Mathematical modeling and optimal control of the COVID-19 dynamics

Learning from the COVID-19 pandemic: a systematic review of mathematical vaccine prioritization models

As the world becomes ever more connected, the chance of pandemics increases as well. The recent COVID-19 pandemic and the concurrent global mass vaccine roll-out provides an ideal setting to learn from and refine our understanding of infectious disease models for better future preparedness. In this review, we systematically analyze and categorize mathematical models that have been developed to design optimal vaccine prioritization strategies of an initially limited vaccine. As older individuals are disproportionately affected by COVID-19, the focus is on models that take age explicitly into account. The lower mobility and activity level of older individuals gives rise to non-trivial trade-offs. Secondary research questions concern the optimal time interval between vaccine doses and spatial vaccine distribution. This review showcases the effect of various modeling assumptions on model outcomes. A solid understanding of these relationships yields better infectious disease models and thus public health decisions during the next pandemic.

Vaccination impact on impending HIV-COVID-19 dual epidemic with autogenous behavior modification: Hill-type functional response and premeditated optimization technique

An HIV-COVID-19 co-infection dynamics is modeled mathematically assimilating the vaccination mechanism that incorporates endogenous modification of human practices generated by the COVID-19 prevalence, absorbing the relevance of the treatment mechanism in suppressing the co-infection burden. Envisaging a COVID-19 situation, the HIV-subsystem is analyzed by introducing COVID-19 vaccination for the HIV-infected population as a prevention, and the "vaccination influenced basic reproduction number" of HIV is derived. The mono-infection systems experience forward bifurcation that evidences the persistence of diseases above unit epidemic thresholds. Delicate simulation methodologies are employed to explore the impacts of baseline vaccination, prevalence-dependent spontaneous behavioral change that induces supplementary vaccination, and medication on the dual epidemic. Captivatingly, a paradox is revealed showing that people start to get vaccinated at an additional rate with the increased COVID-19 prevalence, which ultimately diminishes the dual epidemic load. It suggests increasing the baseline vaccination rate and the potency of propagated awareness. Co-infection treatment needs to be emphasized parallelly with single infection medication under dual epidemic situations. Further, an optimization technique is introduced to the co-infection model integrating vaccination and treatment control mechanisms, which approves the strategy combining vaccination with awareness and medication as the ideal one for epidemic and economic gain. Conclusively, it is manifested that waiting frivolously for any anticipated outbreak, depending on autogenous behavior modification generated by the increased COVID-19 prevalence, instead of elevating vaccination campaigns and the efficacy of awareness beforehand, may cause devastation to the population under future co-epidemic conditions.

[Formula: see text] model for analyzing [Formula: see text]-19 pandemic process via [Formula: see text]-Caputo fractional derivative and numerical simulation

The objective of this study is to develop the [Formula: see text] epidemic model for [Formula: see text]-[Formula: see text] utilizing the [Formula: see text]-Caputo fractional derivative. The reproduction number ([Formula: see text]) is calculated utilizing the next generation matrix method. The equilibrium points of the model are computed, and both the local and global stability of the disease-free equilibrium point are demonstrated. Sensitivity analysis is discussed to describe the importance of the parameters and to demonstrate the existence of a unique solution for the model by applying a fixed point theorem. Utilizing the fractional Euler procedure, an approximate solution to the model is obtained. To study the transmission dynamics of infection, numerical simulations are conducted by using MatLab. Both numerical methods and simulations can provide valuable insights into the behavior of the system and help in understanding the existence and properties of solutions. By placing the values [Formula: see text], [Formula: see text] and [Formula: see text] instead of [Formula: see text], the derivatives of the Caputo and Caputo-Hadamard and Katugampola appear, respectively, to compare the results of each with real data. Besides, these simulations specifically with different fractional orders to examine the transmission dynamics. At the end, we come to the conclusion that the simulation utilizing Caputo derivative with the order of 0.95 shows the prevalence of the disease better. Our results are new which provide a good contribution to the current research on this field of research.

