Modeling and optimal control analysis of transmission dynamics of COVID-19: The case of Ethiopia

A novel collocation method with a coronavirus optimization algorithm for the optimal control of COVID-19: A case study of Wuhan, China

In this study, we develop a numerical optimization approach to address the challenge of optimal control in the spread of COVID-19. We evaluate the impact of various control strategies aimed at reducing the number of exposed and infectious individuals. Our novel approach employs Legendre wavelets, their derivative operational matrix, and a collocation method to transform the COVID-19 transmission optimal control model into a nonlinear programming (NLP) problem. To solve this problem, we employ a coronavirus optimization algorithm (COVIDOA) to determine the optimal control, state variables, and objective value. We investigate three control plans for this highly contagious disease, focusing on individual protection, rapid detection and treatment, detection with delay in treatment, and environmental viral dispersion as time-based control functions. These strategies are applied within an SEIR-type control model specific to COVID-19 in China, designed to mitigate disease spread. Lastly, we analyze the effects of various parameters within the COVID-19 spread model. Our numerical results highlight the significant impact of strategies that minimize the number of exposed and infectious individuals, particularly those related to rapid detection, detection delay, and environmental viral dispersion, in controlling and preventing the transmission of the COVID-19 virus.

A fractional-order mathematical model for malaria and COVID-19 co-infection dynamics

This study proposes a fractional-order mathematical model for malaria and COVID-19 co-infection using the Atangana-Baleanu Derivative. We explain the various stages of the diseases together in humans and mosquitoes, and we also establish the existence and uniqueness of the fractional order co-infection model solution using the fixed point theorem. We conduct the qualitative analysis along with an epidemic indicator, the basic reproduction number R0 of this model. We investigate the global stability at the disease and endemic free equilibrium of the malaria-only, COVID-19-only, and co-infection models. We run different simulations of the fractional-order co-infection model using a two-step Lagrange interpolation polynomial approximate method with the aid of the Maple software package. The results reveal that reducing the risk of malaria and COVID-19 by taking preventive measures will reduce the risk factor for getting COVID-19 after contracting malaria and will also reduce the risk factor for getting malaria after contracting COVID-19 even to the point of extinction.

Mathematical modeling for Delta and Omicron variant of SARS-CoV-2 transmission dynamics in Greece

A compartmental, epidemiological, mathematical model was developed in order to analyze the transmission dynamics of Delta and Omicron variant, of SARS-CoV-2, in Greece. The model was parameterized twice during the 4th and 5th wave of the pandemic. The 4th wave refers to the period during which the Delta variant was dominant (approximately July to December of 2021) and the 5th wave to the period during which the Omicron variant was dominant (approximately January to May of 2022), in accordance with the official data from the National Public Health Organization (NPHO). Fitting methods were applied to evaluate important parameters in connection with the transmission of the variants, as well as the social behavior of population during these periods of interest. Mathematical models revealed higher numbers of contagiousness and cases of asymptomatic disease during the Omicron variant period, but a decreased rate of hospitalization compared to the Delta period. Also, parameters related to the behavior of the population in Greece were also assessed. More specifically, the use of protective masks and the abidance of social distancing measures. Simulations revealed that over 5,000 deaths could have been avoided, if mask usage and social distancing were 20% more efficient, during the short period of the Delta and Omicron outbreak. Furthermore, the spread of the variants was assessed using viral load data. The data were recorded from PCR tests at 417 Army Equity Fund Hospital (NIMTS), in Athens and the Ct values from 746 patients with COVID-19 were processed, to explain transmission phenomena and disease severity in patients. The period when the Delta variant prevailed in the country, the average Ct value was calculated as 25.19 (range: 12.32-39.29), whereas during the period when the Omicron variant prevailed, the average Ct value was calculated as 28 (range: 14.41-39.36). In conclusion, our experimental study showed that the higher viral load, which is related to the Delta variant, may interpret the severity of the disease. However, no correlation was confirmed regarding contagiousness phenomena. The results of the model, Ct analysis and official data from NPHO are consistent.

