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This paper proposes a novel fractional-order delayed Ross-Macdonald model for malaria transmission. This paper aims to systematically investigate the effect of both the incubation periods of Plasmodium and the order on the dynamic behavior of diseases. Utilizing inequality techniques, contraction mapping theory, fractional linear stability theorem, and bifurcation theory, several sufficient conditions for the existence and uniqueness of solutions, the local stability of the positive equilibrium point, and the existence of fractional-order Hopf bifurcation are obtained under different time delays cases. The results show that time delay can change the stability of system. System becomes unstable and generates a Hopf bifurcation when the delay increases to a certain value. Besides, the value of order influences the stability interval size. Thus, incubation periods and the order have a major effect on the dynamic behavior of the model. The effectiveness of the theoretical results is shown through numerical simulations.

The linearisation procedure used in the modelling of the hydro-pneumatic suspension system by Moreau et al. contains an inaccuracy. Hence, the conclusion that the weight of the sprung mass is normally absent of the dynamic behaviour of the sprung mass is demonstrated to be false.

This paper investigates the guaranteed cost positive consensus for linear multi-agent systems (MASs) with multiple time-varying delays and switching topologies. First, necessary and sufficient conditions for the positivity of the focused system are given. The positive consensus criterion and switching signals are derived by combining multiple Lyapunov-Krasovskii functionals with mode-dependent average dwell time approaches. Then, the guaranteed cost positive consensus is studied via the state errors among neighbouring agents and control inputs. It realizes a trade-off between the consensus regulation performance and the control energy consumption of MASs. Particularly, the determined upper bound of the cost function depends merely on the initial conditions of the agents and is independent of the switching movements. Finally, simulation results are given to validate the proposed control scheme.

COVID-19 is a public health emergency for human beings and brings some very harmful consequences in social and economic fields. In order to model COVID-19 and develop the effective control measures, this paper proposes an SEIR-type epidemic model with the contacting distance between the healthy individuals and the asymptomatic or symptomatic infected individuals, and the immigration rate of the healthy individuals since the contacting distance and the immigration rate are two critical factors which determine the transmission of COVID-19. Firstly, the threshold values of the contacting distance and the immigration rate are obtained to analyze the presented model. Secondly, based on the data from January 10, 2020, to March 18, 2020, for Wuhan city, all parameters are estimated. Finally, based on the estimated parameters, the sensitivity analysis and the numerical study are conducted. The results show that the contacting distance and the immigration rate play an important role in controlling COVID-19. Meanwhile, the extinct lag decreases as the contacting distance increases and/or the immigration rate decreases. Our study could give some reasonable suggestions for the health officials and the public and provide a theoretical issue for globally controlling the COVID-19 pandemic.

In this paper, we consider a stochastic SIR epidemic model with general disease incidence rate and perturbation caused by nonlinear white noise and L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\acute{e}$$\end{document}e´vy jumps. First of all, we study the existence and uniqueness of the global positive solution of the model. Then, we establish a threshold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}λ by investigating the one-dimensional model to determine the extinction and persistence of the disease. To verify the model has an ergodic stationary distribution, we adopt a new method which can obtain the sufficient and almost necessary conditions for the extinction and persistence of the disease. Finally, some numerical simulations are carried out to illustrate our theoretical results.

Did the pattern of US stock market volatility change due to COVID-19 or have the US stock markets been less volatile despite the pandemic shock? And as for tech stocks, are they even less volatile than the market overall? In this paper, we provide evidence in favor of a “quietness” in the stock markets, interrupted by COVID-19, by analyzing dispersion, skewness and kurtosis characteristics of the empirical distribution of nine returns series that include individual FATANG stocks (FAANG: Facebook, Amazon, Apple, Netflix and Google; plus Tesla) and US indices (S&P 500, DJIA and NASDAQ). In comparison with the years before, the daily average return after COVID-19 was 6.48, 2.58 and 2.34 times higher for Tesla, Apple and NASDAQ, respectively. In terms of volatility, the increase was more pronounced in the three stock indices when compared to the individual FATANG stocks. This paper also puts forward a new methodology based on semi-variance and semi-kurtosis. While the value of the ratio between semi-kurtosis and kurtosis is always higher than 70% for the three US stock indices, in the case of stocks the opposite is true, which highlights the importance of large positive returns when compared to negative ones. Structural breaks and conditional heteroskedasticity are also analyzed by considering the traditional symmetrical and asymmetrical GARCH models. We show that in the most recent past, despite the COVID-19 pandemic, the FATANG tech stocks are characterized mostly by conditional homoskedasticity, while the returns of US stock indices are characterized mainly by conditional heteroskedasticity.

SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2) has been causing an outbreak of a new type of pneumonia globally, and repeated outbreaks have already appeared. Among the studies on the spread of the COVID-19, few studies have investigated the repeated outbreaks in stages, and the quantitative condition of a controllable spread has not been revealed. In this paper, a brief compartmental model is developed. The effective reproduction number (ERN) of the model is interpreted by the ratio of net newly infectious individuals to net isolation infections to assess the controllability of the spread of COVID-19. It is found that the value of the ERN at the inflection point of the pandemic is equal to one. The effectiveness of the quarantine, even the treatment, is parametrized in various stages with Gompertz functions to increase modeling accuracy. The impacts of the vaccinations are discussed by adding a vaccinated compartment. The results show that the sufficient vaccinations can make the inflection point appear early and significantly reduce subsequent increases in newly confirmed cases. The analysis of the ERNs of COVID-19 in the United States, Spain, France, and Peru confirms that the condition of a repeated outbreak is to relax or lift the interventions related to isolation and quarantine interventions to a level where the ERN is greater than one.

The currently ongoing COVID-19 pandemic confronts governments and their health systems with great challenges for disease management. Epidemiological models play a crucial role, thereby assisting policymakers to predict the future course of infections and hospitalizations. One difficulty with current models is the existence of exogenous and unmeasurable variables and their significant effect on the infection dynamics. In this paper, we show how a method from nonlinear control theory can complement common compartmental epidemiological models. As a result, one can estimate and predict these exogenous variables requiring the reported infection cases as the only data source. The method allows to investigate how the estimates of exogenous variables are influenced by non-pharmaceutical interventions and how imminent epidemic waves could already be predicted at an early stage. In this way, the concept can serve as an “epidemometer” and guide the optimal timing of interventions. Analyses of the COVID-19 epidemic in various countries demonstrate the feasibility and potential of the proposed approach. The generic character of the method allows for straightforward extension to different epidemiological models.

Given a data-set of Ribonucleic acid (RNA) sequences we can infer the phylogenetics of the samples and tackle the information for scientific purposes. Based on current data and knowledge, the SARS-CoV-2 seemingly mutates much more slowly than the influenza virus that causes seasonal flu. However, very recent evolution poses some doubts about such conjecture and shadows the out-coming light of people vaccination. This paper adopts mathematical and computational tools for handling the challenge of analyzing the data-set of different clades of the severe acute respiratory syndrome virus-2 (SARS-CoV-2). On one hand, based on the mathematical paraphernalia of tools, the concept of distance associated with the Kolmogorov complexity and Shannon information theories, as well as with the Hamming scheme, are considered. On the other, advanced data processing computational techniques, such as, data compression, clustering and visualization, are borrowed for tackling the problem. The results of the synergistic approach reveal the complex time dynamics of the evolutionary process and may help to clarify future directions of the SARS-CoV-2 evolution.

Although deterministic compartmental models are useful for predicting the general trend of a disease's spread, they are unable to describe the random daily fluctuations in the number of new infections and hospitalizations, which is crucial in determining the necessary healthcare capacity for a specified level of risk. In this paper, we propose a stochastic SEIHR (sSEIHR) model to describe such random fluctuations and provide sufficient conditions for stochastic stability of the disease-free equilibrium, based on the basic reproduction number that we estimated. Our extensive numerical results demonstrate strong threshold behavior near the estimated basic reproduction number, suggesting that the necessary conditions for stochastic stability are close to the sufficient conditions derived. Furthermore, we found that increasing the noise level slightly reduces the final proportion of infected individuals. In addition, we analyze COVID-19 data from various regions worldwide and demonstrate that by changing only a few parameter values, our sSEIHR model can accurately describe both the general trend and the random fluctuations in the number of daily new cases in each region, allowing governments and hospitals to make more accurate caseload predictions using fewer compartments and parameters than other comparable stochastic compartmental models.

Recently, considering the temporary immunity of individuals who have recovered from certain infectious diseases, Liu et al. (Phys A Stat Mech Appl 551:124152, 2020) proposed and studied a stochastic susceptible-infected-recovered-susceptible model with logistic growth. For a more realistic situation, the effects of quarantine strategies and stochasticity should be taken into account. Hence, our paper focuses on a stochastic susceptible-infected-quarantined-recovered-susceptible epidemic model with temporary immunity. First, by means of the Khas’minskii theory and Lyapunov function approach, we construct a critical value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0^S$$\end{document}R0S corresponding to the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0$$\end{document}R0 of the deterministic system. Moreover, we prove that there is a unique ergodic stationary distribution if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0^S>1$$\end{document}R0S>1. Focusing on the results of Zhou et al. (Chaos Soliton Fractals 137:109865, 2020), we develop some suitable solving theories for the general four-dimensional Fokker–Planck equation. The key aim of the present study is to obtain the explicit density function expression of the stationary distribution under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0^S>1$$\end{document}R0S>1. It should be noted that the existence of an ergodic stationary distribution together with the unique exact probability density function can reveal all the dynamical properties of disease persistence in both epidemiological and statistical aspects. Next, some numerical simulations together with parameter analyses are shown to support our theoretical results. Last, through comparison with other articles, results are discussed and the main conclusions are highlighted.

