scholarly journals A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed A. Aba Oud ◽  
Aatif Ali ◽  
Hussam Alrabaiah ◽  
Saif Ullah ◽  
Muhammad Altaf Khan ◽  
...  
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Model Parameters ◽  
Disease Dynamics ◽  
Threshold Parameter ◽  
Fractional Model ◽  
Infected People ◽  
Fractional Models ◽  
Disease Epidemics ◽  
The Impact

AbstractCOVID-19 or coronavirus is a newly emerged infectious disease that started in Wuhan, China, in December 2019 and spread worldwide very quickly. Although the recovery rate is greater than the death rate, the COVID-19 infection is becoming very harmful for the human community and causing financial loses to their economy. No proper vaccine for this infection has been introduced in the market in order to treat the infected people. Various approaches have been implemented recently to study the dynamics of this novel infection. Mathematical models are one of the effective tools in this regard to understand the transmission patterns of COVID-19. In the present paper, we formulate a fractional epidemic model in the Caputo sense with the consideration of quarantine, isolation, and environmental impacts to examine the dynamics of the COVID-19 outbreak. The fractional models are quite useful for understanding better the disease epidemics as well as capture the memory and nonlocality effects. First, we construct the model in ordinary differential equations and further consider the Caputo operator to formulate its fractional derivative. We present some of the necessary mathematical analysis for the fractional model. Furthermore, the model is fitted to the reported cases in Pakistan, one of the epicenters of COVID-19 in Asia. The estimated value of the important threshold parameter of the model, known as the basic reproduction number, is evaluated theoretically and numerically. Based on the real fitted parameters, we obtained $\mathcal{R}_{0} \approx 1.50$ R 0 ≈ 1.50 . Finally, an efficient numerical scheme of Adams–Moulton type is used in order to simulate the fractional model. The impact of some of the key model parameters on the disease dynamics and its elimination are shown graphically for various values of noninteger order of the Caputo derivative. We conclude that the use of fractional epidemic model provides a better understanding and biologically more insights about the disease dynamics.

scholarly journals Dynamics of the rumor-spreading model with hesitation mechanism in heterogenous networks and bilingual environment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuai Yang ◽  
Haijun Jiang ◽  
Cheng Hu ◽  
Juan Yu ◽  
Jiarong Li
Keyword(s):  
Lyapunov Method ◽  
Reproduction Number ◽  
Model Parameters ◽  
Rumor Spreading ◽  
Spreading Model ◽  
Reductio Ad Absurdum ◽  
The Stability ◽  
The Impact ◽  
Theoretical Results ◽  
The Basic Reproduction Number

Abstract In this paper, a novel rumor-spreading model is proposed under bilingual environment and heterogenous networks, which considers that exposures may be converted to spreaders or stiflers at a set rate. Firstly, the nonnegativity and boundedness of the solution for rumor-spreading model are proved by reductio ad absurdum. Secondly, both the basic reproduction number and the stability of the rumor-free equilibrium are systematically discussed. Whereafter, the global stability of rumor-prevailing equilibrium is explored by utilizing Lyapunov method and LaSalle’s invariance principle. Finally, the sensitivity analysis and the numerical simulation are respectively presented to analyze the impact of model parameters and illustrate the validity of theoretical results.

scholarly journals Threshold Dynamics in a Model for Zika Virus Disease with Seasonality

2021 ◽  
Vol 83 (4) ◽  
Author(s):  
Mahmoud A. Ibrahim ◽  
Attila Dénes
Keyword(s):  
Zika Virus ◽  
Reproduction Number ◽  
Virus Disease ◽  
Threshold Dynamics ◽  
Global Dynamics ◽  
Threshold Parameter ◽  
Globally Asymptotically Stable ◽  
Linear Integral ◽  
The Impact ◽  
Disease Free