Implementation of computationally efficient numerical approach to analyze a Covid-19 pandemic model

Corona virus disease (Covid-19) which has caused frustration in the human community remains the concern of the globe as every government struggles to defeat the pandemic. To deal with the situation, we have extensively studied a deadly Covid-19 model to provide a deep insight into the disease dynamics. A mathematical analysis of the model utilizing preventive measures is performed with the aim to reduce the disease burden. Some comprehensive mathematical techniques are employed to demonstrate several essential properties of solutions. To start with, we proved the existence and uniqueness of solutions. Equilibrium points are stated both in the absence and presence of the pandemic. Biologically important quantity known as threshold parameter is computed to handle the future disease dynamics and analyzed for its sensitivity. We proved the stability of the proposed model at equilibrium points by employing necessary conditions on threshold parameter. A reliable and competitive numerical analysis is conducted to observe the effectiveness of implemented strategies and to verify obtained analytical results. The most sensitive parameters are determined through sensitivity analysis. An important feature of this study is to employ Non-Standard Finite Difference (NSFD) numerical scheme to solve the system instead of other standard methods like Runge–Kutta method of order 4 (RK4). Finally, several numerical simulations are performed to validate our former theoretical analysis. Numerical results exhibiting dynamical behavior of Covid-19 system under the influence of involved parameters suggest that both the implemented strategies, especially quarantine of exposed individuals, are effective for the substantial reduction in the diseased population and to achieve the herd immunity.

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.

A Mathematical Modelling and Analysis of COVID-19 Transmission Dynamics with Optimal Control Strategy

We proposed a deterministic compartmental model for the transmission dynamics of COVID-19 disease. We performed qualitative and quantitative analysis of the deterministic model concerning the local and global stability of the disease-free and endemic equilibrium points. We found that the disease-free equilibrium is locally asymptotically stable when the basic reproduction number is less than unity, while the endemic equilibrium point becomes locally asymptotically stable if the basic reproduction number is above unity. Furthermore, we derived the global stability of both the disease-free and endemic equilibriums of the system by constructing some Lyapunov functions. If $R_0\le 1$ , it is found that the disease-free equilibrium is globally asymptotically stable, while the endemic equilibrium point is globally asymptotically stable when $R_0>1$ . The numerical results of the general dynamics are in agreement with the theoretical solutions. We established the optimal control strategy by using Pontryagin’s maximum principle. We performed numerical simulations of the optimal control system to investigate the impact of implementing different combinations of optimal controls in controlling and eradicating COVID-19 disease. From this, a significant difference in the number of cases with and without controls was observed. We observed that the implementation of the combination of the control treatment rate, $u_2$ , and the control treatment rate, $u_3$ , has shown effective and efficient results in eradicating COVID-19 disease in the community relative to the other strategies.

Modeling and Analysis of COVID-19 Spread: The Impacts of Nonpharmaceutical Protocols

In this study, the extended SEIR dynamical model is formulated to investigate the spread of coronavirus disease (COVID-19) via a special focus on contact with asymptomatic and self-isolated infected individuals. Furthermore, a mathematical analysis of the model, including positivity, boundedness, and local and global stability of the disease-free and endemic equilibrium points in terms of the basic reproduction number, is presented. The sensitivity analysis indicates that reducing the disease contact rate and the transmissibility factor related to asymptomatic individuals, along with increasing the quarantine/self-isolation rate and the contact-tracing process, from the view of flattening the curve for novel coronavirus, are crucial to the reduction in disease-related deaths.

