Global stability and optimal control of a two-patch tuberculosis epidemic model using fractional-order derivatives

2020 ◽  
Vol 13 (03) ◽  
pp. 2050008
Author(s):  
Hossein Kheiri ◽  
Mohsen Jafari
Keyword(s):  
Optimal Control ◽  
Effective Treatment ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Maximum Level ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
Fractional Differential ◽  
The Impact

In this paper, we propose a fractional-order and two-patch model of tuberculosis (TB) epidemic, in which susceptible, slow latent, fast latent and infectious individuals can travel freely between the patches, but not under treatment infected individuals, due to medical reasons. We obtain the basic reproduction number [Formula: see text] for the model and extend the classical LaSalle’s invariance principle for fractional differential equations. We show that if [Formula: see text], the disease-free equilibrium (DFE) is locally and globally asymptotically stable. If [Formula: see text] we obtain sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. We extend the model by inclusion the time-dependent controls (effective treatment controls in both patches and controls of screening on travel of infectious individuals between patches), and formulate a fractional optimal control problem to reduce the spread of the disease. The numerical results show that the use of all controls has the most impact on disease control, and decreases the size of all infected compartments, but increases the size of susceptible compartment in both patches. We, also, investigate the impact of the fractional derivative order [Formula: see text] on the values of the controls ([Formula: see text]). The results show that the maximum levels of effective treatment controls in both patches increase when [Formula: see text] is reduced from 1, while the maximum level of the travel screening control of infectious individuals from patch 2 to patch 1 increases when [Formula: see text] limits to 1.

Mathematical analysis and optimal control for Chikungunya virus with two routes of infection with nonlinear incidence rate

2021 ◽  
pp. 2150088 ◽  
Author(s):  
Miled El Hajji ◽  
Abdelhamid Zaghdani ◽  
Sayed Sayari
Keyword(s):  
Optimal Control ◽  
Steady State ◽  
Chikungunya Virus ◽  
Optimal Solution ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
State Formula ◽  
Number Formula ◽  
Global Threat

Chikungunya fever, caused by Chikungunya virus (CHIKV) and transmitted to humans by infected Aedes mosquitoes, has posed a global threat in several countries. In this paper, we investigated a modified within-host Chikungunya virus (CHIKV) infection model with antibodies where two routes of infection are considered. In a first step, the basic reproduction number [Formula: see text] was calculated and the local and global stability analysis of the steady states is carried out using the local linearization and the Lyapunov method. It is proven that the CHIKV-free steady-state [Formula: see text] is globally asymptotically stable when [Formula: see text], and the infected steady-state [Formula: see text] is globally asymptotically stable when [Formula: see text]. In a second step, we applied an optimal strategy in order to optimize the infected compartment and to maximize the uninfected one. For this, we formulated a nonlinear optimal control problem. Existence of the optimal solution was discussed and characterized using some adjoint variables. Thus, an algorithm based on competitive Gauss–Seidel-like implicit difference method was applied in order to resolve the optimality system. The theoretical results are confirmed by some numerical simulations.

TRANSMISSION DYNAMICS AND OPTIMAL CONTROL OF AN INFLUENZA MODEL WITH QUARANTINE AND TREATMENT

2012 ◽  
Vol 05 (03) ◽  
pp. 1260011 ◽  
Author(s):  
WEI-WEI SHI ◽  
YUAN-SHUN TAN
Keyword(s):  
Optimal Control ◽  
Disease Transmission ◽  
Reproduction Number ◽  
Influenza Pandemic ◽  
Control Measures ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
Parameter Values ◽  
The Impact ◽  
Disease Free

We develop an influenza pandemic model with quarantine and treatment, and analyze the dynamics of the model. Analytical results of the model show that, if basic reproduction number [Formula: see text], the disease-free equilibrium (DFE) is globally asymptotically stable, if [Formula: see text], the disease is uniformly persistent. The model is then extended to assess the impact of three anti-influenza control measures, precaution, quarantine and treatment, by re-formulating the model as an optimal control problem. We focus primarily on controlling disease with a possible minimal the systemic cost. Pontryagin's maximum principle is used to characterize the optimal levels of the three controls. Numerical simulations of the optimality system, using a set of reasonable parameter values, indicate that the precaution measure is more effective in reducing disease transmission than the other two control measures. The precaution measure should be emphasized.