Mathematical analysis and optimal control of an epidemic model with vaccination and different infectivity

This paper aims to explore the complex dynamics and impact of vaccinations on controlling epidemic outbreaks. An epidemic transmission model which considers vaccinations and two different infection statuses with different infectivity is developed. In terms of a dynamic analysis, we calculate the basic reproduction number and control reproduction number and discuss the stability of the disease-free equilibrium. Additionally, a numerical simulation is performed to explore the effects of vaccination rate, immune waning rate and vaccine ineffective rate on the epidemic transmission. Finally, a sensitivity analysis revealed three factors that can influence the threshold: transmission rate, vaccination rate, and the hospitalized rate. In terms of optimal control, the following three time-related control variables are introduced to reconstruct the corresponding control problem: reducing social distance, enhancing vaccination rates, and enhancing the hospitalized rates. Moreover, the characteristic expression of optimal control problem. Four different control combinations are designed, and comparative studies on control effectiveness and cost effectiveness are conducted by numerical simulations. The results showed that Strategy C (including all the three controls) is the most effective strategy to reduce the number of symptomatic infections and Strategy A (including reducing social distance and enhancing vaccination rate) is the most cost-effective among the three strategies.

Towards Optimal Control of Amyloid Fibrillation

Epigallocatechin-3-gallate, as a representative amyloid inhibitors, has shown a promising ability against A[Formula: see text] fibrillation by directly degradating the mature fibrils. Most previous studies have been focusing on its functional mechanisms, meanwhile its optimal dosage has been seldom considered. To solve this critical issue, we refer to the generalized Logistic model for amyloid fibrillation and inhibition and adopt the optimal control theory to balance the effectiveness and cost (or toxicity) of inhibitors. The optimal control trajectory of inhibitors is analytically solved, based on which the influence of model parameters, the difference between the optimal control strategy and several other traditional drug dosing strategies are systematically compared and validated through experiments. It is found that the strategy of multiple-times adding is more suitable for a long-term disease treatment, while single high-dose therapy is preferred for a short-term treatment. We hope our findings can shed light on the rational usage of amyloid inhibitors in clinic.

Optimal age-specific vaccination control for COVID-19: An Irish case study

The outbreak of a novel coronavirus causing severe acute respiratory syndrome in December 2019 has escalated into a worldwide pandemic. In this work, we propose a compartmental model to describe the dynamics of transmission of infection and use it to obtain the optimal vaccination control. The model accounts for the various stages of the vaccination, and the optimisation is focused on minimising the infections to protect the population and relieve the healthcare system. As a case study, we selected the Republic of Ireland. We use data provided by Ireland's COVID-19 Data-Hub and simulate the evolution of the pandemic with and without the vaccination in place for two different scenarios, one representative of a national lockdown situation and the other indicating looser restrictions in place. One of the main findings of our work is that the optimal approach would involve a vaccination programme where the older population is vaccinated in larger numbers earlier while simultaneously part of the younger population also gets vaccinated to lower the risk of transmission between groups. We compare our simulated results with those of the vaccination policy taken by the Irish government to explore the advantages of our optimisation method. Our comparison suggests that a similar reduction in cases may have been possible even with a reduced set of vaccinations available for use.

Stability analysis and optimal control of a fractional-order generalized SEIR model for the COVID-19 pandemic

In view of the spread of corona virus disease 2019 (COVID-19), this paper proposes a fractional-order generalized SEIR model. The non-negativity of the solution of the model is discussed. Based on the established threshold R 0 , the existence of the disease-free equilibrium and endemic equilibrium is analyzed. Then, sufficient conditions are established to ensure the local asymptotic stability of the equilibria. The parameters of the model are identified based on the statistical data of COVID-19 cases. Furthermore, the validity of the model for describing the COVID-19 outbreak is verified. Meanwhile, the accuracy of the relevant theoretical results are also verified. Considering the relevant strategies of COVID-19 prevention and control, the fractional optimal control problem (FOCP) is proposed. Numerical schemes for Riemann-Liouville (R-L) fractional-order adjoint system with transversal conditions is presented. Based on the relevant statistical data, the corresponding FOCP is numerically solved, and the control effect of the COVID-19 outbreak under the optimal control strategy is discussed.