AbstractWe present a compartmental population model for the spread of Zika virus disease including sexual and vectorial transmission as well as asymptomatic carriers. We apply a non-autonomous model with time-dependent mosquito birth, death and biting rates to integrate the impact of the periodicity of weather on the spread of Zika. We define the basic reproduction number $${\mathscr {R}}_{0}$$ R 0 as the spectral radius of a linear integral operator and show that the global dynamics is determined by this threshold parameter: If $${\mathscr {R}}_0 < 1,$$ R 0 < 1 , then the disease-free periodic solution is globally asymptotically stable, while if $${\mathscr {R}}_0 > 1,$$ R 0 > 1 , then the disease persists. We show numerical examples to study what kind of parameter changes might lead to a periodic recurrence of Zika.

scholarly journals Spatial dynamics of a diffusive SIRI model with distinct dispersal rates and heterogeneous environment

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lian Duan ◽  
Lihong Huang ◽  
Chuangxia Huang
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Spatial Dynamics ◽  
Heterogeneous Environment ◽  
Threshold Parameter ◽  
Dispersal Rate ◽  
Siri Model ◽  
The Basic Reproduction Number

<p style='text-indent:20px;'>In this paper, we are concerned with the dynamics of a diffusive SIRI epidemic model with heterogeneous parameters and distinct dispersal rates for the susceptible and infected individuals. We first establish the basic properties of solutions to the model, and then identify the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathscr{R}_{0} $\end{document}</tex-math></inline-formula> which serves as a threshold parameter that predicts whether epidemics will persist or become globally extinct. Moreover, we study the asymptotic profiles of the positive steady state as the dispersal rate of the susceptible or infected individuals approaches zero. Our analytical results reveal that the epidemics can be extinct by limiting the movement of the susceptible individuals, and the infected individuals concentrate on certain points in some circumstances when limiting their mobility.</p>

scholarly journals Analysis of a Diffusive Heroin Epidemic Model in a Heterogeneous Environment

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Jinliang Wang ◽  
Hongquan Sun
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Threshold Dynamics ◽  
Heterogeneous Environment ◽  
Threshold Parameter ◽  
Simulation Results ◽  
Heroin Model

This paper is concerned with a reaction-diffusion heroin model in a bound domain. The objective of this paper is to explore the threshold dynamics based on threshold parameter and basic reproduction number (BRN) ℜ0, and it is proved that if ℜ0<1, heroin spread will be extinct, while if ℜ0>1, heroin spread is uniformly persistent and there exists a positive heroin-spread steady state. We also obtain that the explicit formula of ℜ0 and global attractiveness of constant positive steady state (PSS) when all parameters are positive constants. Our simulation results reveal that compared to the homogeneous setting, the spatial heterogeneity has essential impacts on increasing the risk of heroin spread.

scholarly journals Threshold parameters for a model of epidemic spread among households and workplaces

2009 ◽  
Vol 6 (40) ◽  
pp. 979-987 ◽  
Author(s):  
L. Pellis ◽  
N. M. Ferguson ◽  
C. Fraser
Keyword(s):  
Infectious Disease ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Model Parameters ◽  
Epidemic Spread ◽  
Threshold Parameter ◽  
Disease Epidemiology ◽  
Infectious Disease Epidemiology ◽  
Complex Models ◽  
The Basic Reproduction Number

The basic reproduction number R 0 is one of the most important concepts in modern infectious disease epidemiology. However, for more realistic and more complex models than those assuming homogeneous mixing in the population, other threshold quantities can be defined that are sometimes more useful and easily derived in terms of model parameters. In this paper, we present a model for the spread of a permanently immunizing infection in a population socially structured into households and workplaces/schools, and we propose and discuss a new household-to-household reproduction number R H for it. We show how R H overcomes some of the limitations of a previously proposed threshold parameter, and we highlight its relationship with the effort required to control an epidemic when interventions are targeted at randomly selected households.

