Stability analysis and optimal control of covid-19 with convex incidence rate in Khyber Pakhtunkhawa (Pakistan)

Evaluating the impact of double dose vaccination on SARS-CoV-2 spread through optimal control analysis

This paper presents a new nonlinear epidemic model for the spread of SARS-CoV-2 that incorporates the effect of double dose vaccination. The model is analyzed using qualitative, stability, and sensitivity analysis techniques to investigate the impact of vaccination on the spread of the virus. We derive the basic reproduction number and perform stability analysis of the disease-free and endemic equilibrium points. The model is also subjected to sensitivity analysis to identify the most influential model parameters affecting the disease dynamics. The values of the parameters are estimated with the help of the least square curve fitting tools. Finally, the model is simulated numerically to assess the effectiveness of various control strategies, including vaccination and quarantine, in reducing the spread of the virus. Optimal control techniques are employed to determine the optimal allocation of resources for implementing control measures. Our results suggest that increasing the vaccination coverage, adherence to quarantine measures, and resource allocation are effective strategies for controlling the epidemic. The study provides valuable insights into the dynamics of the pandemic and offers guidance for policymakers in formulating effective control measures.

Optimal control strategies for toxoplasmosis disease transmission dynamics via harmonic mean-type incident rate

Toxoplasma infection in humans is considered due to direct contact with infected cats. Toxoplasma infection (an endemic disease) has the potential to affect various organs and systems (brain, eyes, heart, lungs, liver, and lymph nodes). Bilinear incidence rate and constant population (birth rate is equal to death rate) are used in the literature to explain the dynamics of Toxoplasmosis disease transmission in humans and cats. The goal of this study is to consider the mathematical model of Toxoplasma disease with harmonic mean type incident rate and also consider that the population of humans and cats is not equal (birth rate and the death rate are not equal). In examining Toxoplasma transmission dynamics in humans and cats, harmonic mean incidence rates are better than bilinear incidence rates. The disease dynamics are first schematically illustrated, and then the law of mass action is applied to obtain nonlinear ordinary differential equations (ODEs). Analysis of the boundedness, positivity, and equilibrium points of the system has been analyzed. The reproduction number is calculated using the next-generation matrix technique. The stability of disease-free and endemic equilibrium are analyzed. Sensitivity analysis is also done for reproduction number. Numerical simulation shows that the infection is spread in the population when the contact rate β h and β c increases while the infection is reduced when the recovery rate δ h increases. This study investigates the impact of various optimal control strategies, such as vaccinations for the control of disease and the awareness of disease awareness, on the management of disease.

A vigorous study of fractional order mathematical model for SARS-CoV-2 epidemic with Mittag-Leffler kernel

SARS-CoV-2 and its variants have been investigated using a variety of mathematical models. In contrast to multi-strain models, SARS-CoV-2 models exhibit a memory effect that is often overlooked and more realistic. Atangana-Baleanu’s fractional-order operator is discussed in this manuscript for the analysis of the transmission dynamics of SARS-CoV-2. We investigated the transmission mechanism of the SARS-CoV-2 virus using the non-local Atangana-Baleanu fractional-order approach taking into account the different phases of infection and transmission routes. Using conventional ordinary derivative operators, our first step will be to develop a model for the proposed study. As part of the extension, we will incorporate fractional order derivatives into the model where the used operator is the fractional order operator of order Ψ1. Additionally, some basic aspects of the proposed model are examined in addition to calculating the reproduction number and determining the possible equilibrium. Stability analysis of the model is conducted to determine the necessary equilibrium conditions as they are also required in developing a numerical setup. Utilizing the theory of nonlinear functional analysis, for the model, Ulam-Hyers’ stability is established. We present a numerical scheme based on Newton’s polynomial in order to set up an iterative algorithm for the proposed ABC system. The application of this scheme to a variety of values of Φ1 indicates that there is a relationship between infection dynamics and the derivative’s order. We present further simulations which demonstrate the importance and cruciality of different parameters, as well as their effect on the dynamics and administer the disease. Furthermore, this study will provide a better understanding of the mechanisms underlying contagious diseases, thus supporting the development of policies to control them.