34,354,966 active cases and 460,787 deaths because of COVID-19 pandemic were recorded on November 06, 2021, in India. To end this ongoing global COVID-19 pandemic, there is an urgent need to implement multiple population-wide policies like social distancing, testing more people and contact tracing. To predict the course of the pandemic and come up with a strategy to control it effectively, a compartmental model has been established. The following six stages of infection are taken into consideration: susceptible (S), asymptomatic infected (A), clinically ill or symptomatic infected (I), quarantine (Q), isolation (J) and recovered (R), collectively termed as SAIQJR. The qualitative behavior of the model and the stability of biologically realistic equilibrium points are investigated in terms of the basic reproduction number. We performed sensitivity analysis with respect to the basic reproduction number and obtained that the disease transmission rate has an impact in mitigating the spread of diseases. Moreover, considering the non-pharmaceutical and pharmaceutical intervention strategies as control functions, an optimal control problem is implemented to mitigate the disease fatality. To reduce the infected individuals and to minimize the cost of the controls, an objective functional has been constructed and solved with the aid of Pontryagin’s maximum principle. The implementation of optimal control strategy at the start of a pandemic tends to decrease the intensity of epidemic peaks, spreading the maximal impact of an epidemic over an extended time period. Extensive numerical simulations show that the implementation of intervention strategy has an impact in controlling the transmission dynamics of COVID-19 epidemic. Further, our numerical solutions exhibit that the combination of three controls are more influential when compared with the combination of two controls as well as single control. Therefore, the implementation of all the three control strategies may help to mitigate novel coronavirus disease transmission at this present epidemic scenario. Supplementary Information The online version supplementary material available at 10.1007/s11071-022-07235-7.

The COVID-19 pandemic shows to have a huge impact on people's health and countries' infrastructures around the globe. Iran was one of the first countries that experienced the vast prevalence of the coronavirus outbreak. The Iranian authorities applied various non-pharmaceutical interventions to eradicate the epidemic in different periods. This study aims to investigate the effectiveness of non-pharmaceutical interventions in managing the current Coronavirus pandemic and to predict the next wave of infection in Iran. To achieve the research objective, the number of cases and deaths before and after the interventions was studied and the effective reproduction number of the infection was analyzed under various scenarios. The SEIR generic model was applied to capture the dynamic of the pandemic in Iran. To capture the effects of different interventions, the corresponding reproduction number was considered. Depending on how people are responsive to interventions, the effectiveness of each intervention has been investigated. Results show that the maximum number of the total of infected individuals will occur around the end of May and the start of June 2021. It is concluded that the outbreak could be smoothed if full lockdown and strict quarantine continue. The proposed modeling could be used as an assessment tool to evaluate the effects of different interventions in new outbreaks.

Since the earliest outbreak of COVID-19, the disease continues to obstruct life normalcy in many parts of the world. The present work proposes a mathematical framework to improve non-pharmaceutical interventions during the new normal before vaccination settles herd immunity. The considered approach is built from the viewpoint of decision makers in developing countries where resources to tackle the disease from both a medical and an economic perspective are scarce. Spatial auto-correlation analysis via global Moran’s index and Moran’s scatter is presented to help modulate decisions on hierarchical-based priority for healthcare capacity and interventions (including possible vaccination), finding a route for the corresponding deployment as well as landmarks for appropriate border controls. These clustering tools are applied to sample data from Sri Lanka to classify the 26 Regional Director of Health Services (RDHS) divisions into four clusters by introducing convenient classification criteria. A metapopulation model is then used to evaluate the intra- and inter-cluster contact restrictions as well as testing campaigns under the absence of confounding factors. Furthermore, we investigate the role of the basic reproduction number to determine the long-term trend of the regressing solution around disease-free and endemic equilibria. This includes an analytical bifurcation study around the basic reproduction number using Brouwer Degree Theory and asymptotic expansions as well as related numerical investigations based on path-following techniques. We also introduce the notion of average policy effect to assess the effectivity of contact restrictions and testing campaigns based on the proposed model’s transient behavior within a fixed time window of interest.

This paper studies an SEIR-type epidemic model with time delay and vaccination control. The vaccination control is applied when the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0>1$$\end{document}R0>1. The vaccination strategy is expressed as a state delayed feedback which is related to the current and previous state of the epidemic model, and makes the model become a linear system in new coordinates. For the presence and absence of vaccination control, we investigate the nonnegativity and boundedness of the model, respectively. We obtain some sufficient conditions for the eigenvalues of the linear system such that the nonnegativity of the epidemic model can be guaranteed when the vaccination strategy is applied. In addition, we study the stability of disease-free equilibrium when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_01$$\end{document}R0>1. Finally, we use the obtained theoretical results to simulate the vaccination strategy to control the spread of COVID-19.

