ABSTRACT
Mathematical modeling techniques have been used extensively during the human immunodeficiency virus (HIV) epidemic. Drug injection causes increased HIV spread in most countries globally. The media is crucial in spreading health awareness by changing mixing behavior. The published studies show some of the ways that differential equation models can be employed to explain how media awareness programs influence the spread and containment of disease (Greenhalgh et al. Appl Math Comput. 2015;251:539–563). Here we build a differential equation model which shows how disease awareness programs can alter the HIV prevalence in a group of people who inject drugs (PWIDs). This builds on previous work by Greenhalgh and Hay (1997) and Liang et al. (2016). We have constructed a mathematical model to describe the improved model that reduces the spread of the diseases through the effect of awareness of disease on sharing needles and syringes among the PWID population. The model supposes that PWIDs clean their needles before use rather than after. We carry out a steady state analysis and examine local stability. Our discussion has been focused on two ways of studying the influence of awareness of infection levels in epidemic modeling. The key biological parameter of our model is the basic reproductive number R0$$ {R}_0 $$. R0$$ {R}_0 $$ is a crucial number which determines the behavior of the infection. We find that if R0$$ {R}_0 $$ is less than one then the disease-free steady state is the unique steady state and moreover whatever the initial fraction of infected individuals then the disease will die out as time becomes large. If R0$$ {R}_0 $$ exceeds one there is the disease-free steady state and a unique steady state with disease present. We also showed that the disease-free steady state is locally asymptotically stable if R0$$ {R}_0 $$ is less than one, neutrally stable if R0$$ {R}_0 $$ is equal to one and unstable if R0$$ {R}_0 $$ exceeds one. In the last case, when R0$$ {R}_0 $$ is greater than one the endemic steady state was locally asymptotically stable. Our analytical results are confirmed by using simulation with realistic parameter values. In nontechnical terms, the number R0$$ {R}_0 $$ is a critical value describing how the disease will spread. If R0$$ {R}_0 $$ is less than or equal to one then the disease will always die out but if R0$$ {R}_0 $$ exceeds one and disease is present the disease will sustain itself and moreover the numbers of PWIDs with disease will tend to a unique nonzero value.
ABSTRACT
In this paper, a non-singular SIR model with the Mittag-Leffler law is proposed. The nonlinear Beddington-DeAngelis infection rate and Holling type II treatment rate are used. The qualitative properties of the SIR model are discussed in detail. The local and global stability of the model are analyzed. Moreover, some conditions are developed to guarantee local and global asymptotic stability. Finally, numerical simulations are provided to support the theoretical results and used to analyze the impact of face masks, social distancing, quarantine, lockdown, immigration, treatment rate of the disease, and limitation in treatment resources on COVID-19. The graphical results show that face masks, social distancing, quarantine, lockdown, immigration, and effective treatment rates significantly reduce the infected population over time. In contrast, limitation in the availability of treatment raises the infected population. © 2023 The Author(s). Published by IOP Publishing Ltd.
ABSTRACT
In this paper, we propose a COVID-19 epidemic model with quarantine class. The model contains 6 sub-populations, namely the susceptible (S), exposed (E), infected (I), quarantined (Q), recovered (R), and death (D) sub-populations. For the proposed model, we show the existence, uniqueness, non-negativity, and boundedness of solution. We obtain two equilibrium points, namely the disease-free equilibrium (DFE) point and the endemic equilibrium (EE) point. Applying the next generation matrix, we get the basic reproduction number (R0). It is found that R0 is inversely proportional to the quarantine rate as well as to the recovery rate of infected subpopulation. The DFE point always exists and if R0 < 1 then the DFE point is asymptotically stable, both locally and globally. On the other hand, if R0 > 1 then there exists an EE point, which is globally asymptotically stable. Here, there occurs a forward bifurcation driven by R0 . The dynamical properties of the proposed model have been verified our numerical simulations. © 2023 the author(s).
ABSTRACT
In this paper, we propose a COVID-19 epidemic model with quarantine class. The model contains 6 sub-populations, namely the susceptible (S), exposed (E), infected (I), quarantined (Q), recovered (R), and death (D) sub-populations. For the proposed model, we show the existence, uniqueness, non-negativity, and boundedness of solution. We obtain two equilibrium points, namely the disease-free equilibrium (DFE) point and the endemic equilibrium (EE) point. Applying the next generation matrix, we get the basic reproduction number (R0). It is found that R0 is inversely proportional to the quarantine rate as well as to the recovery rate of infected subpopulation. The DFE point always exists and if R0 < 1 then the DFE point is asymptotically stable, both locally and globally. On the other hand, if R0 > 1 then there exists an EE point, which is globally asymptotically stable. Here, there occurs a forward bifurcation driven by R0 . The dynamical properties of the proposed model have been verified our numerical simulations. © 2023 the author(s).
ABSTRACT
In this paper, a mathematical model of the COVID-19 pandemic with lockdown that provides a more accurate representation of the infection rate has been analyzed. In this model, the total population is divided into six compartments: the susceptible class, lockdown class, exposed class, asymptomatic infected class, symptomatic infected class, and recovered class. The basic reproduction number (R0) is calculated using the next-generation matrix method and presented graphically based on different progression rates and effective contact rates of infective individuals. The COVID-19 epidemic model exhibits the disease-free equilibrium and endemic equilibrium. The local and global stability analysis has been done at the disease-free and endemic equilibrium based on R0. The stability analysis of the model shows that the disease-free equilibrium is both locally and globally stable when R0<1, and the endemic equilibrium is locally and globally stable when R0>1 under some conditions. A control strategy including vaccination and treatment has been studied on this pandemic model with an objective functional to minimize. Finally, numerical simulation of the COVID-19 outbreak in India is carried out using MATLAB, highlighting the usefulness of the COVID-19 pandemic model and its mathematical analysis.
ABSTRACT
This current work studies a new mathematical model for SARS-CoV-2. We show how immigration, protection, death rate, exposure, cure rate and interaction of infected people with healthy people affect the population. Our model is SIR model, which has three classes including susceptible, infected and recovered respectively. Here, we find the basic reproduction number and local stability through jacobean matrix. Lyapunvo function theory is used to calculate the global stability for the problem under investigation. Also a nonstandard finite difference sachem (NSFDS) is used to simulate the results.
ABSTRACT
The fact that no there exists yet an absolute treatment or vaccine for COVID-19, which was declared as a pandemic by the World Health Organization (WHO) in 2020, makes very important spread out over time of the epidemic in order to burden less on hospitals and prevent collapsing of the health care system. This case is a consequence of limited resources and is valid for all countries in the world facing with this serious threat. Slowing the speed of spread will probably make that the outbreak last longer, but it will cause lower total death count. In this study, a new SEIR epidemic model formed by taking into account the impact of health care capacity has been examined and local and global stability of the model has been analyzed. In addition, the model has been also supported by some numerical simulations.