Optimal strategies for coordinating infection control and socio-economic activities

It becomes challenging to identify feasible control strategies for simultaneously relaxing the countermeasures and containing the Covid-19 pandemic, given China's huge population size, high susceptibility, persist vaccination waning, and relatively weak strength of health systems. We propose a novel mathematical model with waning of immunity and solve the optimal control problem, in order to provide an insight on how much detecting and social distancing are required to coordinate socio-economic activities and epidemic control. We obtain the optimal intensity of countermeasures, i.e., the dynamic nucleic acid screening and social distancing, under which the health system is functioning normally and people can engage in a certain level of socio-economic activities. We find that it is the isolation capacity or the restriction of the case fatality rate (CFR) rather than the hospital capacity that mainly determines the optimal strategies. And the solved optimal controls under quarterly CFR restrictions exhibit oscillations. It is worth noticing that, if without considering booster or very low booster rate, the optimal strategy is a "on-off" mode, alternating between lock down and opening with certain social distancing, which reflects the importance and necessity of China's static management on a certain area during Covid-19 outbreak. The findings suggest some feasible paths to smoothly transit from the Covid-19 pandemic to an endemic phase.

A memory effect model to predict COVID-19: analysis and simulation

On 19 September 2020, the Centers for Disease Control and Prevention (CDC) recommended that asymptomatic individuals, those who have close contact with infected person, be tested. Also, American society for biological clinical comments on testing of asymptomatic individuals. So, we proposed a new mathematical model for evaluating the population-level impact of contact rates (social-distancing) and the rate at which asymptomatic people are hospitalized (isolated) following testing due to close contact with documented infected people. The model is a deterministic system of nonlinear differential equations that is fitted and parameterized by least square curve fitting using COVID-19 pandemic data of Pakistan from 1 October 2020 to 30 April 2021. The fractional derivative is used to understand the biological process with crossover behavior and memory effect. The reproduction number and conditions for asymptotic stability are derived diligently. The most common non-integer Caputo derivative is used for deeper analysis and transmission dynamics of COVID-19 infection. The fractional-order Adams-Bashforth method is used for the solution of the model. In light of the dynamics of the COVID-19 outbreak in Pakistan, non-pharmaceutical interventions (NPIs) in terms of social distancing and isolation are being investigated. The reduction in the baseline value of contact rates and enhancement in hospitalization rate of symptomatic can lead the elimination of the pandemic.

The measles epidemic model assessment under real statistics: an application of stochastic optimal control theory

A stochastic epidemic model with random noise transmission is taken into account, describing the dynamics of the measles viral infection. The basic reproductive number is calculated corresponding to the stochastic model. It is determined that, given initial positive data, the model has bounded, unique, and positive solution. Additionally, utilizing stochastic Lyapunov functional theory, we study the extinction of the disease. Stationary distribution and extinction of the infection are examined by providing sufficient conditions. We employed optimal control principles and examined stochastic control systems to regulate the transmission of the virus using environmental factors. Graphical representations have been offered for simplicity of comprehending in order to further verify the acquired analytical findings.

Modelling the impact of some control strategies on the transmission dynamics of Ebola virus in human-bat population: An optimal control analysis

Qualitative research and comprehensive public awareness to nip the transmission of Ebola virus in the bud before it becomes a global threat is fast becoming imperative especially now that the Gambia Ebola virus is mutated. It is therefore necessary to consider and investigate a vector-host transmission model for possible control strategy of this deadly disease. Hence, in this study, we presented a novel and feasible human-bat (host-vector) S h E h I h R h i s R h n i - S b E b I b model which foretells the spread and severity of the Ebola virus from bats to humans to investigate the combined effects of three control strategies viz: (1) allowing specialized and designated agencies to bury deceased from Ebola infection without relatives touching or curdling the remains as usually practiced in most part of Africa as last respect for their departed love ones ( k 1 ), (2) systematic and deliberate depopulation of bats in the metropolis (through persecution with pesticide exposure, pre capturing, chemical timber treatment for roosts destruction) to discourage hunting them for food by virtue of their proximity ( k 2 ) and (3) immediate treatment of infected individuals in isolation ( k 3 ). We established, among others, the endemic equilibrium, disease-free equilibrium, global and local stability, non-negativity, and boundedness of the model to prove the epidemiological feasibility of the model. The reality of the presence of optimal control remarkably influences the dynamics of transmission of the virus and simulated results also confirm the great effect of the combination of the control strategies k 1 , k 2 and k 3 in flattening the curve of Ebola transmission (fig 1 - fig 8). Health workers and policy makers are better informed with fundamental precautions that could help eradicate Ebola from the populace.