Nowadays, humanity is facing one of the most dangerous pandemics known as COVID-19. Due to its high inter-person contagiousness, COVID-19 is rapidly spreading across the world. Positive patients are often suffering from different symptoms that can vary from mild to severe including cough, fever, sore throat, and body aches. In more dire cases, infected patients can experience severe symptoms that can cause breathing difficulties which lead to stern organ failure and die. The medical corps all over the world are overloaded because of the exponentially myriad number of contagions. Therefore, screening for the disease becomes overwrought with the limited tools of test. Additionally, test results may take a long time to acquire, leaving behind a higher potential for the prevalence of the virus among other individuals by the patients. To reduce the chances of infection, we suggest a prediction model that distinguishes the infected COVID-19 cases based on clinical symptoms and features. This model can be helpful for citizens to catch their infection without the need for visiting the hospital. Also, it helps the medical staff in triaging patients in case of a deficiency of medical amenities. In this paper, we use the non-dominated sorting genetic algorithm (NSGA-II) to select the interesting features by finding the best trade-offs between two conflicting objectives: minimizing the number of features and maximizing the weights of selected features. Then, a classification phase is conducted using an AdaBoost classifier. The proposed model is evaluated using two different datasets. To maximize results, we performed a natural selection of hyper-parameters of the classifier using the genetic algorithm. The obtained results prove the efficiency of NSGA-II as a feature selection algorithm combined with AdaBoost classifier. It exhibits higher classification results that outperformed the existing methods.

Coronavirus disease 2019 is a recent strong challenge for the world. In this paper, an epidemiology model is investigated as a model for the development of COVID-19. The propagation of COVID-19 through various sub-groups of society is studied. Some critical parameters, such as the background of mortality without considering the disease state and the speed of moving people from infected to resistance, affect the conditions of society. In this paper, early warning indicators are used to predict the bifurcation points in the system. In the interaction of various sub-groups of society, each sub-group can have various parameters. Six cases of the sub-groups interactions are studied. By coupling these sub-groups, various dynamics of the whole society are investigated.

In view of the facts in the infection and propagation of COVID-19, a stochastic reaction–diffusion epidemic model is presented to analyse and control this infectious diseases. Stationary distribution and Turing instability of this model are discussed for deriving the sufficient criteria for the persistence and extinction of disease. Furthermore, the amplitude equations are derived by using Taylor series expansion and weakly nonlinear analysis, and selection of Turing patterns for this model can be determined. In addition, the optimal quarantine control problem for reducing control cost is studied, and the differences between the two models are compared. By applying the optimal control theory, the existence and uniqueness of the optimal control and the optimal solution are obtained. Finally, these results are verified and illustrated by numerical simulation.

India is one of the countries in the world which is badly affected by the Covid-19 second wave. Assembly election in four states and a union territory of India was taken place during March-May 2021 when the Covid-19 second wave was close to its peak and affected a huge number of people. We studied the impact of assembly election on the effective contact rate and the effective reproduction number of Covid-19 using different epidemiological models like SIR, SIRD, and SEIR. We also modeled the effective reproduction number for all election-bound states using different mathematical functions. We separately studied the case of all election-bound states and found all the states shown a distinct increase in the effective contact rate and the effective reproduction number during the election-bound time and just after that compared to pre-election time. States, where elections were conducted in single-phase, showed less increase in the effective contact rate and the reproduction number. The election commission imposed extra measures from the first week of April 2021 to restrict big campaign rallies, meetings, and different political activities. The effective contact rate and the reproduction number showed a trend to decrease for few states due to the imposition of the restrictions. We also compared the effective contact rate, and the effective reproduction number of all election-bound states and the rest of India and found all the parameters related to the spread of virus for election-bound states are distinctly high compared to the rest of India.

The current investigation is related to the design of novel integrated neuroswarming heuristic paradigm using Gudermannian artificial neural networks (GANNs) optimized with particle swarm optimization (PSO) aid with active-set (AS) algorithm, i.e., GANN-PSOAS, for solving the nonlinear third-order Emden–Fowler model (NTO-EFM) involving single as well as multiple singularities. The Gudermannian activation function is exploited to construct the GANNs-based differential mapping for NTO-EFMs, and these networks are arbitrary integrated to formulate the fitness function of the system. An objective function is optimized using hybrid heuristics of PSO with AS, i.e., PSOAS, for finding the weights of GANN. The correctness, effectiveness and robustness of the designed GANN-PSOAS are verified through comparison with the exact solutions on three problems of NTO-EFMs. The assessments on statistical observations demonstrate the performance on different measures for the accuracy, consistency and stability of the proposed GANN-PSOAS solver.

In this study, a four-dimensional fractional hyperchaotic model is analyzed based on general Riemann–Liouville–Caputo (RLC) fractal–fractional derivative (FFD). A series of new operators are constructed using three different elements, namely, the general Mittag–Leffler function, exponential decay, and power law. The operators have two parameters: One is considered as fractional order and the other as fractal dimension. The Qi hyperchaotic fractional attractor is modeled by using these operators, and the models are solved numerically using a very efficient numerical scheme. Meanwhile, the existence and uniqueness of solutions have been investigated to justify the physical adequacy of the model and the numerical scheme proposed in the resolution. The numerical simulations for some specific fractional order and fractal dimension are presented. Furthermore, these results obtained via generalized Caputo–Fabrizio and fractal–fractional derivative show some crossover effects, which is due to non-index law property. Finally, these obtained from generalized fractal–fractional derivative show very strange and new attractors with self-similarities.

