Stability Analysis and Optimal Control Strategy for Prevention of Pine Wilt Disease

We propose a mathematical model of pine wilt disease (PWD) which is caused by pine sawyer beetles carrying the pinewood nematode (PWN). We calculate the basic reproduction numberR0and investigate the stability of a disease-free and endemic equilibrium in a given mathematical model. We show that the stability of the equilibrium in the proposed model can be controlled through the basic reproduction numberR0. We then discuss effective optimal control strategies for the proposed PWD mathematical model. We demonstrate the existence of a control problem, and then we apply both analytical and numerical techniques to demonstrate effective control methods to prevent the transmission of the PWD. In order to do this, we apply two control strategies: tree-injection of nematicide and the eradication of adult beetles through aerial pesticide spraying. Optimal prevention strategies can be determined by solving the corresponding optimality system. Numerical simulations of the optimal control problem using a set of reasonable parameter values suggest that reducing the number of pine sawyer beetles is more effective than the tree-injection strategy for controlling the spread of PWD.

Download Full-text

Analysis of the Mathematical Model for the Spread of Pine Wilt Disease

Journal of Applied Mathematics ◽

10.1155/2013/184054 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 7

Author(s):

Xiangyun Shi ◽

Guohua Song

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Pine Wilt Disease ◽

Wilt Disease ◽

Endemic Equilibrium ◽

Feasible Region ◽

Global Dynamics ◽

Pine Wilt ◽

Theoretical Results ◽

The Basic Reproduction Number

This paper formulates and analyzes a pine wilt disease model. Mathematical analyses of the model with regard to invariance of nonnegativity, boundedness of the solutions, existence of nonnegative equilibria, permanence, and global stability are presented. It is proved that the global dynamics are determined by the basic reproduction numberℛ0and the other valueℛcwhich is larger thanℛ0. Ifℛ0andℛcare both less than one, the disease-free equilibrium is asymptotically stable and the pine wilt disease always dies out. If one is between the two values, though the pine wilt disease could occur, the outbreak will stop. If the basic reproduction number is greater than one, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at the endemic equilibrium state if it initially exists. Numerical simulations are carried out to illustrate the theoretical results, and some disease control measures are especially presented by these theoretical results.

Download Full-text

Simple MSEIR Model for Measles Transmission

Asian Research Journal of Mathematics ◽

10.9734/arjom/2019/v12i330090 ◽

2019 ◽

pp. 1-11

Author(s):

Mojeeb Al-Rahman EL-Nor Osman ◽

Appiagyei Ebenezer ◽

Isaac Kwasi Adu

Keyword(s):

Mathematical Model ◽

Basic Reproduction Number ◽

Recovery Rate ◽

Birth Rate ◽

Reproduction Number ◽

Progression Rate ◽

Asymptotically Stable ◽

The Stability ◽

Disease Free ◽

The Basic Reproduction Number

In this paper, an Immunity-Susceptible-Exposed-Infectious-Recovery (MSEIR) mathematical model was used to study the dynamics of measles transmission. We discussed that there exist a disease-free and an endemic equilibria. We also discussed the stability of both disease-free and endemic equilibria. The basic reproduction number is obtained. If , then the measles will spread and persist in the population. If , then the disease will die out. The disease was locally asymptotically stable if and unstable if . ALSO, WE PROVED THE GLOBAL STABILITY FOR THE DISEASE-FREE EQUILIBRIUM USING LASSALLE'S INVARIANCE PRINCIPLE OF Lyaponuv function. Furthermore, the endemic equilibrium was locally asymptotically stable if , under certain conditions. Numerical simulations were conducted to confirm our analytic results. Our findings were that, increasing the birth rate of humans, decreasing the progression rate, increasing the recovery rate and reducing the infectious rate can be useful in controlling and combating the measles.

Download Full-text

Optimal Control of a Dengue-Dengvaxia Model: Comparison Between Vaccination and Vector Control

Computational and Mathematical Biophysics ◽

10.1515/cmb-2020-0124 ◽

2021 ◽

Vol 9 (1) ◽

pp. 198-212

Author(s):

Cheryl Q. Mentuda

Keyword(s):

Optimal Control ◽

Vector Control ◽

Basic Reproduction Number ◽

Human Population ◽

Model Comparison ◽

Reproduction Number ◽

Control Strategies ◽

Minimum Principle ◽

Transmitted Disease ◽

The Basic Reproduction Number

Abstract Dengue is the most common mosquito-borne viral infection transmitted disease. It is due to the four types of viruses (DENV-1, DENV-2, DENV-3, DENV-4), which transmit through the bite of infected Aedes aegypti and Aedes albopictus female mosquitoes during the daytime. The first globally commercialized vaccine is Dengvaxia, also known as the CYD-TDV vaccine, manufactured by Sanofi Pasteur. This paper presents a Ross-type epidemic model to describe the vaccine interaction between humans and mosquitoes using an entomological mosquito growth population and constant human population. After establishing the basic reproduction number ℛ0, we present three control strategies: vaccination, vector control, and the combination of vaccination and vector control. We use Pontryagin’s minimum principle to characterize optimal control and apply numerical simulations to determine which strategies best suit each compartment. Results show that vector control requires shorter time applications in minimizing mosquito populations. Whereas vaccinating the primary susceptible human population requires a shorter time compared to the secondary susceptible human.