THEORETICAL INVESTIGATION OF THE IMPACT ON EPIDEMIC THRESHOLD OF TRAVEL BETWEEN COMMUNITIES OF RESIDENT POPULATIONS

2013 ◽  
Vol 21 (02) ◽  
pp. 1350010 ◽  
Author(s):  
KLOT PATANARAPEELERT ◽  
D. GARCIA LOPEZ ◽  
I-MING TANG ◽  
MARC A. DUBOIS
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Initial Phase ◽  
Reproduction Number ◽  
Travel Patterns ◽  
Epidemic Threshold ◽  
Contact Patterns ◽  
Resident Populations ◽  
The Impact ◽  
The Basic Reproduction Number

During the initial phase of an epidemic, individual displacements between different regions modify the contact patterns. Understanding mobility processes and their consequences is necessary to predict the subsequent spread of the disease in order to optimize control policies. The basic reproduction number is commonly used to determine the threshold between extinction and expansion of the disease. Once it is derived for an epidemic model that includes the travel of population between distinct localities, the dependence of the diseases dynamics upon travel rates becomes explicit. In this study, we examine the effects of travel on the epidemic threshold for a model of two communities. The travel rates are treated as varying subject to two scenarios. We show theoretically that if the transmission potentials within communities are moderate, the epidemic threshold can be modified by travel. The conditions for the presence of the threshold induced by travel is determined and the critical values of travel at which the basic reproduction number is equal to one are derived. We show further that these results can also be applied to a model of three communities under specific travel patterns and that the derived basic reproduction number has a form similar to that of the two communities problem.

scholarly journals Mathematical analysis of an age structured epidemic model with a quarantine class

2021 ◽  
Author(s):  
Bedreddine AINSEBA ◽  
Tarik Touaoula ◽  
Zakia Sari
Keyword(s):  
Reproduction Number ◽  
Asymptotic Behavior Of Solutions ◽  
Model Parameters ◽  
Qualitative Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Structured ◽  
The Stability ◽  
The Impact

In this paper, an age structured epidemic Susceptible-Infected-Quarantined-Recovered-Infected (SIQRI) model is proposed, where we will focus on the role of individuals that leave their class of quarantine before being completely recovered and thus will participate again to the transmission of the disease. We investigate the asymptotic behavior of solutions by studying the stability of both trivial and positive equilibria. In order to see the impact of the different model parameters like the relapse rate on the qualitative behavior of our system, we firstly, give the explicit expression of the epidemic reproduction number $R_{0}.$ This number is a combination of the classical epidemic reproduction number for the SIQR model and a new epidemic reproduction number corresponding to the individuals infected by a relapsed person from the R-class. It is shown that, if $R_{0}\leq 1$, the disease free equilibrium is globally asymptotically stable and becomes unstable for $R_{0}>1$. Secondly, while $R_{0}>1$, a suitable Lyapunov functional is constructed to prove that the unique endemic equilibrium is globally asymptotically stable on some subset $\Omega_{0}.$

scholarly journals Modeling analysis of a SEIQR epidemic model to assess the impact of undetected cases , and predict the early peack of the COVID-19 outbreak in Cameroon

2020 ◽  
Author(s):  
NKAGUE NKAMBA LEONTINE ◽  
MAYOMBE MANN MARTIN LUTHER ◽  
MANGA THOMAS TIMOTHEE ◽  
J. Mbang
Keyword(s):  
Severe Acute Respiratory Syndrome ◽  
Lyapunov Functions ◽  
Reproduction Number ◽  
Benign Diseases ◽  
Infected People ◽  
Modeling Analysis ◽  
Containment Measures ◽  
The Impact ◽  
Case Data ◽  
Disease Free

Abstract COVID-19 is a highly contagious disease, and the strain is severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). It belongs to the coronavirus family, which can result in benign diseases in humans, such as a cold, and can also cause serious pathologies such as Severe Acute Respiratory Syndrome (SARS). In this study, we have modeled the COVID-19 epidemic in Cameroon. We used early reported case data to predict the peak, assess the impact of containment measures, and the impact of undetected infected people on the epidemic trend and characteristics of COVID-19. The basic reproduction number is computed using Lyapunov functions, and the global stability of disease-free and endemic equilibrium are demonstrated.