Despite the huge relevance of vaccines for preventing COVID-19, physical isolation and quarantine of infected individuals are still the key strategies to fight against the COVID-19 pandemic. Based on a COVID-19 transmission epidemiological model governed by ordinary differential equations, here we propose an intermittent non-pharmacological protocol to control the fraction of infected individuals. In our approach, unlike what generically happens for numerical simulation models, we provide a global analysis of the model, giving qualitative information about every initial condition. Under some simple hypothesis and variations of parameters, we present some bifurcations and we are able to predict the minimum social distancing effort that do not collapse the health system.

We propose a new epidemic model considering the partial mapping relationship in a two-layered time-varying network, which aims to study the influence of information diffusion on epidemic spreading. In the model, one layer represents the epidemic-related information diffusion in the social networks, while the other layer denotes the epidemic spreading in physical networks. In addition, there just exist mapping relationships between partial pairs of nodes in the two-layered network, which characterizes the interaction between information diffusion and epidemic spreading. Meanwhile, the information and epidemics can only spread in their own layers. Afterwards, starting from the microscopic Markov chain (MMC) method, we can establish the dynamic equation of epidemic spreading and then analytically deduce its epidemic threshold, which demonstrates that the ratio of correspondence between two layers has a significant effect on the epidemic threshold of the proposed model. Finally, it is found that MMC method can well match with Monte Carlo (MC) simulations, and the relevant results can be helpful to understand the epidemic spreading properties in depth.

A novel approach to link the environmental stresses with the COVID-19 cases is adopted during this research. The time-dependent data are extracted from the online repositories that are freely available for knowledge and research. Since the time series data analysis is desired for the COVID-19 time-dependent frequent waves, here in this manuscript, we have developed a time series model with the aid of "nonlinear autoregressive network with exogenous inputs (NARX)" approach. The distribution of infectious agent-containing droplets from an infected person to an uninfected person is a common form of respiratory disease transmission. SARS-CoV-2 has mainly spread via short-range respiratory droplet transmission. Airborne transmission of SARS-CoV-2 seems to have occurred over long distances or times in unusual conditions; SARS-CoV-2 RNA was found in PM10 collected in Italy. This research shows that SARS-CoV-2 particles adsorbed to outdoor PM remained viable for a long time, given the epidemiology of COVID-19, outdoor air pollution is unlikely to be a significant route of transmission. In this research, ANN time series is used to analyze the data resulting from the COVID-19 first and second waves and the forecasted results show that air pollution affects people in different areas of Italy and make more people sick with covid-19. The model is developed based on the disease transmission data of Italy.

Multiple new variants of SARS-CoV-2 have been identified as the COVID-19 pandemic spreads across the globe. However, most epidemic models view the virus as static and unchanging and thus fail to address the consequences of the potential evolution of the virus. Here, we built a competitive susceptible-infected-removed (coSIR) model to simulate the competition between virus strains of differing severities or transmissibility under various virus control policies. The coSIR model predicts that although the virus is extremely unlikely to evolve into a “super virus” that causes an increased fatality rate, virus variants with less severe symptoms can lead to potential new outbreaks and can cost more lives over time. The present model also demonstrates that the protocols restricting the transmission of the virus, such as wearing masks and social distancing, are the most effective strategy in reducing total mortality. A combination of adequate testing and strict quarantine is a powerful alternative to policies such as mandatory stay-at-home orders, which may have an enormous negative impact on the economy. In addition, building Mobile Cabin Hospitals can be effective and efficient in reducing the mortality rate of highly infectious virus strains. Supplementary Information The online version contains supplementary material available at 10.1007/s11071-021-06705-8.

The transmission dynamics of COVID-19 is investigated in this study. A SINDy-LM modeling method that can effectively balance model complexity and prediction accuracy is proposed based on data-driven technique. First, the Sparse Identification of Nonlinear Dynamical systems (SINDy) method is used to discover and describe the nonlinear functional relationship between the dynamic terms in the model in accordance with the observation data of the COVID-19 epidemic. Moreover, the Levenberg–Marquardt (LM) algorithm is utilized to optimize the obtained model for improving the accuracy of the SINDy algorithm. Second, the obtained model, which is consistent with the logistic model in mathematical form with small errors and high robustness, is leveraged to review the epidemic situation in China. Otherwise, the evolution of the epidemic in Australia and Egypt is predicted, which demonstrates that this method has universality for constructing the global COVID-19 model. The proposed model is also compared with the extreme learning machine (ELM), which shows that the prediction accuracy of the SINDy-LM method outperforms that of the ELM method and the generated model has higher sparsity.