Download Full-text

Analysis of the SARS-CoV-2 Outbreak in Northern Cyprus using a Mathematical Model

10.21203/rs.3.rs-28213/v1 ◽

2020 ◽

Author(s):

Tamer Sanlidag ◽

Nazife Sultanoglu ◽

Bilgen Kaymakamzade ◽

Evren Hincal ◽

Murat Sayan ◽

...

Keyword(s):

Mathematical Model ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Equilibrium Points ◽

Real Data ◽

Northern Cyprus ◽

The Stability ◽

Disease Free ◽

The Basic Reproduction Number

Abstract The present study studied the dynamics of SARS-CoV-2 in Northern-Cyprus (NC) by using real data and a designed mathematical model. The model consisted of two equilibrium points, which were disease-free and epidemic. The stability of the equilibrium points was determined by the magnitude of the basic reproduction number (𝑹𝟎). If 𝑹𝟎 < 1, the disease eventually disappears, if 𝑹𝟎 ≥ 1, the presence of an epidemic is stated. 𝑹𝟎 has been calculated patient zero, with a range of 2.38 to 0.65. Currently, the 𝑹𝟎 for NC was found to be 0.65, indicating that NC is free from the SARS-CoV-2 epidemic.

Download Full-text

Analysis of the SARS-CoV-2 Outbreak in Northern Cyprus using a Mathematical Model

10.21203/rs.3.rs-28704/v1 ◽

2020 ◽

Author(s):

Tamer Sanlidag ◽

Nazife Sultanoglu ◽

Bilgen Kaymakamzade ◽

Evren Hincal ◽

Murat Sayan ◽

...

Keyword(s):

Mathematical Model ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Equilibrium Points ◽

Real Data ◽

Northern Cyprus ◽

The Stability ◽

Disease Free ◽

The Basic Reproduction Number

Download Full-text

OPTIMAL CONTROL ANALYSIS OF THE EFFECT OF TREATMENT, ISOLATION AND VACCINATION ON HEPATITIS B VIRUS

Journal of Biological System ◽

10.1142/s0218339020400057 ◽

2020 ◽

Vol 28 (02) ◽

pp. 351-376 ◽

Cited By ~ 5

Author(s):

MUHAMMAD ALTAF KHAN ◽

SYED AZHAR ALI SHAH ◽

SAIF ULLAH ◽

KAZEEM OARE OKOSUN ◽

MUHAMMAD FAROOQ

Keyword(s):

Optimal Control ◽

Hepatitis B ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Control Strategies ◽

Control Measures ◽

Hepatitis B Infection ◽

Control Variables ◽

Effect Of Treatment ◽

The Basic Reproduction Number

Hepatitis B infection is a serious health issue and a major cause of deaths worldwide. This infection can be overcome by adopting proper treatment and control strategies. In this paper, we develop and use a mathematical model to explore the effect of treatment on the dynamics of hepatitis B infection. First, we formulate and use a model without control variables to calculate the basic reproduction number and to investigate basic properties of the model such as the existence and stability of equilibria. In the absence of control measures, we prove that the disease free equilibrium is locally asymptotically stable when the basic reproduction number is less than unity. Also, using persistent theorem, it is shown that the infection is uniformly persistent, whenever the basic reproduction number is greater than unity. Using optimal control theory, we incorporate into the model three time-dependent control variables and investigate the conditions required to curtail the spread of the disease. Finally, to illustrate the effectiveness of each of the control strategies on disease control and eradication, we perform numerical simulations. Based on the numerical results, we found that the first two strategies (treatment and isolation strategy) and (vaccination and isolation strategy) are not very effective as a long term control or eradication strategy for HBV. Hence, we recommend that in order to effectively control the disease, all the control measures (isolation, vaccination and treatment) must be implemented at the same time.