Nonlinear optimal control strategies for a mathematical model of COVID-19 and influenza co-infection

Infectious diseases have remained one of humanity's biggest problems for decades. Multiple disease infections, in particular, have been shown to increase the difficulty of diagnosing and treating infected people, resulting in worsening human health. For example, the presence of influenza in the population is exacerbating the ongoing COVID-19 pandemic. We formulate and analyze a deterministic mathematical model that incorporates the biological dynamics of COVID-19 and influenza to effectively investigate the co-dynamics of the two diseases in the public. The existence and stability of the disease-free equilibrium of COVID-19-only and influenza-only sub-models are established by using their respective threshold quantities. The result shows that the COVID-19 free equilibrium is locally asymptotically stable when R C < 1 , whereas the influenza-only model, is locally asymptotically stable when R F < 1 . Furthermore, the existence of the endemic equilibria of the sub-models is examined while the conditions for the phenomenon of backward bifurcation are presented. A generalized analytical result of the COVID-19-influenza co-infection model is presented. We run a numerical simulation on the model without optimal control to see how competitive outcomes between-hosts and within-hosts affect disease co-dynamics. The findings established that disease competitive dynamics in the population are determined by transmission probabilities and threshold quantities. To obtain the optimal control problem, we extend the formulated model by including three time-dependent control functions. The maximum principle of Pontryagin was used to prove the existence of the optimal control problem and to derive the necessary conditions for optimum disease control. A numerical simulation was performed to demonstrate the impact of different combinations of control strategies on the infected population. The findings show that, while single and twofold control interventions can be used to reduce disease, the threefold control intervention, which incorporates all three controls, will be the most effective in reducing COVID-19 and influenza in the population.

Optimal control strategies to combat COVID-19 transmission: A mathematical model with incubation time delay

The coronavirus disease 2019, started spreading around December 2019, still persists in the population all across the globe. Though different countries have been able to cope with the disease to some extent and vaccination for the same has been developed, it cannot be ignored that the disease is still not on the verge of completely eradicating, which in turn creates a need for having deeper insights of the disease in order to understand it well and hence be able to work towards its eradication. Meanwhile, using mitigation strategies like non-pharmaceutical interventions can help in controlling the disease. In this work, our aim is to study the dynamics of COVID-19 using compartmental approach by applying various analytical methods. We obtain formula for important tools like R0 and establish the stability of disease-free equilibrium point for R0<1. Further, based on R0, we discuss the stability and existence of the endemic equilibrium point. We incorporate various control strategies possible and using optimal control theory, study their expected positive impacts on the spread of the disease. Later, using a biologically feasible set of parameters, we numerically analyse the model. We even study the trend of the outbreak in China, for over 120 days, where the active cases rise up to a peak and then the curve flattens.

Attention based parameter estimation and states forecasting of COVID-19 pandemic using modified SIQRD Model

In this work, we propose a new mathematical modeling of the spread of COVID-19 infection in an arbitrary population, by modifying the SIQRD model as m-SIQRD model, while taking into consideration the eight governmental interventions such as cancellation of events, closure of public places etc., as well as the influence of the asymptomatic cases on the states of the model. We introduce robustness and improved accuracy in predictions of these models by utilizing a novel deep learning scheme. This scheme comprises of attention based architecture, alongside with Generative Adversarial Network (GAN) based data augmentation, for robust estimation of time varying parameters of m-SIQRD model. In this regard, we also utilized a novel feature extraction methodology by employing noise removal operation by Spline interpolation and Savitzky-Golay filter, followed by Principal Component Analysis (PCA). These parameters are later directed towards two main tasks: forecasting of states to the next 15 days, and estimation of best policy encodings to control the infected and deceased number within the framework of data driven synergetic control theory. We validated the superiority of the forecasting performance of the proposed scheme over countries of South Korea and Germany and compared this performance with 7 benchmark forecasting models. We also showed the potential of this scheme to determine best policy encodings in South Korea for 15 day forecast horizon.

