scholarly journals Dynamical analysis of a fractional order model incorporating fear in the disease transmission rate of COVID-19

2020 ◽  
Vol 99 (99) ◽  
pp. 1-17
Author(s):  
Debasis Mukherjee Debasis Mukherjee ◽  
Chandan Maji
Keyword(s):  
Fractional Order ◽  
Disease Transmission ◽  
Transmission Rate ◽  
Three Dimensional ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Fear Effect ◽  
And Control ◽  
Disease Free

This paper deals with a fractional-order three-dimensional compartmental model with fear effect. We have investigated whether fear can play an important role or not to spread and control the infectious diseases like COVID-19, SARS etc. in a bounded region. The basic results on uniqueness, non-negativity and boundedness of the solution of the system are investigated. Stability analysis ensures that the disease-free equilibrium point is locally asymptotically stable if carrying capacity greater than a certain threshold value.We have also derived the conditions for which endemic equilibrium is globally asymptotically stable that means the disease persists in the system. Numerical simulation suggests that the fear factor is an important role which is observed through Hopf-bifurcation.

scholarly journals HIV/AIDS-Pneumonia Codynamics Model Analysis with Vaccination and Treatment

2022 ◽  
Vol 2022 ◽  
pp. 1-20
Author(s):  
Shewafera Wondimagegnhu Teklu ◽  
Koya Purnachandra Rao
Keyword(s):  
Disease Transmission ◽  
Transmission Rate ◽  
Progression Rate ◽  
Globally Asymptotically Stable ◽  
Standard Data ◽  
Reproduction Numbers ◽  
Manifold Theory ◽  
And Control ◽  
Disease Free ◽  
Hiv Aids

In this paper, we proposed and analyzed a realistic compartmental mathematical model on the spread and control of HIV/AIDS-pneumonia coepidemic incorporating pneumonia vaccination and treatment for both infections at each infection stage in a population. The model exhibits six equilibriums: HIV/AIDS only disease-free, pneumonia only disease-free, HIV/AIDS-pneumonia coepidemic disease-free, HIV/AIDS only endemic, pneumonia only endemic, and HIV/AIDS-pneumonia coepidemic endemic equilibriums. The HIV/AIDS only submodel has a globally asymptotically stable disease-free equilibrium if R 1 < 1 . Using center manifold theory, we have verified that both the pneumonia only submodel and the HIV/AIDS-pneumonia coepidemic model undergo backward bifurcations whenever R 2 < 1   and R 3 = max R 1 , R 2 < 1 , respectively. Thus, for pneumonia infection and HIV/AIDS-pneumonia coinfection, the requirement of the basic reproduction numbers to be less than one, even though necessary, may not be sufficient to completely eliminate the disease. Our sensitivity analysis results demonstrate that the pneumonia disease transmission rate   β 2 and the HIV/AIDS transmission rate   β 1 play an important role to change the qualitative dynamics of HIV/AIDS and pneumonia coinfection. The pneumonia infection transmission rate β 2 gives rises to the possibility of backward bifurcation for HIV/AIDS and pneumonia coinfection if R 3 = max R 1 , R 2 < 1 , and hence, the existence of multiple endemic equilibria some of which are stable and others are unstable. Using standard data from different literatures, our results show that the complete HIV/AIDS and pneumonia coinfection model reproduction number is R 3 = max R 1 , R 2 = max 1.386 , 9.69   = 9.69   at β 1 = 2 and β 2 = 0.2   which shows that the disease spreads throughout the community. Finally, our numerical simulations show that pneumonia vaccination and treatment against disease have the effect of decreasing pneumonia and coepidemic disease expansion and reducing the progression rate of HIV infection to the AIDS stage.

The impact of media coverage with a piecewise transmission rate in the spread of diseases

2016 ◽  
Vol 10 (01) ◽  
pp. 1750003
Author(s):  
Maoxing Liu ◽  
Lixia Zuo
Keyword(s):  
Basic Reproduction Number ◽  
Media Coverage ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Three Dimensional ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Contact Transmission ◽  
The Impact ◽  
The Basic Reproduction Number

A three-dimensional compartmental model with media coverage is proposed to describe the real characteristics of its impact in the spread of infectious diseases in a given region. A piecewise continuous transmission rate is introduced to describe that media coverage exhibits its effect only when the number of the infected exceeds a certain critical level. Further, it is assumed that the impact of media coverage on the contact transmission is described by an exponential decreasing factor. Stability analysis of the model shows that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity. On the other hand, when the basic reproduction number is greater than unity and media coverage impact is sufficiently small, a unique endemic equilibrium exists, which is globally asymptotically stable.

scholarly journals Mathematical Modeling of Echinococcosis in Humans, Dogs, and Sheep

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Getachew Bitew Birhan ◽  
Justin Manango W. Munganga ◽  
Adamu Shitu Hassan
Keyword(s):  
Cystic Echinococcosis ◽  
Transmission Rate ◽  
Interaction Model ◽  
Transmission Dynamics ◽  
Human Populations ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Prey Interaction ◽  
Disease Free

