scholarly journals Mathematical Study for Chikungunya Virus with Nonlinear General Incidence Rate

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2186
Author(s):  
Salah Alsahafi ◽  
Stephen Woodcock
Keyword(s):  
Local Stability ◽  
Chikungunya Virus ◽  
Equilibrium Points ◽  
Recruitment Strategy ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Infected Cells ◽  
Stability Results ◽  
Disease Free ◽  
Nonlinear Optimal Control Problem

In this article, we examine the dynamics of a Chikungunya virus (CHIKV) infection model with two routes of infection. The model uses four categories, namely, uninfected cells, infected cells, the CHIKV virus, and antibodies. The equilibrium points of the model, which consist of the free point for the CHIKV and CHIKV endemic point, are first analytically determined. Next, the local stability of the equilibrium points is studied, based on the basic reproduction number (R0) obtained by the next-generation matrix. From the analysis, it is found that the disease-free point is locally asymptotically stable if R0≤1, and the CHIKV endemic point is locally asymptotically stable if R0>1. Using the Lyapunov method, the global stability analysis of the steady-states confirms the local stability results. We then describe our design of an optimal recruitment strategy to minimize the number of infected cells, as well as a nonlinear optimal control problem. Some numerical simulations are provided to visualize the analytical results obtained.

scholarly journals A Mathematical Modeling of Tuberculosis Dynamics with Hygiene Consciousness as a Control Strategy

2020 ◽  
pp. 38-48
Author(s):  
Phineas Z. Mawira ◽  
David M. Malonza
Keyword(s):  
Stability Analysis ◽  
Global Stability ◽  
Control Strategy ◽  
Local Stability ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Linear Ordinary Differential Equations ◽  
Global Stability Analysis ◽  
Local Stability Analysis ◽  
Disease Free

Tuberculosis, an airborne infectious disease, remains a major threat to public health in Kenya. In this study, we derived a system of non-linear ordinary differential equations from the SLICR mathematical model of TB to study the effects of hygiene consciousness as a control strategy against TB in Kenya. The effective basic reproduction number (R0) of the model was determined by the next generation matrix approach. We established and analyzed the equilibrium points. Using the Routh-Hurwitz criterion for local stability analysis and comparison theorem for global stability analysis, the disease-free equilibrium (DFE) was found to be locally asymptotically stable given that R0 < 1.  Also by using the Routh-Hurwitz criterion for local stability analysis and Lyapunov function and LaSalle’s invariance principle for global stability analysis, the endemic equilibrium (EE) point was found to be locally asymptotically stable given that R0 > 1. Using MATLAB ode45 solver, we simulated the model numerically and the results suggest that hygiene consciousness can helpin controlling TB disease if incorporated effectively.

scholarly journals Kestabilan Model Epidemi Seir dengan Matriks Hurwitz

2016 ◽  
Vol 11 (2) ◽  
pp. 74
Author(s):  
Roni Tri Putra ◽  
Sukatik - ◽  
Sri Nita
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Local Stability ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Stability Of Equilibrium ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, it will be studied local stability of equilibrium points of  a SEIR epidemic model with infectious force in latent, infected and immune period. From the model it will be found investigated the existence and its stability of points its equilibrium by Hurwitz matrices. The local stability of equilibrium points is depending on the value of the basic reproduction number  If   the disease free equilibrium is local asymptotically stable.

scholarly journals Mathematical Modeling of Typhoid Fever Disease Incorporating Unprotected Humans in the Spread Dynamics

2019 ◽  
pp. 1-11
Author(s):  
Julia Wanjiku Karunditu ◽  
George Kimathi ◽  
Shaibu Osman
Keyword(s):  
Typhoid Fever ◽  
Global Stability ◽  
Equilibrium Point ◽  
Local Stability ◽  
Jacobian Matrix ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Spread Dynamics ◽  
Disease Free

A deterministic mathematical model of typhoid fever incorporating unprotected humans is formulated in this study and employed to study local and global stability of equilibrium points. The model incorporating Susceptible, unprotected, Infectious and Recovered humans which are analyzed mathematically and also result into a system of ordinary differential equations which are used for interpretations and comparison to the qualitative solutions in studying the spread dynamics of typhoid fever. Jacobian matrix was considered in the study of local stability of disease free equilibrium point and Castillo-Chavez approach used to determine global stability of disease free equilibrium point. Lyapunov function was used to study global stability of endemic equilibrium point. Both equilibrium points (DFE and EE) were found to be local and globally asymptotically stable. This means that the disease will be dependent on numbers of unprotected humans and other factors who contributes positively to the transmission dynamics.