Qualitative analysis of a fractional-order two-strain epidemic model with vaccination and general non-monotonic incidence rate

In this paper, a fractional-order two-strain epidemic model with vaccination and general non-monotonic incidence rate is analyzed. The studied problem is formulated using susceptible, infectious and recovered compartmental model. A Caputo fractional operator is incorporated in each compartment to describe the memory effect related to an epidemic evolution. First, the global existence, positivity and boundedness of solutions of the proposed model are proved. The basic reproduction numbers associated with studied problem are calculated. Four steady states are given, namely the disease-free equilibrium, the strain 1 endemic equilibrium, the strain 2 endemic equilibrium, and the endemic equilibrium associated with both strains. By considering appropriate Lyapunov functions, the global stability of the equilibrium points is proven according to the model parameters. Our modeling approach using a generalized non-monotonic incidence functions encloses a variety of fractional-order epidemic models existing in the literature. Finally, the theoretical findings are illustrated using numerical simulations.

Modeling COVID-19 Transmission Dynamics: A Bibliometric Review

A good amount of research has evolved just in three years in COVID-19 transmission, mortality, vaccination, and some socioeconomic studies. A few bibliometric reviews have already been performed in the literature, especially on the broad theme of COVID-19, without any particular area such as transmission, mortality, or vaccination. This paper fills this gap by conducting a bibliometric review on COVID-19 transmission as the first of its kind. The main aim of this study is to conduct a bibliometric review of the literature in the area of COVID-19 transmission dynamics. We have conducted bibliometric analysis using descriptive and network analysis methods to review the literature in this area using RStudio, Openrefine, VOSviewer, and Tableau. We reviewed 1103 articles published in 2020-2022. The result identified the top authors, top disciplines, research patterns, and hotspots and gave us clear directions for classifying research topics in this area. New research areas are rapidly emerging in this area, which needs constant observation by researchers to combat this global epidemic.

The impact of COVID-19 on a Malaria dominated region: A mathematical analysis and simulations

One of society’s major concerns that have continued for a long time is infectious diseases. It has been demonstrated that certain disease infections, in particular multiple disease infections, make it more challenging to identify and treat infected individuals, thus deteriorating human health. As a result, a COVID-19-malaria co-infection model is developed and analyzed to study the effects of threshold quantities and co-infection transmission rate on the two diseases’ synergistic relationship. This allowed us to better understand the co-dynamics of the two diseases in the population. The existence and stability of the disease-free equilibrium of each single infection were first investigated by using their respective reproduction number. The COVID-19 and malaria-free equilibrium are locally asymptotically stable when the individual threshold quantities RC and RM are below unity. Additionally, the occurrence of the malaria prevalent equilibrium is examined, and the requirements for the backward bifurcation’s existence are provided. Sensitivity analysis reveals that the two main parameters that influence the spread of COVID-19 infection are the disease transmission rate (βc) and the fraction of the exposed individuals becoming symptomatic (ψ), while malaria transmission is influenced by the abundance of vector population, which is driven by recruitment rate (πv) with an increase in the effective biting rate (b), probability of malaria transmission per mosquito bite (βm), and probability of malaria transmission from infected humans to vectors (βv). The findings from the numerical simulation of the model show that COVID-19 will predominate in the populace and drives malaria to extinction when RM<1<RC, whereas malaria will dominate in the population and drives COVID-19 into extinction when RC<1<RM. At the disease’s endemic equilibrium, the two diseases will coexist with the one with the highest reproduction number predominating but not eradicating the other. It was demonstrated in particular that COVID-19 will invade a population where malaria is endemic if the invasion reproduction number exceeds unity. The findings also demonstrate that when the two diseases are at endemic equilibrium, the prevalence of co-infection increases COVID-19’s burden on the population while decreasing malaria incidence.

Optimal control and cost-effectiveness analysis for a COVID-19 model with individual protection awareness

This paper is focused on the design of optimal control strategies for COVID-19 and the model containing susceptible individuals with awareness of protection and susceptible individuals without awareness of protection is established. The goal of this paper is to minimize the number of infected people and susceptible individuals without protection awareness, and to increase the willingness of susceptible individuals to take protection measures. We conduct a qualitative analysis of this mathematical model. Based on the sensitivity analysis, the optimal control method is proposed, namely personal protective measures, vaccination and awareness raising programs. It is found that combining the three methods can minimize the number of infected people. Moreover, the introduction of awareness raising program in society will greatly reduce the existence of susceptible individuals without protection awareness. To evaluate the most cost-effective strategy we performed a cost-effectiveness analysis using the ICER method.