A novel image steganography technique in order to hide the ciphered voice data has been suggested in this work. The doctor's voice comments belonging to a coronavirus disease 2019 (COVID-19) patient are hidden in a medical image in order to protect the patient information. The introduced steganography technique is based on chaos theory. Firstly, the voice comments of the doctor are converted to an image and secondly, they are ciphered utilizing the suggested encryption algorithm based on a chaotic system. Then, they are embedded into the cover medical image. A lung angiography dual-energy computed tomography (CT) scan of a COVID-19 patient is used as a cover object. Numerical and security analyses of steganography method have been performed in MATLAB environment. The similarity metrics are calculated for R, G, B components of cover image and stego image as visual quality analysis metrics to examine the performance of the introduced steganography procedure. For a 512 x 512 pixel cover image, SSIM values are obtained as 0.8337, 0.7926, and 0.9273 for R, G, B components, respectively. Moreover, security analyses which are differential attack, histogram, information entropy, correlation of neighboring pixels and the initial condition sensitivity are carried out. The information entropy is calculated as 7.9993 bits utilizing the suggested steganography scheme. The mean value of the ten UACI and NPCR values are obtained as 33.5688% and 99.8069%, respectively. The results of security analysis have revealed that the presented steganography procedure is able to resist statistical attacks and the chaotic system-based steganography scheme shows the characteristics of the sensitive dependence on the initial condition and the secret key. The proposed steganography method which is based on a chaotic system has superior performance in terms of being robust against differential attack and hiding encrypted voice comments of the doctor. Moreover, the introduced algorithm is also resistant against exhaustive, known plaintext, and chosen plaintext attacks.

In this study, a novel reaction-diffusion model for the spread of the new coronavirus (COVID-19) is investigated. The model is a spatial extension of the recent COVID-19 SEIR model with nonlinear incidence rates by taking into account the effects of random movements of individuals from different compartments in their environments. The equilibrium points of the new system are found for both diffusive and non-diffusive models, where a detailed stability analysis is conducted for them. Moreover, the stability regions in the space of parameters are attained for each equilibrium point for both cases of the model and the effects of parameters are explored. A numerical verification for the proposed model using a finite difference-based method is illustrated along with their consistency, stability and proving the positivity of the acquired solutions. The obtained results reveal that the random motion of individuals has significant impact on the observed dynamics and steady-state stability of the spread of the virus which helps in presenting some strategies for the better control of it.

Over the last 9 months, the most prominent global health threat has been COVID-19. It first appeared in Wuhan, China, and then rapidly spread throughout the world. Since no treatment or preventative strategy has been identified until this time, millions of people across the world have been seriously affected by COVID-19. The modelling and prediction of confirmed COVID-19 cases have been given much attention by government policymakers for the purpose of combating it more effectively. For this purpose, the modelling and prediction performances of the linear model (LM), generalized additive model(GAM) and the time-varying linear model (Tv-LM) via Kalman filter are compared. This has never yet been undertaken in the literature. This comparative analysis also evaluates the linear relationship between the confirmed cases of COVID-19 in individual countries with the world. The analysis is implemented using daily COVID-19 confirmed rates of the top 8 most heavily affected countries and that of the world between 11 March and 21 December 2020 and 14-day forward predictions. The empirical findings show that the Tv-LM outperforms others in terms of model fit and predictability, suggesting that the relationship between each country’s rates with the world’s should be locally linear, not globally linear.

At present, more and more countries have entered the parallel stage of fighting the epidemic and restoring the economy after reaching the inflection point. Due to economic pressure, the government of India had to implement a policy of relaxing control during the rising period of the epidemic. This paper proposes a compartment model to study the development of COVID-19 in India after relaxing control. The Sigmoid function reflecting the cumulative effect is used to characterize the model-based diagnosis rate, cure rate and mortality rate. Considering the influence of the lockdown on the model parameters, the data are fitted using the method of least squares before and after the lockdown. According to numerical simulation and model analysis, the impact of India's relaxation of control before and after the inflection point is studied. Research shows that adopting a relaxation policy prematurely will have disastrous consequences. Even if the degree of relaxation is only 5% before the inflection point, it will increase the number of deaths by 15.03%. If the control is relaxed after the inflection point, the higher degree of relaxation, the more likely a secondary outbreak will occur, which will extend the duration of the pandemic, leading to more deaths and put more pressure on the health care system. It is found that after the implementation of the relaxation policy, medical quarantine capability and public cooperation are two vital indicators. The results show that if the supply of kits and detection speed can be increased after the control is relaxed, the secondary outbreak can be effectively avoided. Meanwhile, the increase in public cooperation can significantly reduce the spread of the virus, suppress the second outbreak of the pandemic and reduce the death toll. It is of reference significance to the government's policy formulation.