scholarly journals Meta-analysis of the SARS-CoV-2 serial interval and the impact of parameter uncertainty on the COVID-19 reproduction number

2020 ◽  
Author(s):  
Robert Challen ◽  
Ellen Brooks-Pollock ◽  
Krasimira Tsaneva-Atanasova ◽  
Leon Danon
Keyword(s):  
Incubation Period ◽  
Time Delays ◽  
Hospital Admissions ◽  
Reproduction Number ◽  
Meta Analysis ◽  
Model Parameters ◽  
Small Data ◽  
Generation Interval ◽  
Serial Interval ◽  
The Impact

AbstractThe serial interval of an infectious disease, commonly interpreted as the time between onset of symptoms in sequentially infected individuals within a chain of transmission, is a key epidemiological quantity involved in estimating the reproduction number. The serial interval is closely related to other key quantities, including the incubation period, the generation interval (the time between sequential infections) and time delays between infection and the observations associated with monitoring an outbreak such as confirmed cases, hospital admissions and deaths. Estimates of these quantities are often based on small data sets from early contact tracing and are subject to considerable uncertainty, which is especially true for early COVID-19 data. In this paper we estimate these key quantities in the context of COVID-19 for the UK, including a meta-analysis of early estimates of the serial interval. We estimate distributions for the serial interval with a mean 5.6 (95% CrI 5.1–6.2) and SD 4.2 (95% CrI 3.9–4.6) days (empirical distribution), the generation interval with a mean 4.8 (95% CrI 4.3–5.41) and SD 1.7 (95% CrI 1.0–2.6) days (fitted gamma distribution), and the incubation period with a mean 5.5 (95% CrI 5.1–5.8) and SD 4.9 (95% CrI 4.5–5.3) days (fitted log normal distribution). We quantify the impact of the uncertainty surrounding the serial interval, generation interval, incubation period and time delays, on the subsequent estimation of the reproduction number, when pragmatic and more formal approaches are taken. These estimates place empirical bounds on the estimates of most relevant model parameters and are expected to contribute to modelling COVID-19 transmission.

THE DYNAMICS AND THERAPEUTIC STRATEGIES OF A SEIS EPIDEMIC MODEL

2013 ◽  
Vol 06 (05) ◽  
pp. 1350029 ◽  
Author(s):  
XINZHU MENG ◽  
ZHITAO WU ◽  
TONGQIAN ZHANG
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Therapeutic Strategy ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Epidemic Disease ◽  
Globally Asymptotically Stable ◽  
Effective Therapeutic Strategy ◽  
The Impact ◽  
The Basic Reproduction Number

Based on an epidemic model which Manvendra and Vinay [Mathematical model to simulate infections disease, VSRD-TNTJ3(2) (2012) 60–68] have proposed, we consider the dynamics and therapeutic strategy of a SEIS epidemic model with latent patients and active patients. First, the basic reproduction number is established by applying the method of the next generation matrix. By means of appropriate Lyapunov functions, it is proven that while the basic reproduction number 0 < R0 < 1, the disease-free equilibrium is globally asymptotically stable and the disease eliminates; and if the basic reproduction number R0 > 1, the endemic equilibrium is globally asymptotically stable and therefore the disease becomes endemic. Numerical investigations of their basin of attraction indicate that the locally stable equilibria are global attractors. Second, we consider the impact of treatment on epidemic disease and analytically determine the most effective therapeutic strategy. We conclude that the most effective therapeutic strategy consists of treating both the exposed and the infectious, while treating only the exposed is the least effective therapeutic strategy. Finally, numerical simulations are given to illustrate the effectiveness of the proposed results.

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