A fractional-order model of COVID-19 considering the fear effect of the media and social networks on the community

A fractional-order mathematical model based on vaccinated and infected compartments of SARS-CoV-2 with a real case study during the last stages of the epidemiological event

In 2020 the world faced with a pandemic spread that affected almost everything of humans' social and health life. Regulations to decrease the epidemiological spread and studies to produce the vaccine of SARS-CoV-2 were on one side a hope to return back to the regular life, but on the other side there were also notable criticism about the vaccines itself. In this study, we established a fractional order differential equations system incorporating the vaccinated and re-infected compartments to a S I R frame to consider the expanded and detailed form as an S V I I v R model. We considered in the model some essential parameters, such as the protection rate of the vaccines, the vaccination rate, and the vaccine's lost efficacy after a certain period. We obtained the local stability of the disease-free and co-existing equilibrium points under specific conditions using the Routh-Hurwitz Criterion and the global stability in using a suitable Lyapunov function. For the numerical solutions we applied the Euler's method. The data for the simulations were taken from the World Health Organization (WHO) to illustrate numerically some scenarios that happened.

Vaccination control measures of an epidemic model with long-term memristive effect

COVID-19 is a drastic air-way tract infection that set off a global pandemic recently. Most infected people with mild and moderate symptoms have recovered with naturally acquired immunity. In the interim, the defensive mechanism of vaccines helps to suppress the viral complications of the pathogenic spread. Besides effective vaccination, vaccine breakthrough infections occurred rapidly due to noxious exposure to contagions. This paper proposes a new epidemiological control model in terms of Atangana Baleanu Caputo (ABC) type fractional order differ integrals for the reported cases of COVID-19 outburst. The qualitative theoretical and numerical analysis of the aforesaid mathematical model in terms of three compartments namely susceptible, vaccinated, and infected population are exhibited through non-linear functional analysis. The hysteresis kernel involved in AB integral inherits the long-term memory of the dynamical trajectory of the epidemics. Hyer-Ulam's stability of the system is studied by the dichotomy operator. The most effective approximate solution is derived by numerical interpolation to our proposed model. An extensive analysis of the vigorous vaccination and the proportion of vaccinated individuals are explored through graphical simulations. The efficacious enforcement of this vaccination control mechanism will mitigate the contagious spread and severity.

Conceptual analysis of the combined effects of vaccination, therapeutic actions, and human subjection to physical constraint in reducing the prevalence of COVID-19 using the homotopy perturbation method

Background

The COVID-19 pandemic has put the world's survival in jeopardy. Although the virus has been contained in certain parts of the world after causing so much grief, the risk of it emerging in the future should not be overlooked because its existence cannot be shown to be completely eradicated.

Results

This study investigates the impact of vaccination, therapeutic actions, and compliance rate of individuals to physical limitations in a newly developed SEIQR mathematical model of COVID-19. A qualitative investigation was conducted on the mathematical model, which included validating its positivity, existence, uniqueness, and boundedness. The disease-free and endemic equilibria were found, and the basic reproduction number was derived and utilized to examine the mathematical model's local and global stability. The mathematical model's sensitivity index was calculated equally, and the homotopy perturbation method was utilized to derive the estimated result of each compartment of the model. Numerical simulation carried out using Maple 18 software reveals that the COVID-19 virus's prevalence might be lowered if the actions proposed in this study are applied.

Conclusion

It is the collective responsibility of all individuals to fight for the survival of the human race against COVID-19. We urged that all persons, including the government, researchers, and health-care personnel, use the findings of this research to remove the presence of the dangerous COVID-19 virus.

Fractional stochastic modelling of COVID-19 under wide spread of vaccinations: Egyptian case study

This work predicts the dynamics of the COVID-19 under widespread vaccination to anticipate the virus's current and future waves. We focused on establishing two population-based models for predictions: the fractional-order model and the fractional-order stochastic model. Based on dose efficacy, which is one of the main imposed assumptions in our study, some vaccinated people will probably be exposed to infection by the same viral wave. We validated the generated models by applying them to the current viral wave in Egypt. We assumed that the Egyptian current wave began on 10^th September 2021. Using current actual data and varying our models’ fractional orders, we generate different predicted wave scenarios. The numerical solution of our models is obtained using the fractional Euler method and the fractional Euler Maruyama method. At the end, we compared the current predicted wave under a high vaccination rate with the previous viral wave. Through this comparison, the vaccination control effect is quantified.

