Mathematical modeling approach to the transmission dynamics of pine wilt disease with saturated incidence rate

2018 ◽  
Vol 11 (03) ◽  
pp. 1850035 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Kamil Shah ◽  
Yasir Khan ◽  
Saeed Islam
Keyword(s):  
Incidence Rate ◽  
Local Stability ◽  
Pine Wilt Disease ◽  
Wilt Disease ◽  
Pine Wilt ◽  
Proposed Model ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Disease Free ◽  
Endemic State

The present paper investigates the dynamics of pine wilt disease with saturated incidence rate. The proposed model is stable both locally and globally. The local stability of the disease-free equilibrium is determined by the basic reproduction [Formula: see text]. The disease-free equilibrium is stable locally and globally whenever [Formula: see text]. If [Formula: see text], then the endemic state is stable both locally and globally. Further, a brief discussion with conclusion on the numerical results of the proposed model is presented.

The dynamics of a delayed deterministic and stochastic epidemic SIR models with a saturated incidence rate

2018 ◽  
Vol 26 (4) ◽  
pp. 235-245 ◽  
Author(s):  
Modeste N’zi ◽  
Ilimidi Yattara
Keyword(s):  
Stochastic Model ◽  
White Noise ◽  
Incidence Rate ◽  
Deterministic Model ◽  
Contact Rate ◽  
Almost Sure Stability ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Global Positive Solution ◽  
Disease Free

AbstractWe treat a delayed SIR (susceptible, infected, recovered) epidemic model with a saturated incidence rate and its perturbation through the contact rate using a white noise. We start with a deterministic model and then add a perturbation on the contact rate using a white noise to obtain a stochastic model. We prove the existence and uniqueness of the global positive solution for both deterministic and stochastic delayed differential equations. Under suitable conditions on the parameters, we study the global asymptotic stability of the disease-free equilibrium of the deterministic model and the almost sure stability of the disease-free equilibrium of the stochastic model.

scholarly journals Mathematical Analysis and Stochastic Stability of Nonlinear Epidemic Model with Incidence Rate

2020 ◽  
pp. 8-19
Author(s):  
Laid Chahrazed
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Globally Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Stability ◽  
Disease Free ◽  
Temporary Immunity

In this work, we consider a nonlinear epidemic model with temporary immunity and saturated incidence rate. Size N(t) at time t, is divided into three sub classes, with N(t)=S(t)+I(t)+Q(t); where S(t), I(t) and Q(t) denote the sizes of the population susceptible to disease, infectious and quarantine members with the possibility of infection through temporary immunity, respectively. We have made the following contributions: The local stabilities of the infection-free equilibrium and endemic equilibrium are; analyzed, respectively. The stability of a disease-free equilibrium and the existence of other nontrivial equilibria can be determine by the ratio called the basic reproductive number, This paper study the reduce model with replace S with N, which does not have non-trivial periodic orbits with conditions. The endemic -disease point is globally asymptotically stable if R0 ˃1; and study some proprieties of equilibrium with theorems under some conditions. Finally the stochastic stabilities with the proof of some theorems. In this work, we have used the different references cited in different studies and especially the writing of the non-linear epidemic mathematical model with [1-7]. We have used the other references for the study the different stability and other sections with [8-26]; and sometimes the previous references.

Deterministic and stochastic stability of an SIRS epidemic model with a saturated incidence rate

2017 ◽  
Vol 25 (1) ◽  
Author(s):  
Modeste N’zi ◽  
Jacques Tano
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Deterministic Model ◽  
Random Noise ◽  
Stochastic Version ◽  
Noise Perturbation ◽  
Sirs Epidemic Model ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Disease Free

AbstractIn this paper, we formulate an epidemic model for the spread of an infectious disease in a population of varying size. The total population is divided into three distinct epidemiological subclass of individuals (susceptible, infectious and recovered) and we study a deterministic and stochastic models with saturated incidence rate. The stochastic model is obtained by incorporating a random noise into the deterministic model. In the deterministic case, we briefly discuss the global asymptotic stability of the disease free equilibrium by using a Lyapunov function. For the stochastic version, we study the global existence and positivity of the solution. Under suitable conditions on the intensity of the white noise perturbation, we prove that there are a

Modeling and scientific computing for the transmission dynamics of Avian influenza with half-saturated incidence

2020 ◽  
Vol 11 (04) ◽  
pp. 2050035
Author(s):  
Muhammad Altaf Khan ◽  
Saif Ullah ◽  
Yasir Khan ◽  
Muhammad Farhan
Keyword(s):  
Avian Influenza ◽  
Influenza Infection ◽  
Transmission Dynamics ◽  
Global Dynamics ◽  
Nonlinear Dynamical ◽  
Local And Global Dynamics ◽  
Proposed Model ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Disease Free

This paper presents the mathematical analysis of the dynamical system for avian influenza. The proposed model considers a nonlinear dynamical model of birds and human. The half-saturated incidence rate is used for the transmission of avian influenza infection. Rigorous mathematical results are presented for the proposed models. The local and global dynamics of each model are presented and proven that when [Formula: see text], then the disease-free equilibrium of each model is stable both locally and globally, and when [Formula: see text], then the endemic equilibrium is stable both locally and globally. The numerical results obtained for the proposed model shows that influenza could be eliminated from the community if the threshold is not greater than unity.

scholarly journals On dynamics of fractional incommensurate model of Covid-19 with nonlinear saturated incidence rate