The present novel corona virus (2019-nCoV) infection has created a global emergency situation by spreading all over the world in a large scale within very short time period. The infection induced death rate is also very high. There is no vaccine or anti-viral medicine for such infection. So at this moment a major worldwide problem is that how we can control this pandemic. On the other hand, India is a high population density country, where the corona virus disease (COVID-19) has started to spread from $1^{st}$ week of March, 2020 in a significant number of COVID-19 positive cases. Due to this high population density human to human social contact rate is very high in India. So control of the pandemic COVID-19 in early stage is very urgent and challenging problem. Mathematical models are employed in this paper to study the COVID-19 dynamics, to identify the influential parameters and to find the proper prevention strategies to reduce the outbreak size. In this work, we have formulated a deterministic compartmental model to study the spreading of COVID-19 and estimated the model parameters by fitting the model with reported data of ongoing pandemic in India. Sensitivity analysis has been done to identify the key model parameters. The basic reproduction number has been estimated from actual data and the effective basic reproduction number has been studied on the basis of reported cases. Some effective preventive measures and their impacts on the disease dynamics have also been studied. Future trends of the disease transmission has been Predicted from our model with some control measures. Finally, the positive measures to control the disease have been summarized.

In this paper, a generalized fractional-order SEIR model is proposed, denoted by SEIQRP model, which divided the population into susceptible, exposed, infectious, quarantined, recovered and insusceptible individuals and has a basic guiding significance for the prediction of the possible outbreak of infectious diseases like the coronavirus disease in 2019 (COVID-19) and other insect diseases in the future. Firstly, some qualitative properties of the model are analyzed. The basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{ R }}_{0}$$\end{document}R0 is derived. When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{ R }}_{0}1$$\end{document}R0>1, the endemic equilibrium point is also unique. Furthermore, some conditions are established to ensure the local asymptotic stability of disease-free and endemic equilibrium points. The trend of COVID-19 spread in the USA is predicted. Considering the influence of the individual behavior and government mitigation measurement, a modified SEIQRP model is proposed, defined as SEIQRPD model, which is divided the population into susceptible, exposed, infectious, quarantined, recovered, insusceptible and dead individuals. According to the real data of the USA, it is found that our improved model has a better prediction ability for the epidemic trend in the next two weeks. Hence, the epidemic trend of the USA in the next two weeks is investigated, and the peak of isolated cases is predicted. The modified SEIQRP model successfully capture the development process of COVID-19, which provides an important reference for understanding the trend of the outbreak.

In this paper, we study the dynamics of an infectious disease in the presence of a continuous-imperfect vaccine and latent period. We consider a general incidence rate function with a non-monotonicity property to interpret the psychological effect in the susceptible population. After we propose the model, we provide the well-posedness property and calculate the effective reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_E$$\end{document}RE. Then, we obtain the threshold dynamics of the system with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_E$$\end{document}RE by studying the global stability of the disease-free equilibrium when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_E1$$\end{document}RE>1. For the endemic equilibrium, we use the semi-discretization method to analyze its linear stability. Then, we discuss the critical vaccination coverage rate that is required to eliminate the disease. Numerical simulations are provided to implement a case study regarding data of influenza patients, study the local and global sensitivity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_E

Recently, COVID-19 has attracted a lot of attention of researchers from different fields. Wearing masks is a frequently adopted precautionary measure. In this paper, we investigate the effect of behavior of wearing masks on epidemic dynamics in the context of COVID-19. At each time, every susceptible individual chooses whether to wear a mask or not in the next time step, which depends on an evaluation of the potential costs and perceived risk of infection. When the cost of infection is high, the majority of the population choose to wear masks, where global awareness plays a significant role. However, if the mask source is limited, global awareness may give rise to a negative result. In this case, more mask source should be allocated to the individuals with high risk of infection.

The world is experiencing an ongoing pandemic of coronavirus disease-2019 (COVID-19), which is caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). In attempts to control the pandemic, a range of nonpharmaceutical interventions (NPIs) has been implemented worldwide. However, the effect of synchronized NPIs for the control of COVID-19 at temporal and spatial scales has not been well studied. Therefore, a meta-population model that incorporates essential nonlinear processes was constructed to uncover the transmission characteristics of SARS-CoV-2 and then assess the effectiveness of synchronized NPIs on COVID-19 dynamics in China. Regional synchronization of NPIs was observed in China, and it was found that a combination of synchronized NPIs (the travel restrictions, the social distancing and the infection isolation) prevented 93.7% of SARS-CoV-2 infections. The use of synchronized NPIs at the time of the Wuhan lockdown may have prevented as much as 38% of SARS-CoV-2 infections, compared with the unsynchronized scenario. The interconnectivity of the epicenter, the implementation time of synchronized NPIs, and the number of regions considered all affected the performance of synchronized NPIs. The results highlight the importance of using synchronized NPIs in high-risk regions for the control of COVID-19 and shed light on effective strategies for future pandemic responses. Supplementary Information The online version contains supplementary material available at 10.1007/s11071-021-06505-0.

