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.

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 Modeling the effect of temperature variability on malaria control strategies

2020 ◽  
Vol 15 ◽  
pp. 65
Author(s):  
Salisu M. Garba ◽  
Usman A. Danbaba
Keyword(s):  
Control Strategies ◽  
Effect Of Temperature ◽  
Bed Nets ◽  
Asymptotically Stable ◽  
Temperature Changes ◽  
Offspring Number ◽  
Local Asymptotic Stability ◽  
Reproduction Numbers ◽  
The Impact ◽  
Disease Free

In this study, a non-autonomous (temperature dependent) and autonomous (temperature independent) models for the transmission dynamics of malaria in a population are designed and rigorously analysed. The models are used to assess the impact of temperature changes on various control strategies. The autonomous model is shown to exhibit the phenomenon of backward bifurcation, where an asymptotically-stable disease-free equilibrium (DFE) co-exists with an asymptotically-stable endemic equilibrium when the associated reproduction number is less than unity. This phenomenon is shown to arise due to the presence of imperfect vaccines and disease-induced mortality rate. Threshold quantities (such as the basic offspring number, vaccination and host type reproduction numbers) and their interpretations for the models are presented. Conditions for local asymptotic stability of the disease-free solutions are computed. Sensitivity analysis using temperature data obtained from Kwazulu Natal Province of South Africa [K. Okuneye and A.B. Gumel. Mathematical Biosciences 287 (2017) 72–92] is used to assess the parameters that have the most influence on malaria transmission. The effect of various control strategies (bed nets, adulticides and vaccination) were assessed via numerical simulations.

scholarly journals Modeling Transmission of Tuberculosis with MDR and Undetected Cases

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Yongqi Liu ◽  
Zhendong Sun ◽  
Guiquan Sun ◽  
Qiu Zhong ◽  
Li Jiang ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Theoretical Analysis ◽  
Control Strategies ◽  
Multidrug Resistant ◽  
Vital Role ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Tb Control ◽  
Disease Free

This paper presents a novel mathematical model with multidrug-resistant (MDR) and undetected TB cases. The theoretical analysis indicates that the disease-free equilibrium is globally asymptotically stable ifR0<1; otherwise, the system may exist a locally asymptotically stable endemic equilibrium. The model is also used to simulate and predict TB epidemic in Guangdong. The results imply that our model is in agreement with actual data and the undetected rate plays vital role in the TB trend. Our model also implies that TB cannot be eradicated from population if it continues to implement current TB control strategies.

Modeling transmission dynamics of Ebola virus disease

2017 ◽  
Vol 10 (04) ◽  
pp. 1750057 ◽  
Author(s):  
Mudassar Imran ◽  
Adnan Khan ◽  
Ali R. Ansari ◽  
Syed Touqeer Hussain Shah
Keyword(s):  
Ebola Virus ◽  
Virus Disease ◽  
Control Strategies ◽  
Transmission Dynamics ◽  
Ebola Virus Disease ◽  
Transmission Model ◽  
Globally Asymptotically Stable ◽  
Dynamical Systems Analysis ◽  
Number Formula ◽  
Disease Free

Ebola virus disease (EVD) has emerged as a rapidly spreading potentially fatal disease. Several studies have been performed recently to investigate the dynamics of EVD. In this paper, we study the transmission dynamics of EVD by formulating an SEIR-type transmission model that includes isolated individuals as well as dead individuals that are not yet buried. Dynamical systems analysis of the model is performed, and it is consequently shown that the disease-free steady state is globally asymptotically stable when the basic reproduction number, [Formula: see text] is less than unity. It is also shown that there exists a unique endemic equilibrium when [Formula: see text]. Using optimal control theory, we propose control strategies, which will help to eliminate the Ebola disease. We use data fitting on models, with and without isolation, to estimate the basic reproductive numbers for the 2014 outbreak of EVD in Liberia and Sierra Leone.

scholarly journals Modeling the Parasitic Filariasis Spread by Mosquito in Periodic Environment

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Yan Cheng ◽  
Xiaoyun Wang ◽  
Qiuhui Pan ◽  
Mingfeng He
Keyword(s):  
Sensitivity Analysis ◽  
Periodic Solution ◽  
Numerical Simulations ◽  
Parasitic Infection ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Threshold Parameter ◽  
Globally Asymptotically Stable ◽  
Periodic Environment ◽  
Disease Free

In this paper a mosquito-borne parasitic infection model in periodic environment is considered. Threshold parameterR0is given by linear next infection operator, which determined the dynamic behaviors of system. We obtain that whenR0<1, the disease-free periodic solution is globally asymptotically stable and whenR0>1by Poincaré map we obtain that disease is uniformly persistent. Numerical simulations support the results and sensitivity analysis shows effects of parameters onR0, which provided references to seek optimal measures to control the transmission of lymphatic filariasis.

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.

THE IMPACT OF MEDIA COVERAGE ON THE DYNAMICS OF INFECTIOUS DISEASE

2008 ◽  
Vol 01 (01) ◽  
pp. 65-74 ◽  
Author(s):  
YIPING LIU ◽  
JING-AN CUI
Keyword(s):  
Infectious Disease ◽  
Media Coverage ◽  
Compartment Model ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Impact ◽  
Globally Stable ◽  
Disease Free

In this paper, we give a compartment model to discuss the influence of media coverage to the spreading and controlling of infectious disease in a given region. The model exhibits two equilibria: a disease-free and a unique endemic equilibrium. Stability analysis of the models shows that the disease-free equilibrium is globally asymptotically stable if the reproduction number (ℝ0), which depends on parameters, is less than unity. But if ℝ0 > 1, it is shown that a unique endemic equilibrium appears, which is asymptotically stable. On a special case, the endemic equilibrium is globally stable. We discuss the role of media coverage on the spreading based on the theory results.

Theoretical assessment of the impact of environmental contamination on the dynamical transmission of polio

2019 ◽  
Vol 12 (02) ◽  
pp. 1950012 ◽  
Author(s):  
C. Balde ◽  
M. Lam ◽  
A. Bah ◽  
S. Bowong ◽  
J. J. Tewa
Keyword(s):  
Basic Reproduction Number ◽  
Environmental Contamination ◽  
Reproduction Number ◽  
Control Strategies ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Indirect Transmission ◽  
Populations At Risk ◽  
The Impact ◽  
The Basic Reproduction Number

A mathematical model for the dynamical transmission of polio is considered, with the aim of investigating the impact of environment contamination. The model captures two infection pathways through both direct human-to-human transmission and indirect human-to-environment-to-human transmission by incorporating the environment as a transition and/or reservoir of viruses. We derive the basic reproduction number [Formula: see text]. We show that the disease free equilibrium is globally asymptotically stable (GAS) if [Formula: see text], while if [Formula: see text], there exists a unique endemic equilibrium which is locally asymptotically stable (LAS). Similar results hold for environmental contamination free sub-model (without the incorporation of the indirect transmission). At the endemic level, we show that the number of infected individuals for the model with the environmental-related contagion is greater than the corresponding number for the environmental contamination free sub-model. In conjunction with the inequality [Formula: see text], where [Formula: see text] is the basic reproduction number for the environmental contamination free sub-model, our finding suggests that the contaminated environment plays a detrimental role on the transmission dynamics of polio disease by increasing the endemic level and the severity of the outbreak. Therefore, it is natural to implement control strategies to reduce the severity of the disease by providing adequate hygienic living conditions, educate populations at risk to follow rigorously those basic hygienic rules in order to avoid adequate contacts with suspected contaminated objects. Further, we perform numerical simulations to support the theory.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document