In this paper, a model for the transmission dynamics of cystic echinococcosis in the dog, sheep, and human populations is developed and analyzed. We first model and analyze the predator-prey interaction model in these populations; then, we propose a mathematical model of the transmission dynamics of cystic echinococcosis. We calculate the basic reproduction number R 0 and prove that the disease-free equilibrium is globally asymptotically stable, and hence, the disease dies out if R 0 > 1 . We further show that the endemic equilibrium is globally asymptotically stable, and hence, the disease persists if R 0 < 1 . Numerical simulations are performed to illustrate our analytic results. We give sensitivity analysis of the key parameters and give strategies that are helpful to control the transmission of cystic echinococcosis, from which the most sensitive parameter is the transmission rate of Echinococcus’ eggs from the environment to sheep ( β es ). Thus, the effective controlling strategies are associated with this parameter.

ON THE GLOBAL STABILITY OF CHOLERA MODEL WITH PREVENTION AND CONTROL

2018 ◽  
Vol 3 (1) ◽  
pp. 28
Author(s):  
M O Ibrahim ◽  
A A Ayoade ◽  
O J Peter ◽  
F A Oguntolu
Keyword(s):  
Global Stability ◽  
Prevention And Control ◽  
Control Measures ◽  
Extended Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
First Order ◽  
Prevention And Control Measures ◽  
And Control ◽  
Disease Free

In this study, a system of first order ordinary differential equations is used to analyse the dynamics of cholera disease via a mathematical model extended from Fung (2014) cholera model. The global stability analysis is conducted for the extended model by suitable Lyapunov function and LaSalle’s invariance principle. It is shown that the disease free equilibrium (DFE) for the extended model is globally asymptotically stable if 𝑅0 𝑞 < 1 and the disease eventually disappears in the population with time while there exists a unique endemic equilibrium that is globally asymptotically stable whenever 𝑅0 𝑞 > 1 for the extended model or 𝑅0 > 1 for the original model and the disease persists at a positive level though with mild waves (i.e few cases of cholera) in the case of𝑅0 𝑞 > 1. Numerical simulations for strong, weak, and no prevention and control measures are carried out to verify the analytical results and Maple 18 is used to carry out the computations.

Analysis of epidemic spreading with feedback mechanism in weighted networks

2015 ◽  
Vol 08 (01) ◽  
pp. 1550007 ◽  
Author(s):  
Jiancheng Zhang ◽  
Jitao Sun
Keyword(s):  
Local Stability ◽  
Transmission Rate ◽  
Feedback Mechanism ◽  
Asymptotically Stable ◽  
Epidemic Spreading ◽  
Epidemic Threshold ◽  
Globally Asymptotically Stable ◽  
Weighted Networks ◽  
Disease Free ◽  
Good Agreement

In this paper, we investigate a new model with a generalized feedback mechanism in weighted networks. Compare to previous models, we consider the initiative response of people and the important impact of nodes with different edges on transmission rate as epidemics prevail. Furthermore, by constructing Lyapunov function, we prove that the disease-free equilibrium E0 is globally asymptotically stable as the epidemic threshold R* < 1. When R* > 1, we obtain the permanence of epidemic and the local stability of endemic equilibrium E*. Finally, one can find a good agreement between numerical simulations and our analytical results.

Dynamic Study of SIQR-B Fractional-Order Epidemic Model of Cholera with Optimal Control Strategies in Mayo-Tsanaga Department of Cameroon Far North Region

2020 ◽  
Vol 15 (04) ◽  
pp. 237-273
Author(s):  
Tchule Nguiwa ◽  
Mibaile Justin ◽  
Djaouda Moussa ◽  
Gambo Betchewe ◽  
Alidou Mohamadou
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Control Strategies ◽  
Dynamical Behavior ◽  
Contamination Rate ◽  
Asymptotically Stable ◽  
Invariant Region ◽  
Globally Asymptotically Stable ◽  
Positive Orthant ◽  
Disease Free

In this paper, we investigated the dynamical behavior of a fractional-order model of the cholera epidemic in Mayo-Tsanaga Department. We extended the model of Lemos-Paião et al. [A. P. Lemos-Paião, C. J. Silva and D. F. M. Torres, J. Comput. Appl. Math. 16, 427 (2016)] by incorporating the contact rate [Formula: see text] by handling cholera death and optimal control strategies such as vaccination [Formula: see text], water sanitation [Formula: see text]. We provide a theoretical study of the model. We derive the basic reproduction number [Formula: see text] which determines the extinction and the persistence of the infection. We show that the disease-free equilibrium is globally asymptotically stable whenever [Formula: see text], while when [Formula: see text], the disease-free equilibrium is unstable and there exists a unique endemic equilibrium point which is locally asymptotically stable on a positively invariant region of the positive orthant. Using the sensitivity analysis, we find that the parameter related to vaccination and therapeutic treatment is more influencing the model. Theoretical results are supported by numerical simulations, which further suggest use of vaccination in endemic area. In case of a lack of necessary funding to fight again cholera, Figure 6 revealed that efforts should focus to keep contamination rate [Formula: see text] (susceptible-to-cholera death) in other to die out the disease.