On the qualitative study of a two-trophic plant-herbivore model

The coexistence of plant-herbivore populations in an ecological system is a fundamental topic of research in mathematical ecology. Plant-herbivore interactions are often described by using discrete-time models in the case of non-overlapping generations: such generations have some specific time interval of life and their old generations are replaced by new generations after some regular interval of time. Keeping in mind the dynamical reliability of continuous-time models we presented two discrete-time plant-herbivore models. Mainly, by applying Euler's forward method a discrete-time plant-herbivore model is obtained from a continuous-time plant-herbivore model. In addition, a dynamically consistent discrete-time plant-herbivore model is obtained by applying a nonstandard difference scheme. Moreover, local stability is discussed and the existence of bifurcation about positive equilibrium is shown under some mathematical conditions. To control bifurcation and chaos, a modified hybrid technique is developed. Finally, to support our theocratical results and to show the dynamical reliability of the nonstandard difference scheme some numerical examples are provided.

The fractional-order discrete COVID-19 pandemic model: stability and chaos

This paper presents and investigates a new fractional discrete COVID-19 model which involves three variables: the new daily cases, additional severe cases and deaths. Here, we analyze the stability of the equilibrium point at different values of the fractional order. Using maximum Lyapunov exponents, phase attractors, bifurcation diagrams, the 0-1 test and approximation entropy (ApEn), it is shown that the dynamic behaviors of the model change from stable to chaotic behavior by varying the fractional orders. Besides showing that the fractional discrete model fits the real data of the pandemic, the simulation findings also show that the numbers of new daily cases, additional severe cases and deaths exhibit chaotic behavior without any effective attempts to curb the epidemic.

Mathematical modeling of corona virus (COVID-19) and stability analysis

In this paper, the mathematical modeling of the novel corona virus (COVID-19) is considered. A brief relationship between the unknown hosts and bats is described. Then the interaction among the seafood market and peoples is studied. After that, the proposed model is reduced by assuming that the seafood market has an adequate source of infection that is capable of spreading infection among the people. The reproductive number is calculated and it is proved that the proposed model is locally asymptotically stable when the reproductive number is less than unity. Then, the stability results of the endemic equilibria are also discussed. To understand the complex dynamical behavior, fractal-fractional derivative is used. Therefore, the proposed model is then converted to fractal-fractional order model in Atangana-Baleanu (AB) derivative and solved numerically by using two different techniques. For numerical simulation Adam-Bash Forth method based on piece-wise Lagrangian interpolation is used. The infection cases for Jan-21, 2020, till Jan-28, 2020 are considered. Then graphical consequences are compared with real reported data of Wuhan city to demonstrate the efficiency of the method proposed by us.

Analysis of multi-strain infection of vaccinated and recovered population through epidemic model: Application to COVID-19

In this work, an innovative multi-strain SV EAIR epidemic model is developed for the study of the spread of a multi-strain infectious disease in a population infected by mutations of the disease. The population is assumed to be completely susceptible to n different variants of the disease, and those who are vaccinated and recovered from a specific strain k (k ≤ n) are immune to previous and present strains j = 1, 2, ⋯, k, but can still be infected by newer emerging strains j = k + 1, k + 2, ⋯, n. The model is designed to simulate the emergence and dissemination of viral strains. All the equilibrium points of the system are calculated and the conditions for existence and global stability of these points are investigated and used to answer the question as to whether it is possible for the population to have an endemic with more than one strain. An interesting result that shows that a strain with a reproduction number greater than one can still die out on the long run if a newer emerging strain has a greater reproduction number is verified numerically. The effect of vaccines on the population is also analyzed and a bound for the herd immunity threshold is calculated. The validity of the work done is verified through numerical simulations by applying the proposed model and strategy to analyze the multi-strains of the COVID-19 virus, in particular, the Delta and the Omicron variants, in the United State.