Computational modeling of fractional COVID-19 model by Haar wavelet collocation Methods with real data

This study explores the use of numerical simulations to model the spread of the Omicron variant of the SARS-CoV-2 virus using fractional-order COVID-19 models and Haar wavelet collocation methods. The fractional order COVID-19 model considers various factors that affect the virus's transmission, and the Haar wavelet collocation method offers a precise and efficient solution to the fractional derivatives used in the model. The simulation results yield crucial insights into the Omicron variant's spread, providing valuable information to public health policies and strategies designed to mitigate its impact. This study marks a significant advancement in comprehending the COVID-19 pandemic's dynamics and the emergence of its variants. The COVID-19 epidemic model is reworked utilizing fractional derivatives in the Caputo sense, and the model's existence and uniqueness are established by considering fixed point theory results. Sensitivity analysis is conducted on the model to identify the parameter with the highest sensitivity. For numerical treatment and simulations, we apply the Haar wavelet collocation method. Parameter estimation for the recorded COVID-19 cases in India from 13 July 2021 to 25 August 2021 has been presented.

Analysis and simulation of a stochastic COVID-19 model with large-scale nucleic acid detection and isolation measures: A case study of the outbreak in Urumqi, China in August, 2022

In this paper, a stochastic COVID-19 model with large-scale nucleic acid detection and isolation measures is proposed. Firstly, the existence and uniqueness of the global positive solution is obtained. Secondly, threshold criteria for the stochastic extinction and persistence in the mean with probability one are established. Moreover, a sufficient condition for the existence of unique ergodic stationary distribution for any positive solution is also established. Finally, numerical simulations are carried out in combination with real COVID-19 data from Urumqi, China and the theoretical results are verified.

Modeling and numerical analysis of a fractional order model for dual variants of SARS-CoV-2☆

This paper considers the novel fractional-order operator developed by Atangana-Baleanu for transmission dynamics of the SARS-CoV-2 epidemic. Assuming the importance of the non-local Atangana-Baleanu fractional-order approach, the transmission mechanism of SARS-CoV-2 has been investigated while taking into account different phases of infection and various transmission routes of the disease. To conduct the proposed study, first of all, we shall formulate the model by using the classical operator of ordinary derivatives. We utilize the fractional order derivative and the model will be extended to a model containing fractional order derivatives. The operator being used is the fractional differential operator and has fractional order Φ1. The model is analyzed further and some basic aspects of the model are investigated besides calculating the basic reproduction number and the possible equilibria of the proposed model. The equilibria of the model are examined for stability purposes and necessary conditions for stability are obtained. Stability is also necessary in terms of numerical setup. The theory of non-linear functional analysis is employed and Ulam-Hyers’s stability of the model is presented. The approach of newton’s polynomial is considered and a new numerical scheme is developed which helped in presenting an iterative process for the proposed ABC system. Based on this scheme, sample curves are obtained for various values of Φ1 and a pattern is derived between the dynamics of the infection and the order of the derivative. Further simulations are presented which show the cruciality and importance of various parameters and the impact of such parameters on the dynamics and control of the disease is presented. The findings of this study will also provide strong conceptual insights into the mechanisms of contagious diseases, assisting global professionals in developing control policies.

A generalized distributed delay model of COVID-19: An endemic model with immunity waning

