ◽  
Asymptotically Stable ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Heterogeneous Model ◽  
Nous Montrons ◽  
International Audience ◽  
Disease Free

International audience We consider an heterogeneous model of transmission of bilharzia. We compute the basic reproduction ratio R 0. We prove that if R 0 < 1, then the disease free equilibrium is globally asymptotically stable. If R 0 > 1 then there exists an unique endemic equilibrium, which is globally asymptotically stable. We will then consider possible applications to real data On considère un modèle de transmission de la bilharziose prenant en compte les hétérogénéités. Nous calculons le taux de reproduction de base Nous montrons que si R0 < 1, alors l’équilibre sans maladie est globalement asymptotiquement stable. Si R0 > 1, alors il existe un unique équilibre endémique et celui-ci est globalement asymptotiquement stable. Nous considérons ensuite les applications possibles à des données réelles.

scholarly journals Global Stability of a Virus Infection Model with Time Delay and Absorption

2011 ◽  
Vol 2011 ◽  
pp. 1-20
Author(s):  
Xiaohong Tian ◽  
Rui Xu
Keyword(s):  
Time Delay ◽  
Virus Infection ◽  
Global Stability ◽  
Sufficient Conditions ◽  
Infection Model ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Comparison Arguments ◽  
Virus Infection Model

In this paper, a virus infection model with time delay and absorption is studied. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the model is established. By using comparison arguments, it is shown that the infection free equilibrium is globally asymptotically stable when the basic reproduction ratio is less than unity. When the basic reproduction ratio is greater than unity, sufficient conditions are derived for the global stability of the virus-infected equilibrium. Numerical simulations are carried out to illustrate the theoretical results.

scholarly journals PEMODELAN MATEMATIKA DENGAN METODE RUNGE KUTTA UNTUK PENYAKIT CAMPAK MENGGUNAKAN MATLAB R2010a

2018 ◽  
Vol 4 (2) ◽  
Author(s):  
Ketut Queena Fredlina ◽  
Komang Tri Werthi
Keyword(s):  
Mathematical Model ◽  
Mathematical Models ◽  
Measles Virus ◽  
Immunization Coverage ◽  
Living Environment ◽  
Basic Reproduction Ratio ◽  
Reproduction Ratio ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Mathematical Model

ABSTRACT<br />Mathematical models have important roles in various fields of science. By using several assumptions, problems that exist in the living environment can be transformed in mathematical models. From the existing mathematical model, the parameters that affect the model can then be analyzed. An epidemic is an event that can be transformed into a mathematical model. Epidemic events are the occurrence of the spread or outbreak of an illness in a region. Measles is one of the causes of death in developing countries caused by the measles virus, the Paramixovirus group. In 1982 a measles immunization program in Indonesia was conducted. Based on data from the 2015 Ministry of Health, Indonesia has a medium immunization coverage in Southeast Asia, which is 84%. In 2020 Indonesia has a target rate of measles immunization coverage of 95%. Measles is a concern of the Bali Provincial Health Office because the spread of this disease is always high. Specifically in this study we will discuss mathematical models for the incidence of measles epidemics. The problem is how to construct the model and what parameters are the most significant influences in the mathematical model of measles. In making mathematical models for the spread of measles, the population is divided into 3 parts: Susceptible, Infectious, and Recovered. Furthermore, analyze the parameters and determine the basic reproduction ratio (𝑹𝟎), then numerical simulations were carried out using the Order 4 Runge Kutta method.<br />Keywords : Mathematics , Measles, basic reproduction ratio (𝑹𝟎), Runge-Kutta Methods<br />ABSTRAK<br />Model matematika memiliki peran yang cukup penting dalam berbagai bidang ilmu. Dengan menggunakan beberapa asumsi, permasalahan yang ada dalam lingkungan kehidupan dapat ditransformasikan dalam model matematika. Dari model matematika yang ada selanjutnya dapat dianalisis parameter-parameter yang mempengaruhi model tersebut. Kejadian epidemi merupakan salah satu kejadian yang dapat ditransformasikan dalam model matematika. Kejadian epidemi adalah kejadian penyebaran atau mewabahnya suatu penyakit dalam suatu wilayah. Penyakit campak merupakan salah satu penyakit penyebab kematian penduduk di negara-negara berkembang yang disebabkan oleh virus campak golongan Paramixovirus. Pada tahun 1982 program imunisasi campak di Indonesia telah dilakukan. Berdasarkan data dari Departemen Kesehatan 2015, Indonesia memiliki cakupan imunisasi kategori sedang di Asia Tenggara yakni 84%. Pada tahun 2020 Indonesia memiliki target angka cakupan imunisasi campak sebesar 95%. Penyakit campak menjadi perhatian Dinas Kesehatan Profinsi Bali karena penyebaran penyakit ini selalu ada. Secara khusus dalam penelitian ini akan membahas model matematika untuk kejadian epidemi penyakit campak. Yang menjadi permasalahan adalah bagaimana mengontruksi model dan parameter apakah yang berpengaruh paling signifikan dalam model matematika penyakit campak. Dalam pembuatan model matematika untuk penyebaran penyakit campak, populasi manusia dibagi menjadi 3 bagian yaitu : Susceptible, Infectious, dan Recovered. Selanjutnya menganalisis parameter dan menentukan nilai basic reproduction ratio (R0), kemudian dilakukan simulasi numerik dengan metode Runge Kutta Orde 4.<br />Kata kunci : model matematika, campak, basic reproduction ratio (𝑹𝟎),metode Runge-Kutta

scholarly journals Stability and Hopf Bifurcation in an HIV-1 Infection Model with Latently Infected Cells and Delayed Immune Response

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Haibin Wang ◽  
Rui Xu
Keyword(s):  
Immune Response ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Latently Infected ◽  
Infected Cells ◽  
Ctl Immune Response ◽  
Hiv 1

An HIV-1 infection model with latently infected cells and delayed immune response is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria is established and the existence of Hopf bifurcations at the CTL-activated infection equilibrium is also studied. By means of suitable Lyapunov functionals and LaSalle’s invariance principle, it is proved that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio for viral infectionR0≤1; if the basic reproduction ratio for viral infectionR0>1and the basic reproduction ratio for CTL immune responseR1≤1, the CTL-inactivated infection equilibrium is globally asymptotically stable. If the basic reproduction ratio for CTL immune responseR1>1, the global stability of the CTL-activated infection equilibrium is also derived when the time delayτ=0. Numerical simulations are carried out to illustrate the main results.

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