Dynamics of an ecological system

2019 ◽  
Vol 10 (4) ◽  
pp. 355-376
Author(s):  
Shashi Kant
Keyword(s):  
Saddle Point ◽  
Numerical Simulations ◽  
Equilibrium Point ◽  
Local Stability ◽  
Deterministic Model ◽  
Equilibrium Points ◽  
Prey Refuge ◽  
Trivial Equilibrium ◽  
Stability Results ◽  
Predator System

AbstractIn this paper, we investigate the deterministic and stochastic prey-predator system with refuge. The basic local stability results for the deterministic model have been performed. It is found that all the equilibrium points except the positive coexisting equilibrium point of the deterministic model are independent of the prey refuge. The trivial equilibrium point, predator free equilibrium point and prey free equilibrium point are always unstable (saddle point). The existence and local stability of the coexisting equilibrium point is related to the prey refuge. The permanence and extinction conditions of the proposed biological model have been studied rigourously. It is observed that the stochastic effect may be seen in the form of decaying of the species. The numerical simulations for different values of the refuge values have also been included for understanding the behavior of the model graphically.

THE EFFECTS OF VACCINATION AND TREATMENT ON THE SPREAD OF HIV/AIDS

2004 ◽  
Vol 12 (04) ◽  
pp. 399-417 ◽  
Author(s):  
M. KGOSIMORE ◽  
E. M. LUNGU
Keyword(s):  
Equilibrium Point ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Treatment Programs ◽  
Asymptotically Stable ◽  
Reproduction Numbers ◽  
Threshold Conditions ◽  
Existence And Stability ◽  
Disease Free ◽  
Hiv Aids

This study investigates the effects of vaccination and treatment on the spread of HIV/AIDS. The objectives are (i) to derive conditions for the success of vaccination and treatment programs and (ii) to derive threshold conditions for the existence and stability of equilibria in terms of the effective reproduction number R. It is found, firstly, that the success of a vaccination and treatment program is achieved when R0t<R0, R0t<R0v and γeRVT(σ)<RUT(α), where R0t and R0v are respectively the reproduction numbers for populations consisting entirely of treated and vaccinated individuals, R0 is the basic reproduction number in the absence of any intervention, RUT(α) and RVT(σ) are respectively the reproduction numbers in the presence of a treatment (α) and a combination of vaccination and treatment (σ) strategies. Secondly, that if R<1, there exists a unique disease free equilibrium point which is locally asymptotically stable, while if R>1 there exists a unique locally asymptotically stable endemic equilibrium point, and that the two equilibrium points coalesce at R=1. Lastly, it is concluded heuristically that the stable disease free equilibrium point exists when the conditions R0t<R0, R0t<R0v and γeRVT(σ)<RUT(α) are satisfied.

scholarly journals Stability Analysis and Clinic Phenomenon Simulation of a Fractional-Order HBV Infection Model

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Yongmei Su ◽  
Sinuo Liu ◽  
Shurui Song ◽  
Xiaoke Li ◽  
Yongan Ye
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Mass Action ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Infection Model ◽  
Order Model ◽  
Infected Cells ◽  
Set Up ◽  
Hbv Model

In this paper, a fractional-order HBV model was set up based on standard mass action incidences and quasisteady assumption. The basic reproductive number R0 and the cytotoxic T lymphocytes’ immune-response reproductive number R1 were derived. There were three equilibrium points of the model, and stable analysis of each equilibrium point was given with corresponding hypothesis about R0 or R1. Some numerical simulations were also given based on HBeAg clinical data, and the simulation showed that there existed positive logarithmic correlation between the number of infected cells and HBeAg, which was consistent with the clinical facts. The simulation also showed that the clinical individual differences should be reflected by the fractional-order model.