COVID-19 vaccination policies under uncertain transmission characteristics using stochastic programming

We develop a new stochastic programming methodology for determining optimal vaccination policies for a multi-community heterogeneous population. An optimal policy provides the minimum number of vaccinations required to drive post-vaccination reproduction number to below one at a desired reliability level. To generate a vaccination policy, the new method considers the uncertainty in COVID-19 related parameters such as efficacy of vaccines, age-related variation in susceptibility and infectivity to SARS-CoV-2, distribution of household composition in a community, and variation in human interactions. We report on a computational study of the new methodology on a set of neighboring U.S. counties to generate vaccination policies based on vaccine availability. The results show that to control outbreaks at least a certain percentage of the population should be vaccinated in each community based on pre-determined reliability levels. The study also reveals the vaccine sharing capability of the proposed approach among counties under limited vaccine availability. This work contributes a decision-making tool to aid public health agencies worldwide in the allocation of limited vaccines under uncertainty towards controlling epidemics through vaccinations.

Assessing the potential impact of COVID-19 Omicron variant: Insight through a fractional piecewise model

We consider a new mathematical model for the COVID-19 disease with Omicron variant mutation. We formulate in details the modeling of the problem with omicron variant in classical differential equations. We use the definition of the Atangana-Baleanu derivative and obtain the extended fractional version of the omicron model. We study mathematical results for the fractional model and show the local asymptotical stability of the model for infection-free case if R 0 < 1 . We show the global asymptotically stable of the model for the disease free case when R 0 ≤ 1 . We show the existence and uniqueness of solution of the fractional model. We further extend the fractional order model into piecewise differential equation system and give a numerical algorithm for their numerical simulation. We consider the real cases of COVID-19 in South Africa of the third wave March 2021-Sep 2021 and estimate the model parameters and get R 0 ≈ 1 . 4004 . The real parameters values are used to show the graphical results for the fractional and piecewise model.

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.

Global Stability of a Humoral Immunity COVID-19 Model with Logistic Growth and Delays

The mathematical modeling and analysis of within-host or between-host coronavirus disease 2019 (COVID-19) dynamics are considered robust tools to support scientific research. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the cause of COVID-19. This paper proposes and investigates a within-host COVID-19 dynamics model with latent infection, the logistic growth of healthy epithelial cells and the humoral (antibody) immune response. Time delays can affect the dynamics of SARS-CoV-2 infection predicted by mathematical models. Therefore, we incorporate four time delays into the model: (i) delay in the formation of latent infected epithelial cells, (ii) delay in the formation of active infected epithelial cells, (iii) delay in the activation of latent infected epithelial cells, and (iv) maturation delay of new SARS-CoV-2 particles. We establish that the model’s solutions are non-negative and ultimately bounded. This confirms that the concentrations of the virus and cells should not become negative or unbounded. We deduce that the model has three steady states and their existence and stability are perfectly determined by two threshold parameters. We use Lyapunov functionals to confirm the global stability of the model’s steady states. The analytical results are enhanced by numerical simulations. The effect of time delays on the SARS-CoV-2 dynamics is investigated. We observe that increasing time delay values can have the same impact as drug therapies in suppressing viral progression. This offers some insight useful to develop a new class of treatment that causes an increase in the delay periods and then may control SARS-CoV-2 replication.

Stochastic Optimal Control Analysis of a Mathematical Model: Theory and Application to Non-Singular Kernels

Some researchers believe fractional differential operators should not have a non-singular kernel, while others strongly believe that due to the complexity of nature, fractional differential operators can have either singular or non-singular kernels. This contradiction in thoughts has led to the publication of a few papers that are against differential operators with non-singular kernels, causing some negative impacts. Thus, publishers and some Editors-in-Chief are concerned about the future of fractional calculus, which has generally brought confusion among the vibrant and innovative young researchers who desire to apply fractional calculus within their respective fields. Thus, the present work aims to develop a model based on a stochastic process that could be utilized to portray the effect of arbitrary-order derivatives. A nonlinear perturbation is used to study the proposed stochastic model with the help of white noises. The required condition(s) for the existence of an ergodic stationary distribution is obtained via Lyapunov functional theory. The finding of the study indicated that the proposed noises have a remarkable impact on the dynamics of the system. To reduce the spread of a disease, we imposed some control measures on the stochastic model, and the optimal system was achieved. The models both with and without control were coded in MATLAB, and at the conclusion of the research, numerical solutions are provided.