Numerical analysis of Atangana-Baleanu fractional model to understand the propagation of a novel corona virus pandemic

In this manuscript, we formulated a new nonlinear SEIQR fractional order pandemic model for the Corona virus disease (COVID-19) with Atangana-Baleanu derivative. Two main equilibrium points F0∗,F1∗ of the proposed model are stated. Threshold parameter R0 for the model using next generation technique is computed to investigate the future dynamics of the disease. The existence and uniqueness of solution is proved using a fixed point theorem. For the numerical solution of fractional model, we implemented a newly proposed Toufik-Atangana numerical scheme to validate the importance of arbitrary order derivative ρ and our obtained theoretical results. It is worth mentioning that fractional order derivative provides much deeper information about the complex dynamics of Corona model. Results obtained through the proposed scheme are dynamically consistent and good in agreement with the analytical results. To draw our conclusions, we explore a complete quantitative analysis of the given model for different quarantine levels. It is claimed through numerical simulations that pandemic could be eradicated faster if a human community selfishly adopts mandatory quarantine measures at various coverage levels with proper awareness. Finally, we have executed the joint variability of all classes to understand the effectiveness of quarantine policy on human population.

Modeling the effect of time delay in the increment of number of hospital beds to control an infectious disease

One of the key factors to control the spread of any infectious disease is the health care facilities, especially the number of hospital beds. To assess the impact of number of hospital beds and control of an emerged infectious disease, we have formulated a mathematical model by considering population (susceptible, infected, hospitalized) and newly created hospital beds as dynamic variables. In formulating the model, we have assumed that the number of hospital beds increases proportionally to the number of infected individuals. It is shown that on a slight change in parameter values, the model enters to different kinds of bifurcations, e.g., saddle-node, transcritical (backward and forward), and Hopf bifurcation. Also, the explicit conditions for these bifurcations are obtained. We have also shown the occurrence of Bogdanov-Takens (BT) bifurcation using the Normal form. To set up a new hospital bed takes time, and so we have also analyzed our proposed model by incorporating time delay in the increment of newly created hospital beds. It is observed that the incorporation of time delay destabilizes the system, and multiple stability switches arise through Hopf-bifurcation. To validate the results of the analytical analysis, we have carried out some numerical simulations.

Unravelling the dynamics of the COVID-19 pandemic with the effect of vaccination, vertical transmission and hospitalization

The coronavirus disease 2019 (COVID-19) is caused by a newly emerged virus known as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), transmitted through air droplets from an infected person. However, other transmission routes are reported, such as vertical transmission. Here, we propose an epidemic model that considers the combined effect of vertical transmission, vaccination and hospitalization to investigate the dynamics of the virus's dissemination. Rigorous mathematical analysis of the model reveals that two equilibria exist: the disease-free equilibrium, which is locally asymptotically stable when the basic reproduction number ( R 0 ) is less than 1 (unstable otherwise), and an endemic equilibrium, which is globally asymptotically stable when R 0 > 1 under certain conditions, implying the plausibility of the disease to spread and cause large outbreaks in a community. Moreover, we fit the model using the Saudi Arabia cases scenario, which designates the incidence cases from the in-depth surveillance data as well as displays the epidemic trends in Saudi Arabia. Through Caputo fractional-order, simulation results are provided to show dynamics behaviour on the model parameters. Together with the non-integer order variant, the proposed model is considered to explain various dynamics features of the disease. Further numerical simulations are carried out using an efficient numerical technique to offer additional insight into the model's dynamics and investigate the combined effect of vaccination, vertical transmission, and hospitalization. In addition, a sensitivity analysis is conducted on the model parameters against the R 0 and infection attack rate to pinpoint the most crucial parameters that should be emphasized in controlling the pandemic effectively. Finally, the findings suggest that adequate vaccination coupled with basic non-pharmaceutical interventions are crucial in mitigating disease incidences and deaths.