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has been spreading worldwide for over two years, with millions of reported cases and deaths. The deployment of mathematical modeling in the fight against COVID-19 has recorded tremendous success. However, most of these models target the epidemic phase of the disease. The development of safe and effective vaccines against SARS-CoV-2 brought hope of safe reopening of schools and businesses and return to pre-COVID normalcy, until mutant strains like the Delta and Omicron variants, which are more infectious, emerged. A few months into the pandemic, reports of the possibility of both vaccine- and infection-induced immunity waning emerged, thereby indicating that COVID-19 may be with us for longer than earlier thought. As a result, to better understand the dynamics of COVID-19, it is essential to study the disease with an endemic model. In this regard, we developed and analyzed an endemic model of COVID-19 that incorporates the waning of both vaccine- and infection-induced immunities using distributed delay equations. Our modeling framework assumes that the waning of both immunities occurs gradually over time at the population level. We derived a nonlinear ODE system from the distributed delay model and showed that the model could exhibit either a forward or backward bifurcation depending on the immunity waning rates. Having a backward bifurcation implies that $ R_c < 1 $ is not sufficient to guarantee disease eradication, and that the immunity waning rates are critical factors in eradicating COVID-19. Our numerical simulations show that vaccinating a high percentage of the population with a safe and moderately effective vaccine could help in eradicating COVID-19.

A Mathematical Evaluation of the Cost-Effectiveness of Self-Protection, Vaccination, and Disinfectant Spraying for COVID-19 Control

The world is on its path from the post-COVID period, but a fresh wave of the coronavirus infection engulfing most European countries makes the pandemic catastrophic. Mathematical models are of significant importance in unveiling strategies that could stem the spread of the disease. In this paper, a deterministic mathematical model of COVID-19 is studied to characterize a range of feasible control strategies to mitigate the disease. We carried out an analytical investigation of the model’s dynamic behaviour at its equilibria and observed that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number, ${\mathcal{R}}_0$ is less than unity. The endemic equilibrium is also shown to be globally asymptotically stable when ${\mathcal{R}}_0>1$ . Further, we showed that the model exhibits forward bifurcation around ${\mathcal{R}}_0=1$ . Sensitivity analysis was carried out to determine the impact of various factors on the basic reproduction number ${\mathcal{R}}_0$ and consequently, the spread of the disease. An optimal control problem was formulated from the sensitivity analysis. Cost-effectiveness analysis is conducted to determine the most cost-effective strategy that can be adopted to control the spread of COVID-19. The investigation revealed that combining self-protection and environmental control is the most cost-effective control strategy among the enlisted strategies.

Optimal control and cost-effectiveness analysis of a new COVID-19 model for Omicron strain

Omicron, a mutant strain of COVID-19, has been sweeping the world since November 2021. A major characteristic of Omicron transmission is that it is less harmful to healthy adults, but more dangerous for people with underlying disease, the elderly, or children. To simulate the spread of Omicron in the population, we developed a new 9-dimensional mathematical model with high-risk and low-risk exposures. Then we analyzed its dynamic properties and obtain the basic reproduction numberR 0 . With the data of confirmed cases from March 1, 2022 published on the official website of Shanghai, China, we used the weighted nonlinear least square estimation method to estimate the parameters, and get the basic reproduction numberR 0 ≈ 1 . 5118 . Finally, we considered three control measures (isolation, detection and treatment), and studied the optimal control strategy and cost-effectiveness analysis of the model. The control strategy G is determined to be the optimal control strategy from the purpose of making fewer people infected. In strategy G, the three human control measures contain six control variables, and the control strength of these variables needs to be varied according to the pattern shown in Figure 11, so that the number of infections can be minimized and the percentage of reduction of infections can reach more than 95%.

Theoretical and numerical analysis of COVID-19 pandemic model with non-local and non-singular kernels

The global consequences of Coronavirus (COVID-19) have been evident by several hundreds of demises of human beings; hence such plagues are significantly imperative to predict. For this purpose, the mathematical formulation has been proved to be one of the best tools for the assessment of present circumstances and future predictions. In this article, we propose a fractional epidemic model of coronavirus (COVID-19) with vaccination effects. An arbitrary order model of COVID-19 is analyzed through three different fractional operators namely, Caputo, Atangana-Baleanu-Caputo (ABC), and Caputo-Fabrizio (CF), respectively. The fractional dynamics are composed of the interaction among the human population and the external environmental factors of infected peoples. It gives an extra description of the situation of the epidemic. Both the classical and modern approaches have been tested for the proposed model. The qualitative analysis has been checked through the Banach fixed point theory in the sense of a fractional operator. The stability concept of Hyers-Ulam idea is derived. The Newton interpolation scheme is applied for numerical solutions and by assigning values to different parameters. The numerical works in this research verified the analytical results. Finally, some important conclusions are drawn that might provide further basis for in-depth studies of such epidemics.