Assessing the dynamic impacts of non-pharmaceutical and pharmaceutical intervention measures on the containment results against COVID-19 in Ethiopia

The rapid spread of COVID-19 in Ethiopia was attributed to joint effects of multiple factors such as low adherence to face mask-wearing, failure to comply with social distancing measures, many people attending religious worship activities and holiday events, extensive protests, country election rallies during the pandemic, and the war between the federal government and Tigray Region. This study built a system dynamics model to capture COVID-19 characteristics, major social events, stringencies of containment measures, and vaccination dynamics. This system dynamics model served as a framework for understanding the issues and gaps in the containment measures against COVID-19 in the past period (16 scenarios) and the spread dynamics of the infectious disease over the next year under a combination of different interventions (264 scenarios). In the counterfactual analysis, we found that keeping high mask-wearing adherence since the outbreak of COVID-19 in Ethiopia could have significantly reduced the infection under the condition of low vaccination level or unavailability of the vaccine supply. Reducing or canceling major social events could achieve a better outcome than imposing constraints on people's routine life activities. The trend analysis found that increasing mask-wearing adherence and enforcing more stringent social distancing were two major measures that can significantly reduce possible infections. Higher mask-wearing adherence had more significant impacts than enforcing social distancing measures in our settings. As the vaccination rate increases, reduced efficacy could cause more infections than shortened immunological periods. Offsetting effects of multiple interventions (strengthening one or more interventions while loosening others) could be applied when the levels or stringencies of one or more interventions need to be adjusted for catering to particular needs (e.g., less stringent social distancing measures to reboot the economy or cushion insufficient resources in some areas).

Modeling nosocomial infection of COVID-19 transmission dynamics

COVID-19 epidemic has posed an unprecedented threat to global public health. The disease has alarmed the healthcare system with the harm of nosocomial infection. Nosocomial spread of COVID-19 has been discovered and reported globally in different healthcare facilities. Asymptomatic patients and super-spreaders are sough to be among of the source of these infections. Thus, this study contributes to the subject by formulating a S E I H R mathematical model to gain the insight into nosocomial infection for COVID-19 transmission dynamics. The role of personal protective equipment θ is studied in the proposed model. Benefiting the next generation matrix method,R 0 was computed. Routh-Hurwitz criterion and stable Metzler matrix theory revealed that COVID-19-free equilibrium point is locally and globally asymptotically stable wheneverR 0 < 1 . Lyapunov function depicted that the endemic equilibrium point is globally asymptotically stable whenR 0 > 1 . Further, the dynamics behavior ofR 0 was explored when varying θ . In the absence of θ , the value ofR 0 was 8.4584 which implies the expansion of the disease. When θ is introduced in the model,R 0 was 0.4229, indicating the decrease of the disease in the community. Numerical solutions were simulated by using Runge-Kutta fourth-order method. Global sensitivity analysis is performed to present the most significant parameter. The numerical results illustrated mathematically that personal protective equipment can minimizes nosocomial infections of COVID-19.

Mathematical modeling and analysis of the SARS-Cov-2 disease with reinfection

The COVID-19 infection which is still infecting many individuals around the world and at the same time the recovered individuals after the recovery are infecting again. This reinfection of the individuals after the recovery may lead the disease to worse in the population with so many challenges to the health sectors. We study in the present work by formulating a mathematical model for SARS-CoV-2 with reinfection. We first briefly discuss the formulation of the model with the assumptions of reinfection, and then study the related qualitative properties of the model. We show that the reinfection model is stable locally asymptotically when R0<1. For R0≤1, we show that the model is globally asymptotically stable. Further, we consider the available data of coronavirus from Pakistan to estimate the parameters involved in the model. We show that the proposed model shows good fitting to the infected data. We compute the basic reproduction number with the estimated and fitted parameters numerical value is R0≈1.4962. Further, we simulate the model using realistic parameters and present the graphical results. We show that the infection can be minimized if the realistic parameters (that are sensitive to the basic reproduction number) are taken into account. Also, we observe the model prediction for the total infected cases in the future fifth layer of COVID-19 in Pakistan that may begin in the second week of February 2022.