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.

Mathematical modeling and stability analysis of the COVID-19 with quarantine and isolation

The present paper focuses on the modeling of the COVID-19 infection with the use of hospitalization, isolation and quarantine. Initially, we construct the model by spliting the entire population into different groups. We then rigorously analyze the model by presenting the necessary basic mathematical features including the feasible region and positivity of the problem solution. Further, we evaluate the model possible equilibria. The theoretical expression of the most important mathematical quantity of major public health interest called the basic reproduction number is presented. We are taking into account to study the disease free equilibrium by studying its local and global asymptotical analysis. We considering the cases of the COVID-19 infection of Pakistan population and find the parameters using the estimation with the help of nonlinear least square and have R 0 ≈ 1 . 95 . Further, to determine the influence of the model parameters on disease dynamics we perform the sensitivity analysis. Simulations of the model are presented using estimated parameters and the impact of various non-pharmaceutical interventions on disease dynamics is shown with the help of graphical results. The graphical interpretation justify that the effective utilization of keeping the social-distancing, making the quarantine of people (or contact-tracing policy) and to make hospitalization of confirmed infected people that dramatically reduces the number of infected individuals (enhancing the quarantine or contact-tracing by 50% from its baseline reduces 84% in the predicted number of confirmed infected cases). Moreover, it is observed that without quarantine and hospitalization the scenario of the disease in Pakistan is very worse and the infected cases are raising rapidly. Therefore, the present study suggests that still, a proper and effective application of these non-pharmaceutical interventions are necessary to curtail or minimize the COVID-19 infection in Pakistan.

Model of Epidemic Kinetics with a Source on the Example of Moscow

A new theoretical model of epidemic kinetics is considered, which uses elements of the physical model of the kinetics of the atomic level populations of an active laser medium as follows: a description of states and their populations, transition rates between states, an integral operator, and a source of influence. It is shown that to describe a long-term epidemic, it is necessary to use the concept of the source of infection. With a model constant source of infection, the epidemic, in terms of the number of actively infected people, goes to a stationary regime, which does not depend on the population size and the characteristics of quarantine measures. Statistics for Moscow daily increase in infected is used to determine the real source of infection. An interpretation of the waves generated by the source is given. It is shown that more accurate statistics of excess mortality can only be used to clarify the frequency rate of mortality of the epidemic, but not to determine the source of infection.

A mathematical model for SARS-CoV-2 in variable-order fractional derivative

A new coronavirus mathematical with hospitalization is considered with the consideration of the real cases from March 06, 2021 till the end of April 30, 2021. The essential mathematical results for the model are presented. We show the model stability when R 0 < 1 in the absence of infection. We show that the system is stable locally asymptotically when R 0 < 1 at infection free state. We also show that the system is globally asymptotically stable in the disease absence when R 0 < 1 . Data have been used to fit accurately to the model and found the estimated basic reproduction number to be R 0 = 1.2036 . Some graphical results for the effective parameters are drawn for the disease elimination. In addition, a variable-order model is introduced, and so as to handle the outbreak effectively and efficiently, a genetic algorithm is used to produce high-quality control. Numerical simulations clearly show that decision-makers may develop helpful and practical strategies to manage future waves by implementing optimum policies.

Lockdown policies: A macrodynamic perspective for COVID-19☆

The COVID-19 pandemic has produced a global health and economic crisis. The entire world has faced a trade-off between health and recessionary effects. This paper investigates this trade-off according to a macro-dynamic perspective. We set up and simulate a Dynamic Stochastic General Equilibrium model to analyze the COVID-19 contagion within an economy with endogenous dynamics for the pandemic, variable labor utilization, and four lockdown policies with different degrees of size and duration. There are three main results in this study. First, the model matches rather well with the main European economies’ preliminary stylized facts during the COVID-19 pandemic. In particular, a temporary lockdown policy reduces the epidemic’s size but exacerbates the recession’s severity. The negative peak in aggregate production ranges from 10% with a soft containment measure to 25% with a strong containment measure; second, recovery from recession emerges when the lockdown policy is relaxed. On that basis, the output return to its pre-lockdown level after about 50 weeks. Third, sectors characterized by flexible and capital-intensive technology suffer a more severe slowdown.