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.

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.

A mathematical model of cholera in a periodic environment with control actions

2020 ◽  
Vol 13 (04) ◽  
pp. 2050025
Author(s):  
G. Kolaye ◽  
I. Damakoa ◽  
S. Bowong ◽  
R. Houe ◽  
D. Békollè
Keyword(s):  
Control Strategies ◽  
Impulsive Differential Equations ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Periodic Environment ◽  
Control Actions ◽  
Number Formula ◽  
Classical Control ◽  
The Impact ◽  
Disease Free

In this paper, we studied the impact of sensitization and sanitation as possible control actions to curtail the spread of cholera epidemic within a human community. Firstly, we combined a model of Vibrio Cholerae with a generic SIRS cholera model. Classical control strategies in terms of the sensitization of population and sanitation are integrated through the impulsive differential equations. Then we presented the theoretical analysis of the model. More precisely, we computed the disease free equilibrium. We derive the basic reproduction number [Formula: see text] which determines the extinction and the persistence of the infection. We show that the trivial disease-free equilibrium is globally asymptotically stable whenever [Formula: see text], while when [Formula: see text], the trivial disease-free equilibrium is unstable and there exists a unique endemic equilibrium point which is globally asymptotically stable. Theoretical results are supported by numerical simulations, which further suggest that the control of cholera should consider both sensitization and sanitation, with a strong focus on the latter.

scholarly journals Asymptotic Stability for Some Types of Nonlinear Fractional Order Differential-Algebraic Control Systems

2021 ◽  
pp. 623-638
Author(s):  
Sameer Qasim Hasan
Keyword(s):  
Feedback Control ◽  
Asymptotic Stability ◽  
Control Systems ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Stable Solution ◽  
Asymptotically Stable ◽  
Fractional Differential ◽  
Necessary And Sufficient ◽  
Asymptotically Stable Solution

The aim of this paper is to study the asymptotically stable solution of nonlinear single and multi fractional differential-algebraic control systems, involving feedback control inputs, by an effective approach that depends on necessary and sufficient conditions.

scholarly journals Global Stability and Optimal Control Analysis of Malaria Dynamics in the Presence of Human Travelers

2018 ◽  
Vol 10 (1) ◽  
pp. 166-186 ◽  
Author(s):  
Samson Olaniyi ◽  
Kazeem O. Okosun ◽  
Samuel O. Adesanya ◽  
Emmanuel A. Areo
Keyword(s):  
Optimal Control ◽  
Malaria Transmission ◽  
Autonomous System ◽  
Control Strategies ◽  
Global Dynamics ◽  
Control Analysis ◽  
Asymptotically Stable ◽  
Short Term ◽  
Globally Asymptotically Stable ◽  
The Impact

Introduction:The impact of unguarded human movement on the spread of infectious disease like malaria cannot be underestimated. Therefore, this study examines the significance of short term human travelers on malaria transmission dynamics.Methods:A non-autonomous system of ordinary differential equations incorporating four control strategies, namely personal protection, chemo-prophylaxis, chemotherapy and mosquito-reduction effort is presented to describe the dynamics of malaria transmission between two interacting populations. Suitable Lyapunov functions are constructed to analyze the global dynamics of the autonomous version. Moreover, the model which incorporates time-dependent vigilant controls is qualitatively analyzed with the overall goal of minimizing the spread of malaria and the associated costs of control implementation using the optimal control theory. An iterative method of forward-backward Runge-Kutta fourth order scheme is used to simulate the optimality system in order to investigate the effects of the control strategies on the magnitude of infected individuals in the population.Results:Analysis of the autonomous system shows that the disease-free equilibrium is globally asymptotically stable whenever the basic reproduction is less than unity and a uniquely determined endemic equilibrium is shown to be globally asymptotically stable whenever the associated basic reproduction number exceeds unity. In the case of non-autonomous system, necessary conditions for the optimal control of malaria are derived. It is shown that adherence to the combination of the control strategies by short term human travelers helps in curtailing the spread of malaria in the population.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