Modeling and analysis of a fractional anthroponotic cutaneous leishmania model with Atangana-Baleanu derivative

Very recently, Atangana and Baleanu defined a novel arbitrary order derivative having a kernel of non-locality and non-singularity, known as AB derivative. We analyze a non-integer order Anthroponotic Leshmania Cutaneous (ACL) problem exploiting this novel AB derivative. We derive equilibria of the model and compute its threshold quantity, i.e. the so-called reproduction number. Conditions for the local stability of the no-disease as well as the disease endemic states are derived in terms of the threshold quantity. The qualitative analysis for solution of the proposed problem have derived with the aid of the theory of fixed point. We use the predictor corrector numerical approach to solve the proposed fractional order model for approximate solution. We also provide, the numerical simulations for each of the compartment of considered model at different fractional orders along with comparison with integer order to elaborate the importance of modern derivative. The fractional investigation shows that the non-integer order derivative is more realistic about the inner dynamics of the Leishmania model lying between integer order.

Fractal fractional based transmission dynamics of COVID-19 epidemic model

We investigate the dynamical behavior of Coronavirus (COVID-19) for different infections phases and multiple routes of transmission. In this regard, we study a COVID-19 model in the context of fractal-fractional order operator. First, we study the COVID-19 dynamics with a fractal fractional-order operator in the framework of Atangana-Baleanu fractal-fractional operator. We estimated the basic reduction number and the stability results of the proposed model. We show the data fitting to the proposed model. The system has been investigated for qualitative analysis. Novel numerical methods are introduced for the derivation of an iterative scheme of the fractal-fractional Atangana-Baleanu order. Finally, numerical simulations are performed for various orders of fractal-fractional dimension.

Modeling and optimal control of mutated COVID-19 (Delta strain) with imperfect vaccination

As people around the world work to stop the COVID-19 pandemic, mutated COVID-19 (Delta strain) that are more contagious are emerging in many places. How to develop effective and reasonable plans to prevent the spread of mutated COVID-19 is an important issue. In order to simulate the transmission of mutated COVID-19 (Delta strain) in China with a certain proportion of vaccination, we selected the epidemic situation in Jiangsu Province as a case study. To solve this problem, we develop a novel epidemic model with a vaccinated population. The basic properties of the model is analyzed, and the expression of the basic reproduction number R 0 is obtained. We collect data on the Delta strain epidemic in Jiangsu Province, China from July 20, to August 5, 2021. The weighted nonlinear least square estimation method is used to fit the daily asymptomatic infected people, common infected people and severe infected people. The estimated parameter values are obtained, the approximate values of the basic reproduction number are calculated R 0 ≈ 1.378 . Through the global sensitivity analysis, we identify some parameters that have a greater impact on the prevalence of the disease. Finally, according to the evaluation results of parameter influence, we consider three control measures (vaccination, isolation and nucleic acid testing) to control the spread of the disease. The results of the study found that the optimal control measure is to dynamically adjust the three control measures to achieve the lowest number of infections at the lowest cost. The research in this paper can not only enrich theoretical research on the transmission of COVID-19, but also provide reliable control suggestions for countries and regions experiencing mutated COVID-19 epidemics.

Fractional dynamics and stability analysis of COVID-19 pandemic model under the harmonic mean type incidence rate

In this research, COVID-19 model is formulated by incorporating harmonic mean type incidence rate which is more realistic in average speed. Basic reproduction number, equilibrium points, and stability of the proposed model is established under certain conditions. Runge-Kutta fourth order approximation is used to solve the deterministic model. The model is then fractionalized by using Caputo-Fabrizio derivative and the existence and uniqueness of the solution are proved by using Banach and Leray-Schauder alternative type theorems. For the fractional numerical simulations, we use the Adam-Moulton scheme. Sensitivity analysis of the proposed deterministic model is studied to identify those parameters which are highly influential on basic reproduction number.