Mathematical modeling for COVID-19 transmission dynamics: A case study in Ethiopia

In this paper, we proposed a nonlinear deterministic mathematical model for the transmission dynamics of COVID-19. First, we analyzed the system properties such as boundedness of the solutions, existence of disease-free and endemic equilibria, local and global stability of equilibrium points. Besides, we computed the basic reproduction number R 0 and studied its normalized sensitivity for model parameters to identify the most influencing parameter. The local stability of the disease-free equilibrium point is also verified via the help of the Jacobian matrix and Routh Hurwitz criteria. Moreover, the global stability of the disease-free equilibrium point is proved by using the approach of Castillo-Chavez and Song. We also proved the existence of the forward bifurcation using the center manifold theory. Then the model is fitted with COVID-19 infected cases reported from March 13, 2020, to July 31, 2021, in Ethiopia. The values of model parameters are then estimated from the data reported using the least square method together with the fminsearch function in the MATLAB optimization toolbox. Finally, different simulation cases were performed using PYTHON software to compare with analytical results. The simulation results suggest that the spread of COVID-19 can be managed via minimizing the contact rate of infected and increasing the quarantine of exposed individuals.

FRACTIONAL ORDER MODEL FOR THE CORONAVIRUS (COVID-19) IN WUHAN, CHINA

In this paper, the mathematical modeling of five different classes for coronavirus disease-19 (COVID-19) is considered using the fractional arbitrary order derivative in Atangana–Baleanu sense. We use nonlinear analysis for the existence theory of the solution for the suggested model. Additionally, the modified Adam–Bashforth method is used for the numerical approximation of the assumed model. Finally, we simulate the results for 100 days with the help of data from the literature to display the excellency of the suggested model.

Management of and Revitalization Strategy for Megacities Under Major Public Health Emergencies: A Case Study of Wuhan

The outbreak of the COVID-19 pandemic in late 2019 has meant an uphill battle for city management. However, due to deficiencies in facilities and management experience, many megacities are less resilient when faced with such major public health events. Therefore, we chose Wuhan for a case study to examine five essential modules of urban management relevant to addressing the pandemic: (1) the medical and health system, (2) lifeline engineering and infrastructure, (3) community and urban management, (4) urban ecology and (5) economic development. The experience and deficiencies of each module in fighting the pandemic are analyzed, and strategies for revitalization and sustainable development in the future are proposed. The results show that in response to large-scale public health events, a comprehensive and coordinated medical system and good urban ecology can prevent the rapid spread of the epidemic. Additionally, good infrastructure and community management can maintain the operation of the city under the pandemic, and appropriate support policies are conducive to the recovery and development of the urban economy. These precedents provide insights and can serve as a reference for how to change the course of the pandemic in megacities that are still at risk, and they provide experience for responding to other pandemics.

Global Analysis and Optimal Control Model of COVID-19

COVID-19 remains the concern of the globe as governments struggle to defeat the pandemic. Understanding the dynamics of the epidemic is as important as detecting and treatment of infected individuals. Mathematical models play a crucial role in exploring the dynamics of the outbreak by deducing strategies paramount for curtailing the disease. The research extensively studies the SEQIAHR compartmental model of COVID-19 to provide insight into the dynamics of the disease by underlying tailored strategies designed to minimize the pandemic. We first studied the noncontrol model's dynamic behaviour by calculating the reproduction number and examining the two nonnegative equilibria' existence. The model utilizes the Castillo-Chavez method and Lyapunov function to investigate the global stability of the disease at the disease-free and endemic equilibrium. Sensitivity analysis was carried on to determine the impact of some parameters on R₀. We further examined the COVID model to determine the type of bifurcation that it exhibits. To help contain the spread of the disease, we formulated a new SEQIAHR compartmental optimal control model with time-dependent controls: personal protection and vaccination of the susceptible individuals. We solved it by utilizing Pontryagin's maximum principle after studying the dynamical behaviour of the noncontrol model. We solved the model numerically by considering different simulation controls' pairing and examined their effectiveness.