An eco-epidemiological model with the impact of fear

In this study, we propose and analyze an eco-epidemiological model with disease in prey and incorporated the effect of fear on prey species due to predator population. We assume that the prey population grows logistically in the absence of predator species, and the disease is limited to the prey population only. We divide the total prey population into two distinct classes: susceptible prey and infected prey. Predator populations are not infected by the diseases, though feed both the susceptible and infected prey. Due to the fear of predators, the prey population becomes more vigilant and moves away from suspected predators. Such a foraging activity of prey reduces the chance of infection among susceptible prey by lowering the contact with infected prey. We assume that the fear of predators has no effect on infected prey as they are more vigilant. Positivity, boundedness, and uniform persistence of the proposed model are investigated. The biologically feasible equilibrium points and their stability are analyzed. We establish the conditions for the Hopf bifurcation of the proposed model around the endemic steady state. As the level of fear increases, the system moves toward the steady state from a limit cycle oscillation. The increasing level of fear cannot wipe out the diseases from the system, but the amplitude of the infected prey decreases as the level of fear is increased. The system changes its stability as the rate of infection increases, and the predator becomes extinct when the rate of infection in prey is high enough though predators are not infected by the disease.

Fractional Dynamics of a Measles Epidemic Model

In this work, we replaced the integer derivative with Caputo derivative to model the transmission dynamics of measles in an epidemic situation. We began by recalling some results on the local and global stability of the measles-free equilibrium point as well as the local stability of the endemic equilibrium point. We computed the basic reproduction number of the fractional model and found that is it equal to the one in the integer model when the fractional order ν = 1. We then performed a sensitivity analysis using the global method. Indeed, we computed the partial rank correlation coefficient (PRCC) between each model parameter and the basic reproduction number R0 as well as each variable state. We then demonstrated that the fractional model admits a unique solution and that it is globally stable using the Ulam–Hyers stability criterion. Simulations using the Adams-type predictor–corrector iterative scheme were conducted to validate our theoretical results and to see the impact of the variation of the fractional order on the quantitative disease dynamics.

Global Experiences of Community Responses to COVID-19: A Systematic Literature Review

Objective

This study aimed to conduct a systematic review of the global experiences of community responses to the COVID-19 epidemic.

Method

Five electronic databases (PubMed, Embase, CINAHL, ScienceDirect, and Web of Science) were searched for peer-reviewed articles published in English, from inception to October 10, 2021. Two reviewers independently reviewed titles, abstracts, and full texts. A systematic review (with a scientific strategy for literature search and selection in the electronic databases applied to data collection) was used to investigate the experiences of community responses to the COVID-19 pandemic.

Results

This review reported that community responses to COVID-19 consisted mainly of five ways. On the one hand, community-based screening and testing for Coronavirus was performed; on the other hand, the possible sources of transmission in communities were identified and cut off. In addition, communities provided medical aid for patients with mild cases of COVID-19. Moreover, social support for community residents, including material and psychosocial support, was provided to balance epidemic control and prevention and its impact on residents' lives. Last and most importantly, special care was provided to vulnerable residents during the epidemic.

Conclusion

This study systematically reviewed how communities to respond to COVID-19. The findings presented some practical and useful tips for communities still overwhelmed by COVID-19 to deal with the epidemic. Also, some community-based practices reported in this review could provide valuable experiences for community responses to future epidemics.

Modeling and analysis on the transmission of covid-19 Pandemic in Ethiopia

The newest infection is a novel coronavirus named COVID-19, that initially appeared in December 2019, in Wuhan, China, and is still challenging to control. The main focus of this paper is to investigate a novel fractional-order mathematical model that explains the behavior of COVID-19 in Ethiopia. Within the proposed model, the entire population is divided into nine groups, each with its own set of parameters and initial values. A nonlinear system of fractional differential equations for the model is represented using Caputo fractional derivative. Legendre spectral collocation method is used to convert this system into an algebraic system of equations. An inexact Newton iterative method is used to solve the model system. The effective reproduction number (R0) is computed by the next-generation matrix approach. Positivity and boundedness, as well as the existence and uniqueness of solution, are all investigated. Both endemic and disease-free equilibrium points, as well as their stability, are carefully studied. We calculated the parameters and starting conditions (ICs) provided for our model using data from the Ethiopian Public Health Institute (EPHI) and the Ethiopian Ministry of Health from 22 June 2020 to 28 February 2021. The model parameters are determined using least squares curve fitting and MATLAB R2020a is used to run numerical results. The basic reproduction number is R0=1.4575. For this value, disease free equilibrium point is asymptotically unstable and endemic equilibrium point is asymptotically stable, both locally and globally.