Modeling the dynamics of novel coronavirus (COVID-19) via stochastic epidemic model

Novel coronavirus disease is a burning issue all over the world. Spreading of the novel coronavirus having the characteristic of rapid transmission whenever the air molecules or the freely existed virus includes in the surrounding and therefore the spread of virus follows a stochastic process instead of deterministic. We assume a stochastic model to investigate the transmission dynamics of the novel coronavirus. To do this, we formulate the model according to the charectersitics of the corona virus disease and then prove the existence as well as the uniqueness of the global positive solution to show the well posed-ness and feasibility of the problem. Following the theory of dynamical systems as well as by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions of the extinction and the existence of stationary distribution. Finally, we carry out the large scale numerical simulations to demonstrate the verification of our analytical results.

Nonlinear Dynamics of the Introduction of a New SARS-CoV-2 Variant with Different Infectiousness

Several variants of the SARS-CoV-2 virus have been detected during the COVID-19 pandemic. Some of these new variants have been of health public concern due to their higher infectiousness. We propose a theoretical mathematical model based on differential equations to study the effect of introducing a new, more transmissible SARS-CoV-2 variant in a population. The mathematical model is formulated in such a way that it takes into account the higher transmission rate of the new SARS-CoV-2 strain and the subpopulation of asymptomatic carriers. We find the basic reproduction number R0 using the method of the next generation matrix. This threshold parameter is crucial since it indicates what parameters play an important role in the outcome of the COVID-19 pandemic. We study the local stability of the infection-free and endemic equilibrium states, which are potential outcomes of a pandemic. Moreover, by using a suitable Lyapunov functional and the LaSalle invariant principle, it is proved that if the basic reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable. Our study shows that the new more transmissible SARS-CoV-2 variant will prevail and the prevalence of the preexistent variant would decrease and eventually disappear. We perform numerical simulations to support the analytic results and to show some effects of a new more transmissible SARS-CoV-2 variant in a population.

Scenario analysis of COVID-19 transmission dynamics in Malaysia with the possibility of reinfection and limited medical resources scenarios

COVID-19 is a major health threat across the globe, which causes severe acute respiratory syndrome (SARS), and it is highly contagious with significant mortality. In this study, we conduct a scenario analysis for COVID-19 in Malaysia using a simple universality class of the SIR system and extensions thereof (i.e., the inclusion of temporary immunity through the reinfection problems and limited medical resources scenarios leads to the SIRS-type model). This system has been employed in order to provide further insights on the long-term outcomes of COVID-19 pandemic. As a case study, the COVID-19 transmission dynamics are investigated using daily confirmed cases in Malaysia, where some of the epidemiological parameters of this system are estimated based on the fitting of the model to real COVID-19 data released by the Ministry of Health Malaysia (MOH). We observe that this model is able to mimic the trend of infection trajectories of COVID-19 pandemic in Malaysia and it is possible for transmission dynamics to be influenced by the reinfection force and limited medical resources problems. A rebound effect in transmission could occur after several years and this situation depends on the intensity of reinfection force. Our analysis also depicts the existence of a critical value in reinfection threshold beyond which the infection dynamics persist and the COVID-19 outbreaks are rather hard to eradicate. Therefore, understanding the interplay between distinct epidemiological factors using mathematical modelling approaches could help to support authorities in making informed decisions so as to control the spread of this pandemic effectively.

A new mathematical model of COVID-19 using real data from Pakistan

We propose a new mathematical model to investigate the recent outbreak of the coronavirus disease (COVID-19). The model is studied qualitatively using stability theory of differential equations and the basic reproductive number that represents an epidemic indicator is obtained from the largest eigenvalue of the next-generation matrix. The global asymptotic stability conditions for the disease free equilibrium are obtained. The real COVID-19 incidence data entries from 01 July, 2020 to 14 August, 2020 in the country of Pakistan are used for parameter estimation thereby getting fitted values for the biological parameters. Sensitivity analysis is performed in order to determine the most sensitive parameters in the proposed model. To view more features of the state variables in the proposed model, we perform numerical simulations by using different values of some essential parameters. Moreover, profiles of the reproduction number through contour plots have been biologically explained.