2021 ◽  
Author(s):  
Abdelouahed ALLA HAMOU ◽  
ELHOUSSINE AZROUL ◽  
Zakia Hammouch ◽  
Abdelilah Lamrani alaoui
Keyword(s):  
Incidence Rate ◽  
Least Squares Method ◽  
Real Data ◽  
Euler Method ◽  
Proposed Model ◽  
The World ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Fractional Orders ◽  
Theoretical Results

In December 2019, a new virus belonging to the coronavirus strain has been discovered in Wuhan, China, this virus has attracted world-wide attention and it spread rapidly in the world, reaching nearly 216 countries in the world in November 2020. In this chapter, we study the fractional incommensurate SIQR (susceptible, infections, quarantined and removed) COVID-19 model with nonlinear saturated incidence rate using Atangana-Baleanu fractional derivatives. The existence and uniqueness of the solutions for the fractional model is proved using fixed point theorem, the model are shown to have two equilibrium point (disease free and an endemic equilibrium). Some numerical simulations using Euler method are also carried out to support our theoretical results. We estimated the value of the fractional orders and the parameters of the proposed model using the least squares method. Further, the sensitivity analysis of the parameter is performed as a result, our incommensurate model gives a good approximation to real data of COVID-19.

scholarly journals Pulse Roguing Strategy in a Pine Wilt Disease Epidemic Model with General Nonlinear Incidence Rate

2020 ◽  
Vol 08 (12) ◽  
pp. 2943-2953
Author(s):  
Quanben Sun ◽  
Wugui Chen ◽  
Zhicai Guo ◽  
Weiwei Ji ◽  
Jianping Wang
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Pine Wilt Disease ◽  
Wilt Disease ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Pine Wilt

Host vector dynamics of pine wilt disease model with convex incidence rate

2018 ◽  
Vol 113 ◽  
pp. 31-39 ◽  
Author(s):  
Ghaus ur Rahman ◽  
Kamal Shah ◽  
Fazal Haq ◽  
Naveed Ahmad
Keyword(s):  
Incidence Rate ◽  
Disease Model ◽  
Pine Wilt Disease ◽  
Wilt Disease ◽  
Pine Wilt

Distribution of pine wilt disease with respect to temperature in North America, Japan, and Europe

1987 ◽  
Vol 17 (9) ◽  
pp. 1050-1059 ◽  
Author(s):  
T. A. Rutherford ◽  
J. M. Webster
Keyword(s):  
North America ◽  
Pine Wilt Disease ◽  
Wilt Disease ◽  
Temperature Threshold ◽  
Pinewood Nematode ◽  
Pine Wilt ◽  
Air Temperatures ◽  
The Mean ◽  
High Elevations ◽  
Disease Free

In regions of North America and Japan where the pinewood nematode, Bursaphelenchusxylophilus, and its insect vectors occur, pine wilt disease in susceptible pines appears to be expressed only where the mean air temperature exceeds 20 °C for protracted periods. In these warm areas, susceptible pines grow disease-free only at the cooler, high elevations. Pines resistant to pine wilt transcend the 20 °C temperature threshold without becoming diseased. There are no reports of susceptible pines dying of pine wilt in those regions of Europe, North America, or Japan where mean summer air temperatures are less than 20 °C, despite the presence of pinewood nematode and its vectors in these regions. Bursaphelenchusxylophilus is found throughout most of North America; has been reported from Siberia, China, and France; and is regarded as an introduced pathogen in Japan. We hypothesize that it occurs throughout most taiga forests of the northern hemisphere where predominantly cool climates prevent widespread expression of pine wilt disease. The cool climates of much of Europe, North America, and Asia mitigate against the occurrence of pine wilt disease should B. xylophilus be inadvertently introduced. Susceptible pines that are transplanted from cool to warm regions will be at risk to the disease.

scholarly journals Mathematical Analysis of TB Model with Vaccination and Saturated Incidence Rate

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Ashenafi Kelemu Mengistu ◽  
Peter J. Witbooi
Keyword(s):  
Local Stability ◽  
Reproduction Number ◽  
Model System ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Latently Infected ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Disease Free ◽  
The Basic Reproduction Number

The model system of ordinary differential equations considers two classes of latently infected individuals, with different risk of becoming infectious. The system has positive solutions. By constructing a Lyapunov function, it is proved that if the basic reproduction number is less than unity, then the disease-free equilibrium point is globally asymptotically stable. The Routh-Hurwitz criterion is used to prove the local stability of the endemic equilibrium when R 0 > 1 . The model is illustrated using parameters applicable to Ethiopia. A variety of numerical simulations are carried out to illustrate our main results.

scholarly journals Global Stability of a Host-Vector Model for Pine Wilt Disease with Nonlinear Incidence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Kwang Sung Lee ◽  
Abid Ali Lashari
Keyword(s):  
Global Stability ◽  
Pine Wilt Disease ◽  
Geometric Approach ◽  
Wilt Disease ◽  
Endemic Equilibrium ◽  
Vector Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Pine Wilt ◽  
Disease Free

Based on classical epidemic models, this paper considers a deterministic epidemic model for the spread of the pine wilt disease which has vector mediated transmission. The analysis of the model shows that its dynamics are completely determined by the basic reproduction numberR0. Using a Lyapunov function and a LaSalle's invariant set theorem, we proved the global asymptotical stability of the disease-free equilibrium. We find that ifR0≤1, the disease free equilibrium is globally asymptotically stable, and the disease will be eliminated. IfR0>1, a unique endemic equilibrium exists and is shown to be globally asymptotically stable, under certain restrictions on the parameter values, using the geometric approach method for global stability, due to Li and Muldowney and the disease persists at the endemic equilibrium state if it initially exists.

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