scholarly journals MATHEMATICAL ANALYSIS OF THE ROLE OF DETECTION RATE IN THE DYNAMICAL SPREAD OF HIV-TB CO-INFECTION

2016 ◽  
Vol 11 (10) ◽  
pp. 5715-5741
Author(s):  
SUNDAY OLUMUYIWA Adewale ◽  
I.A Olopade ◽  
I.T Mohammed ◽  
S.O Ajao ◽  
O.T Oyedemi
Keyword(s):  
Detection Rate ◽  
Equilibrium Points ◽  
Population Level ◽  
Transformation Method ◽  
Infection Model ◽  
Hiv Model ◽  
Public Health Burden ◽  
Immunodeficiency Virus ◽  
Effect Of Treatment ◽  
Disease Free

Human Immunodeficiency Virus (HIV) co-existing with Tuberculosis (TB) in individuals remains a major global health challenges, with an estimated 1.4 million patients worldwide. These two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. We formulated new fifteen (15) compartmental models to gain more insight into the effect of treatment and detection of infected undetected individuals on the dynamical spread of HIV- TB co-infection. Sub models of HIV and TB only were considered first, followed by the full HIV-TB co-infection model. Existence and uniqueness of HIV and TB only model were analyzed quantitatively, and we shown that HIV model only and TB only model have solutions, moreover, the solutions are unique. Stability of HIV model only, TB model only and full model of HIV-TB co-infection were analyzed for the existence of the disease free and endemic equilibrium points. Basic reproduction number () was analyzed, using next generation matrix method (NGM), and it has been shown that the disease free equilibrium point is locally asymptotically stable whenever  and unstable whenever this threshold exceeds unity. i.e., Numerical simulation was carried out by maple software using differential transformation method, to show the effect of  treatment and detection of infected undetected individuals on the dynamical spread of HIV-TB co-infection. Significantly, all the results obtained from this research show the importance of treatment and detection of infected undetected individuals on the dynamical spread of HIV-TB co-infection. Detection rate of infected undetected individuals reduce the spread of HIV-TB co-infections.

scholarly journals Mathematical Analysis of A Tuberculosis-Lymphatic filariasis Co-infection Model

2021 ◽  
Author(s):  
Egberanmwen Barry Iyare ◽  
Rosemary Uwaila Akhaze ◽  
Ignatius Imariabe Ako
Keyword(s):  
Lymphatic Filariasis ◽  
Reproduction Number ◽  
Recruitment Rate ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Immune Disorder ◽  
Breeding Sites ◽  
Mosquito Mortality ◽  
Well Posed ◽  
Disease Free

Abstract An Ordinary Differential Equation(ODE) co-infection model of Tuberculosis-Lymphatic filariasis is proposed with 17 mutually disjoint compartments. We showed that the model is Mathematically and Epidemiologically well-posed and the disease-free equilibrium(DFE) of the co-infection model is locally asymptotically stable if R_O<1, it is unstable if R_O>1. The numerical results show that lymphatic \textit{filariasis} infection increases susceptibility to tuberculosis infection. This is in agreement with literature that, persons with lowered immunity such as HIV, diabetes, immune disorder etc are at a higher risk of contacting infectious diseases. We also found that increasing the rate of diagnosis and treatment of active tuberculosis and symptomatic lymphatic \textit{filariasis} cases, the incidence of co-infection in the community can be reduced, and that if resources are limited, efforts should be targeted at treating only the co-infected cases. Sensitivity analysis showed that increasing mosquito mortality rate, reducing the probability of mosquito infecting humans, reducing the probability of humans infecting mosquitoes, reducing mosquitoes recruitment rate by destroying mosquitoes breeding sites and reducing the number of times a mosquito bites human will bring down the reproduction number to less than unity.

AN SIQS INFECTION MODEL WITH NONLINEAR AND ISOLATION

2008 ◽  
Vol 01 (02) ◽  
pp. 239-245
Author(s):  
YANG XIUXIANG ◽  
XUE CHUNRONG
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Liapunov Function ◽  
Threshold Value ◽  
Infection Model ◽  
Stable Theory ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Local Disease ◽  
Disease Free

By means of asymptotically stable theory and infection model theory of ordinary differential equation, we do research on SIQS model with nonlinear and isolation. Firstly, we obtain the existence of threshold value R0 of disease-free equilibration point and local disease equilibration point. Secondly, we prove disease-free equilibration point is locally asymptotically stable when R0 < 1, and local disease equilibration point is locally asymptotically stable when R0 > 1. Furthermore, we have disease-free equilibration point and local disease equilibration point are globally asymptotically stable with the help of Liapunov function. Lastly, we explain at the point of biology.

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.

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