scholarly journals Global Stability Analysis of Two-Stage Quarantine-Isolation Model with Holling Type II Incidence Function

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 350 ◽  
Author(s):  
Mohammad A. Safi
Keyword(s):  
Disease Transmission ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Control Measures ◽  
Asymptotically Stable ◽  
Two Stage ◽  
Globally Asymptotically Stable ◽  
Incidence Function ◽  
Special Case ◽  
Disease Free

A new two-stage model for assessing the effect of basic control measures, quarantine and isolation, on a general disease transmission dynamic in a population is designed and rigorously analyzed. The model uses the Holling II incidence function for the infection rate. First, the basic reproduction number ( R 0 ) is determined. The model has both locally and globally asymptotically stable disease-free equilibrium whenever R 0 < 1 . If R 0 > 1 , then the disease is shown to be uniformly persistent. The model has a unique endemic equilibrium when R 0 > 1 . A nonlinear Lyapunov function is used in conjunction with LaSalle Invariance Principle to show that the endemic equilibrium is globally asymptotically stable for a special case.

Assessing the impact of transmissibility on a cluster-based COVID-19 model in India

2021 ◽  
pp. 2141002
Author(s):  
Tanvi ◽  
Mohammad Sajid ◽  
Rajiv Aggarwal ◽  
Ashutosh Rajput
Keyword(s):  
Reproduction Number ◽  
Transmission Rate ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Nonlinear Mathematical Model ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
The Stability ◽  
The Impact ◽  
Disease Free

In this paper, we have proposed a nonlinear mathematical model of different classes of individuals for coronavirus (COVID-19). The model incorporates the effect of transmission and treatment on the occurrence of new infections. For the model, the basic reproduction number [Formula: see text] has been computed. Corresponding to the threshold quantity [Formula: see text], the stability of endemic and disease-free equilibrium (DFE) points are determined. For [Formula: see text], if the endemic equilibrium point exists, then it is locally asymptotically stable, whereas the DFE point is globally asymptotically stable for [Formula: see text] which implies the eradication of the disease. The effects of various parameters on the spread of COVID-19 are discussed in the segment of sensitivity analysis. The model is numerically simulated to understand the effect of reproduction number on the transmission dynamics of the disease COVID-19. From the numerical simulations, it is concluded that if the reproduction number for the coronavirus disease is reduced below unity by decreasing the transmission rate and detecting more number of infectives, then the epidemic can be eradicated from the population.

scholarly journals Dynamics of Fractional-Order Epidemic Models with General Nonlinear Incidence Rate and Time-Delay

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1829
Author(s):  
Ardak Kashkynbayev ◽  
Fathalla A. Rihan
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Incidence Rate ◽  
Fractional Order ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Theoretical Results ◽  
Disease Free

In this paper, we study the dynamics of a fractional-order epidemic model with general nonlinear incidence rate functionals and time-delay. We investigate the local and global stability of the steady-states. We deduce the basic reproductive threshold parameter, so that if R0<1, the disease-free steady-state is locally and globally asymptotically stable. However, for R0>1, there exists a positive (endemic) steady-state which is locally and globally asymptotically stable. A Holling type III response function is considered in the numerical simulations to illustrate the effectiveness of the theoretical results.

scholarly journals A Two-Strain Deterministic Dengue Model With Seasonality Effect

2019 ◽  
Vol 1 (2) ◽  
pp. 13-15
Author(s):  
Afeez Abidemi ◽  
Mohd Ismail Abd Aziz ◽  
Rohanin Ahmad
Keyword(s):  
Disease Transmission ◽  
Birth Rate ◽  
Climatic Factors ◽  
Secondary Infection ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Dengue Disease ◽  
Reproduction Numbers ◽  
Disease Free

A compartmental model is proposed to describe the dynamics of vector-host interations for dengue disease transmission with coexistence of two virus serotypes. The model is modified to incorporate  seasonal-dependent mosquito birth rate in order to examine the influence of climatic factors such as rainfall and temperature on the dynamics of mosquito population and dengue disease transmission. The Next Generation Matrix method is used to obtain the basic reproduction number associated with the model without seasonality effect. The global dynamics of the model is analysed using the Comparison Theorem. The model is simulated in MATLAB with ode45 routine for two cases, namely: the less aggressive case (Case ) and the more aggressive case (Case ).  Analysis of the model shows that the Disease-Free Equilibrium (DFE) is locally asymptotically stable whenever both the basic reproduction numbers  R01 (associated with strain  only) and  R0j (associated with strain  only) are below unity. It is shown that the DFE is globally asymptotically stable when the susceptibility indices for secondary infection in strain  1 ( sigma1) and strain j ( sigmaj), and  are all less than 1.

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