Download Full-text

Modelling the Periodic Outbreak of Measles in Mainland China

Mathematical Problems in Engineering ◽

10.1155/2020/3631923 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Yuyi Xue ◽

Xiaoe Ruan ◽

Yanni Xiao

Keyword(s):

Optimal Control ◽

Numerical Simulations ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Transmission Rate ◽

Control Strategies ◽

Mainland China ◽

Asymptomatic Infection ◽

Protection Measures ◽

The Basic Reproduction Number

In mainland China, measles infection reached the lowest level in 2012 but resurged again after that with a seasonally fluctuating pattern. To investigate the phenomenon of periodic outbreak and identify the crucial parameters that play in the transmission dynamics of measles, we formulate a mathematical model incorporating periodic transmission rate and asymptomatic infection with waning immunity. We define the basic reproduction number as the threshold value to govern whether measles infection dies out or not. Fitting the reported measles cases from 2013 to 2016 to our proposed model, we estimate the basic reproduction number R0 with immunization to be 1.0077. From numerical simulations, we conclude asymptomatic infection does not cause much new infections and the key parameters affecting the transmission of measles are vaccination rate, transmission rate, and recovery rate, which suggests the public to enhance vaccination and protection measures to reduce effective contacts between susceptible and infective individuals and treat infected individuals timely. To minimize the number of infected individuals at a minimal cost, we formulate an optimal control system to design optimal control strategies. Numerical simulations show the effectiveness of optimal control strategies and recommend us to implement the control strategies as soon as possible. In particular, enhancing vaccination is especially effective in lowering the initial outbreak and making disease recurrence less likely.

Download Full-text

Mathematical Model of Schistosomiasis under Flood in Anhui Province

Abstract and Applied Analysis ◽

10.1155/2014/972189 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 4

Author(s):

Longxing Qi ◽

Jing-an Cui ◽

Tingting Huang ◽

Fengli Ye ◽

Longzhi Jiang

Keyword(s):

Mathematical Model ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Anhui Province ◽

Endemic Equilibrium ◽

Observation Data ◽

Schistosomiasis Control ◽

The Stability ◽

The Impact ◽

The Basic Reproduction Number

Based on the real observation data in Tongcheng city, this paper established a mathematical model of schistosomiasis transmission under flood in Anhui province. The delay of schistosomiasis outbreak under flood was considered. Analysis of this model shows that the disease free equilibrium is locally asymptotically stable if the basic reproduction number is less than one. The stability of the unique endemic equilibrium may be changed under some conditions even if the basic reproduction number is larger than one. The impact of flood on the stability of the endemic equilibrium is studied and the results imply that flood can destabilize the system and periodic solutions can arise by Hopf bifurcation. Finally, numerical simulations are performed to support these mathematical results and the results are in accord with the observation data from Tongcheng Schistosomiasis Control Station.

Download Full-text

Dynamical features of pine wilt disease through stability, sensitivity and optimal control

Advances in Difference Equations ◽

10.1186/s13662-021-03411-y ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Riaz Ahmad Khan ◽

Takasar Hussain ◽

Muhammad Ozair ◽

Fatima Tasneem ◽

Muhammad Faizan

Keyword(s):

Lyapunov Functional ◽

Reproduction Number ◽

Control Strategies ◽

Pine Wilt Disease ◽

Wilt Disease ◽

Control Measures ◽

Disease Eradication ◽

Graph Theoretic ◽

Pine Wilt ◽

Dynamical Features

AbstractThis work investigates the dissemination mechanism of pine wilt disease. The basic reproduction number is computed explicitly, and an ultimate invariable level of contagious hosts and vectors, without and with disease, is discussed by using this number. Highly effective techniques, Lyapunov functional and graph theoretic, are utilised to obtain the ultimate constant level of the whole population. The idea of complete disease eradication and reduction of endemic level is explored through the utilisation of two efficient methods. Using sensitivity analysis approach, necessary control measures are suggested to overcome the disease. Using the literature data, the robustness of control strategies is shown graphically.

Download Full-text

Stability analysis of a smoking behavior model

Journal of Physics Conference Series ◽

10.1088/1742-6596/2106/1/012025 ◽

2021 ◽

Vol 2106 (1) ◽

pp. 012025

Author(s):

S M Lestari ◽

Y Yulida ◽

A S Lestia ◽

M A Karim

Keyword(s):

Mathematical Model ◽

Stability Analysis ◽

Equilibrium Point ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Smoking Behavior ◽

Asymptotically Stable ◽

Runge Kutta Method ◽

The Stability ◽

The Basic Reproduction Number

Abstract This research discussed the mathematical model of smoking behavior. The model will be analogous to an epidemic model which will be divided into several compartments/groups. This research aimed to explain the formation of a mathematical model of smoking behavior, to investigate the equilibrium point, the value of the basic reproduction number, to analyze the stability of the model, then to determine and interpret the numerical solutions using the fourth-order Runge-Kutta method. By the results of this research, a mathematical model of smoking behavior which consists of three compartments, namely the population of non-smokers, smokers and ex-smokers, was obtained. Based on the model formed the smoke-free equilibrium point and the smoker equilibrium point, then the basic reproduction number was also obtained using the next generation matrix. Furthermore, the result of the stability analysis of the smoker-free population was asymptotically stable provided that the basic reproduction number is less than one, while the population was asymptotically stable provided that the basic reproduction number is greater than one. The simulation of the model was presented to support the explanation of the stability analysis of the model using the fourth-order Runge-Kutta method based on the parameters that met the requirements of the stability analysis.

Download Full-text