Estimating, monitoring, and forecasting COVID-19 epidemics: a spatiotemporal approach applied to NYC data

We propose a susceptible-exposed-infective-recovered-type (SEIR-type) meta-population model to simulate and monitor the (COVID-19) epidemic evolution. The basic model consists of seven categories, namely, susceptible (S), exposed (E), three infective classes, recovered (R), and deceased (D). We define these categories for n age and sex groups in m different spatial locations. Therefore, the resulting model contains all epidemiological classes for each age group, sex, and location. The mixing between them is accomplished by means of time-dependent infection rate matrices. The model is calibrated with the curve of daily new infections in New York City and its boroughs, including census data, and the proportions of infections, hospitalizations, and deaths for each age range. We finally obtain a model that matches the reported curves and predicts accurate infection information for different locations and age classes.

Mathematical modeling and dynamic analysis of SIQR model with delay for pandemic COVID-19

On the basis of the SIQR epidemic model, we consider the impact of treatment time on the epidemic situation, and we present a differential equation model with time-delay according to the characteristics of COVID-19. Firstly, we analyze the existence and stability of the equilibria in the modified COVID-19 epidemic model. Secondly, we analyze the existence of Hopf bifurcation, and derive the normal form of Hopf bifurcation by using the multiple time scales method. Then, we determine the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, we carry out numerical simulations to verify the correctness of theoretical analysis with actual parameters, and show conclusions associated with the critical treatment time and the effect on epidemic for treatment time.

Analysis of fractional COVID-19 epidemic model under Caputo operator

The article deals with the analysis of the fractional COVID-19 epidemic model (FCEM) with a convex incidence rate. Keeping in view the fading memory and crossover behavior found in many biological phenomena, we study the coronavirus disease by using the noninteger Caputo derivative (CD). Under the Caputo operator (CO), existence and uniqueness for the solutions of the FCEM have been analyzed using fixed point theorems. We study all the basic properties and results including local and global stability. We show the global stability of disease-free equilibrium using the method of Castillo-Chavez, while for disease endemic, we use the method of geometrical approach. Sensitivity analysis is carried out to highlight the most sensitive parameters corresponding to basic reproduction number. Simulations are performed via first-order convergent numerical technique to determine how changes in parameters affect the dynamical behavior of the system.

Fractional optimal control of COVID-19 pandemic model with generalized Mittag-Leffler function

In this paper, we consider a fractional COVID-19 epidemic model with a convex incidence rate. The Atangana-Baleanu fractional operator in the Caputo sense is taken into account. We establish the equilibrium points, basic reproduction number, and local stability at both the equilibrium points. The existence and uniqueness of the solution are proved by using Banach and Leray-Schauder alternative type theorems. For the fractional numerical simulations, we use the Toufik-Atangana scheme. Optimal control analysis is carried out to minimize the infection and maximize the susceptible people.

Stability and optimal control strategies for a novel epidemic model of COVID-19

In this paper, a novel two-stage epidemic model with a dynamic control strategy is proposed to describe the spread of Corona Virus Disease 2019 (COVID-19) in China. Combined with local epidemic control policies, an epidemic model with a traceability process is established. We aim to investigate the appropriate control strategies to minimize the control cost and ensure the normal operation of society under the premise of containing the epidemic. This work mainly includes: (i) propose the concept about the first and the second waves of COVID-19, as well as study the case data and regularity of four cities; (ii) derive the existence and stability of the equilibrium, the parameter sensitivity of the model, and the existence of the optimal control strategy; (iii) carry out the numerical simulation associated with the theoretical results and construct a dynamic control strategy and verify its feasibility.