A Comprehensive Mathematical Model for SARS-CoV-2 in Caputo Derivative

In the present work, we study the COVID-19 infection through a new mathematical model using the Caputo derivative. The model has all the possible interactions that are responsible for the spread of disease in the community. We first formulate the model in classical differential equations and then extend it into fractional differential equations using the definition of the Caputo derivative. We explore in detail the stability results for the model of the disease-free case when R0<1. We show that the model is stable locally when R0<1. We give the result that the model is globally asymptotically stable whenever R0≤1. Further, to estimate the model parameters, we consider the real data of the fourth wave from Pakistan and provide a reasonable fitting to the data. We estimate the basic reproduction number for the proposed data to be R0=1.0779. Moreover, using the real parameters, we present the numerical solution by first giving a reliable scheme that can numerically handle the solution of the model. In our simulation, we give the graphical results for some sensitive parameters that have a large impact on disease elimination. Our results show that taking into consideration all the possible interactions can describe COVID-19 infection.

Modeling the effects of preventive measures and vaccination on the COVID-19 spread in Benin Republic with optimal control

Coronavirus disease (COVID-19) onset in December 2019 is a contagious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Since, the spread of the virus and mortality due to COVID-19 have continued to increase daily leading to a pandemic. In absence of approved medicine and vaccines, many countries imposed policies such as social distancing, mask wearing, hand washing, airport screening, quarantine and others. But rapidly, they were confronted with the high economic and social cost resulting from those policies. Many vaccines have been proposed but their efficiency is still controversial. Now, governments and scholars search for how manage with preventives measures policies and vaccination campaigns to stop the COVID-19 spread. This work studied the effects of these different strategies as time-dependent interventions using mathematical modeling and optimal control approach to ascertain their contribution in the dynamic transmission of COVID-19. The model was proven to have an invariant region and was well-posed. The basic reproduction number was computed with and without respect of preventives measures. The optimal control analysis was carried out using the Pontryagin's maximum principle to figure out the optimal strategy necessary to curtail the disease. The findings revealed that the optimal implementation of preventive measures reduce highly the number of infected individuals but zero infection was not achieved in the population. That was obtained with the optimal implementation of vaccination campaigns which reduce the number of infected individuals. But the optimal and combined implementation of the two interventions performed better with less costs than the two singular implementations.

Mathematical modeling of COVID-19 transmission dynamics between healthcare workers and community

Corona-virus disease 2019 (COVID-19) is an infectious disease that has affected different groups of humankind such as farmers, soldiers, drivers, educators, students, healthcare workers and many others. The transmission rate of the disease varies from one group to another depending on the contact rate. Healthcare workers are at a high risk of contracting the disease due to the high contact rate with patients. So far, there exists no mathematical model which combines both public control measures (as a parameter) and healthcare workers (as an independent compartment). Combining these two in a given mathematical model is very important because healthcare workers are protected through effective use of personal protective equipment, and control measures help to minimize the spread of COVID-19 in the community. This paper presents a mathematical model named SWEI s I a HR; susceptible individuals (S), healthcare workers (W), exposed (E), symptomatic infectious (I s ), asymptomatic infectious (I a ), hospitalized (H), recovered (R). The value of basic reproduction numberR 0 for all parameters in this study is 2.8540. In the absence of personal protective equipment ξ and control measure in the public θ , the value ofR 0 ≈ 4 . 6047 which implies the presence of the disease. When θ and ξ were introduced in the model, basic reproduction number is reduced to 0.4606, indicating the absence of disease in the community. Numerical solutions are simulated by using Runge-Kutta fourth-order method. Sensitivity analysis is performed to presents the most significant parameter. Furthermore, identifiability of model parameters is done using the least square method. The results indicated that protection of healthcare workers can be achieved through effective use of personal protective equipment by healthcare workers and minimization of transmission of COVID-19 in the general public by the implementation of control measures. Generally, this paper emphasizes the importance of using protective measures.