Solution of a dynamical memory effect COVID-19 infection system with leaky vaccination efficacy by non-singular kernel fractional derivatives

In this paper, the recent trends of COVID-19 infection spread have been studied to explore the advantages of leaky vaccination dynamics in SEVR (Susceptible Effected Vaccinated Recovered) compartmental model with the help of Caputo-Fabrizio (CF) and Atangana-Baleanu derivative in the Caputo sense (ABC) non-singular kernel fractional derivative operators with memory effect within the model to show possible long-term approaches of the infection along with limited defensive vaccine efficacy that can be designed numerically over the closed interval ranging from 0 to 1. One of the main goals is to provide a stepping information about the usefulness of the aforementioned non-singular kernel fractional approaches for a lenient case as well as a critical case in COVID-19 infection spread. Another is to investigate the effect of death rate on state variables. The estimation of death rate for state variables with suitable vaccine efficacy has a significant role in the stability of state variables in terms of basic reproduction number that is derived using next generation matrix method, and order of the fractional derivative. For non-integral orders the pandemic modeling sense viz, CF and ABC, has been compared thoroughly. Graphical presentations together with numerical results have proposed that the methodology is powerful and accurate which can provide new speculations for COVID-19 dynamical systems.

Future implications of COVID-19 through Mathematical modeling

COVID-19 is a pandemic respiratory illness. The disease spreads from human to human and is caused by a novel coronavirus SARS-CoV-2. In this study, we formulate a mathematical model of COVID-19 and discuss the disease free state and endemic equilibrium of the model. Based on the sensitivity indexes of the parameters, control strategies are designed. The strategies reduce the densities of the infected classes but do not satisfy the criteria/threshold condition of the global stability of disease free equilibrium. On the other hand, the endemic equilibrium of the disease is globally asymptotically stable. Therefore it is concluded that the disease cannot be eradicated with present resources and the human population needs to learn how to live with corona. For validation of the results, numerical simulations are obtained using fourth order Runge-Kutta method.

A systematic study of autonomous and nonautonomous predator-prey models for the combined effects of fear, refuge, cooperation and harvesting

In the present study, we investigate the roles of fear, refuge and hunting cooperation on the dynamics of a predator-prey system, where the predator population is subject to harvesting at a nonlinear rate. We also focus on the effects of seasonal forcing by letting some of the model parameters to vary with time. We rigorously analyze the autonomous and nonautonomous models mathematically as well as numerically. Our simulation results show that the birth rate of prey and the fear of predators causing decline in it, and harvesting of predators first destabilize and then stabilize the system around the coexistence of prey and predator; if the birth rate of prey is very low, both prey and predator populations extinct from the ecosystem, and for a range of this parameter, only the prey population survive. The fear of predators responsible for increase in the intraspecific competition among the prey species and the refuge behavior of prey have tendency to stabilize the system, whereas the cooperative behavior of predators during the hunting time destroys stability in the ecosystem. Numerical investigations of the seasonally forced model showcase the appearances of periodic solution, higher periodic solutions, bursting patterns and chaotic dynamics.

A systematic study of autonomous and nonautonomous predator-prey models for the combined effects of fear, refuge, cooperation and harvesting

In the present study, we investigate the roles of fear, refuge and hunting cooperation on the dynamics of a predator-prey system, where the predator population is subject to harvesting at a nonlinear rate. We also focus on the effects of seasonal forcing by letting some of the model parameters to vary with time. We rigorously analyze the autonomous and nonautonomous models mathematically as well as numerically. Our simulation results show that the birth rate of prey and the fear of predators causing decline in it, and harvesting of predators first destabilize and then stabilize the system around the coexistence of prey and predator; if the birth rate of prey is very low, both prey and predator populations extinct from the ecosystem, and for a range of this parameter, only the prey population survive. The fear of predators responsible for increase in the intraspecific competition among the prey species and the refuge behavior of prey have tendency to stabilize the system, whereas the cooperative behavior of predators during the hunting time destroys stability in the ecosystem. Numerical investigations of the seasonally forced model showcase the appearances of periodic solution, higher periodic solutions, bursting patterns and chaotic dynamics.