scholarly journals Stability analysis of a dynamical model of tuberculosis with incomplete treatment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ihsan Ullah ◽  
Saeed Ahmad ◽  
Qasem Al-Mdallal ◽  
Zareen A. Khan ◽  
Hasib Khan ◽  
...  
Keyword(s):  
Disease Transmission ◽  
Contact Rate ◽  
Asymptotically Stable ◽  
Treatment Rate ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Efficient Treatment ◽  
Effective Contact ◽  
The Impact ◽  
Incomplete Treatment

Abstract A simple deterministic epidemic model for tuberculosis is addressed in this article. The impact of effective contact rate, treatment rate, and incomplete treatment versus efficient treatment is investigated. We also analyze the asymptotic behavior, spread, and possible eradication of the TB infection. It is observed that the disease transmission dynamics is characterized by the basic reproduction ratio $\Re _{0}$ ℜ 0 ; if $\Re _{0}<1$ ℜ 0 < 1 , there is only a disease-free equilibrium which is both locally and globally asymptotically stable. Moreover, for $\Re _{0}>1$ ℜ 0 > 1 , a unique positive endemic equilibrium exists which is globally asymptotically stable. The global stability of the equilibria is shown via Lyapunov function. It is also obtained that incomplete treatment of TB causes increase in disease infection while efficient treatment results in a reduction in TB. Finally, for the estimated parameters, some numerical simulations are performed to verify the analytical results. These numerical results indicate that decrease in the effective contact rate λ and increase in the treatment rate γ play a significant role in the TB infection control.

scholarly journals Global Stability and Optimal Control of Dengue with Two Coexisting Virus Serotypes

MATEMATIKA ◽  
2019 ◽  
Vol 35 (4) ◽  
pp. 149-170
Author(s):  
Afeez Abidemi ◽  
Rohanin Ahmad ◽  
Nur Arina Bazilah Aziz
Keyword(s):  
Optimal Control ◽  
Global Stability ◽  
Disease Transmission ◽  
Deterministic Model ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Effective Control ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Boundary Equilibrium

This study presents a two-strain deterministic model which incorporates Dengvaxia vaccine and insecticide (adulticide) control strategies to forecast the dynamics of transmission and control of dengue in Madeira Island if there is a new outbreak with a different virus serotypes after the first outbreak in 2012. We construct suitable Lyapunov functions to investigate the global stability of the disease-free and boundary equilibrium points. Qualitative analysis of the model which incorporates time-varying controls with the specific goal of minimizing dengue disease transmission and the costs related to the control implementation by employing the optimal control theory is carried out. Three strategies, namely the use of Dengvaxia vaccine only, application of adulticide only, and the combination of Dengvaxia vaccine and adulticide are considered for the controls implementation. The necessary conditions are derived for the optimal control of dengue. We examine the impacts of the control strategies on the dynamics of infected humans and mosquito population by simulating the optimality system. The disease-freeequilibrium is found to be globally asymptotically stable whenever the basic reproduction numbers associated with virus serotypes 1 and j (j 2 {2, 3, 4}), respectively, satisfy R01,R0j 1, and the boundary equilibrium is globally asymptotically stable when the related R0i (i = 1, j) is above one. It is shown that the strategy based on the combination of Dengvaxia vaccine and adulticide helps in an effective control of dengue spread in the Island.

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