A multi-stage SEIR model to predict the potential of a new COVID-19 wave in KSA after lifting all travel restrictions

The complete lifting of travel restrictions to KSA takes place after 3rd of January 2021. There are fears that KSA will confront a new COVID-19 wave, especially when the most of countries that resumed the international flights are suffering now from the second surge. Fortunately, more than one Covid-19 Vaccine have been rolled out. However, herd immunity could be reached only through widespread vaccination. COVID-19 vaccines need more time to be properly protective, especially in front of people refusing to get vaccinated. A modified multi-stage SEIR model, with distinct reproductive numbers corresponding to before and after lockdown is employed to predict the potential of a new pandemic wave. First, the two-stage model employed to find the best fitting for the reproductive numbers. Then, the model is extended to three-stage one to investigate the relaxation. However, the modified model detects a second wave in early stage from 28th May to 17th June 2020 before even succeeding controlling the first outbreak. Subsequently, the four-stage SEIR model is used to predict the end of the second wave. Moreover, the model is employed to test the potential of a new pandemic surge after the international flights are resumed.

Controlling of pandemic COVID-19 using optimal control theory

In 2019, a new infectious disease called pandemic COVID-19 began to spread from Wuhan, China. In spite of the efforts to stop the disease, being out of the control of the governments it spread rapidly all over the world. From then on, much research has been done in the world with the aim of controlling this contagious disease. A mathematical model for modeling the spread of COVID-19 and also controlling the spread of the disease has been presented in this paper. We find the disease-free equilibrium points as trivial equilibrium (TE), virus absenteeism equilibrium (VAE) and virus incidence equilibrium (VIE) for the proposed model; and at the trivial equilibrium point for the presented dynamic system we obtain the Jacobian matrix so as to be used in finding the largest eigenvalue. Radius spectral method has been used for finding the reproductive number. In the following, by adding a controller to the model and also using the theory of optimal control, we can improve the performance of the model. We must have a correct understanding of the system i.e. how it works, the various variables affecting the system, and the interaction of the variables on each other. To search for the optimal values, we need to use an appropriate optimization method. Given the limitations and needs of the problem, the aim of the optimization is to find the best solutions, to find conditions that result in the maximum of susceptiblity, the minimum of infection, and optimal quarantination.

Mathematical assessment of the role of denial on COVID-19 transmission with non-linear incidence and treatment functions

A mathematical model describing the dynamics of Corona virus disease 2019 (COVID-19) is constructed and studied. The model assessed the role of denial on the spread of the pandemic in the world. Dynamic stability analyzes show that the equilibria, disease-free equilibrium (DFE) and endemic equilibrium point (EEP) of the model are globally asymptotically stable for R0<1 and R0>1, respectively. Again, the model is shown via numerical simulations to possess the backward bifurcation, where a stable DFE co-exists with one or more stable endemic equilibria when the control reproduction number, R0 is less than unity and the rate of denial of COVID-19 is above its upper bound. We then apply the optimal control strategy for controlling the spread of the disease using the controllable variables such as COVID-19 prevention, hospitalization and maximum treatment efforts. Using the Pontryagin maximum principle, we derive analytically the optimal controls of the model. The aforementioned control strategies are performed numerically in the presence of denial and without denial rate. Among such experiments, results without denial have shown to be more productive in ending the pandemic than others where the denial of the disease invalidates the effectiveness of the controls causing the disease to continue ravaging the globe.

Analysis of Atangana-Baleanu fractional-order SEAIR epidemic model with optimal control

We consider a SEAIR epidemic model with Atangana-Baleanu fractional-order derivative. We approximate the solution of the model using the numerical scheme developed by Toufic and Atangana. The numerical simulation corresponding to several fractional orders shows that, as the fractional order reduces from 1, the spread of the endemic grows slower. Optimal control analysis and simulation show that the control strategy designed is operative in reducing the number of cases in different compartments. Moreover, simulating the optimal profile revealed that reducing the fractional-order from 1 leads to the need for quick starting of the application of the designed control strategy at the maximum possible level and maintaining it for the majority of the period of the pandemic.