Pandemic is an unprecedented public health situation, especially for human beings with comorbidity. Vaccination and non-pharmaceutical interventions only remain extensive measures carrying a significant socioeconomic impact to defeating pandemic. Here, we formulate a mathematical model with comorbidity to study the transmission dynamics as well as an optimal control-based framework to diminish COVID-19. This encompasses modeling the dynamics of invaded population, parameter estimation of the model, study of qualitative dynamics, and optimal control problem for non-pharmaceutical interventions (NPIs) and vaccination events such that the cost of the combined measure is minimized. The investigation reveals that disease persists with the increase in exposed individuals having comorbidity in society. The extensive computational efforts show that mean fluctuations in the force of infection increase with corresponding entropy. This is a piece of evidence that the outbreak has reached a significant portion of the population. However, optimal control strategies with combined measures provide an assurance of effectively protecting our population from COVID-19 by minimizing social and economic costs.

Pandemic with mutation and permanent immune spreading in a small-world network described is studied by a modified SIR model, with consideration of mutation-immune mechanism. First, a novel mutation-immune model is proposed to modify the classical SIR model to simulate the transmission of mutable viruses that can be permanently immunized in small-world networks. Then, the influences of the size, coordination number and disorder parameter of the small-world network on the spread of the epidemic are analyzed in detail. Finally, the influences of mutation cycle and infection rate on epidemic transmission in small-world network are investigated further. The results show that the structure of the small-world network and the virus mutation cycle have an important impact on the spread of the epidemic. For viruses that can be permanently immunized, virus mutation is equivalent to making the immune cycle of human beings from infinite to finite. The dynamical behavior of the modified SIR epidemic model changes from an irregular, low-amplitude evolution at small disorder parameter to a spontaneous state of wide amplitude oscillations at large disorder parameter. Moreover, similar transition can also be found in increasing mutation cycle parameter. The maximum valid variation mutation decreases with the increase of disorder parameter and coordination number, but increase with respect to system size. In addition above, as the infection rate increases, the fraction of the infected increases and then decreases. As the mutation cycle increases, the time-average fraction of the infected and the infection rate corresponding to the maximum time-average fraction of the infected also decrease. As one conclusion, the results could give a deep understanding Pandemic with mutation and permanent immune spreading, from viewpoint of small-world network.

The present work is focused on modeling and predicting the cumulative number of deaths from COVID-19 in México by comparing an artificial neural network (ANN) with a Gompertz model applying multiple optimization algorithms for the estimation of coefficients and parameters, respectively. For the modeling process, the data published by the daily technical report COVID-19 in Mexico from March 19th to September 30th were used. The data published in the month of October were included to carry out the prediction. The results show a satisfactory comparison between the real data and those obtained by both models with a R2 > 0.999. The Levenberg–Marquardt and BFGS quasi-Newton optimization algorithm were favorable for fitting the coefficients during learning in the ANN model due to their fast and precision, respectively. On the other hand, the Nelder–Mead simplex algorithm fitted the parameters of the Gompertz model faster by minimizing the sum of squares. Therefore, the ANN model better fits the real data using ten coefficients. However, the Gompertz model using three parameters converges in less computational time. In the prediction, the inverse ANN model was solved by a genetic algorithm obtaining the best precision with a maximum error of 2.22% per day, as opposed to the 5.48% of the Gompertz model with respect to the real data reported from November 1st to 15th. Finally, according to the coefficients and parameters obtained from both models with recent data, a total of 109,724 cumulative deaths for the inverse ANN model and 100,482 cumulative deaths for the Gompertz model were predicted for the end of 2020.

In this work, we study an application of fractional-order Hopfield neural networks for optimization problem solving. The proposed network was simulated using a semi-analytical method based on Adomian decomposition,, and it was applied to the on-line estimation of time-varying parameters of nonlinear dynamical systems. Through simulations, it was demonstrated how fractional-order neurons influence the convergence of the Hopfield network, improving the performance of the parameter identification process if compared with integer-order implementations. Two different approaches for computing fractional derivatives were considered and compared as a function of the fractional-order of the derivatives: the Caputo and the Caputo–Fabrizio definitions. Simulation results related to different benchmarks commonly adopted in the literature are reported to demonstrate the suitability of the proposed architecture in the field of on-line parameter estimation.

An outbreak of the COVID-19 pandemic is a major public health disease as well as a challenging task to people with comorbidity worldwide. According to a report, comorbidity enhances the risk factors with complications of COVID-19. Here, we propose and explore a mathematical framework to study the transmission dynamics of COVID-19 with comorbidity. Within this framework, the model is calibrated by using new daily confirmed COVID-19 cases in India. The qualitative properties of the model and the stability of feasible equilibrium are studied. The model experiences the scenario of backward bifurcation by parameter regime accounting for progress in susceptibility to acquire infection by comorbidity individuals. The endemic equilibrium is asymptotically stable if recruitment of comorbidity becomes higher without acquiring the infection. Moreover, a larger backward bifurcation regime indicates the possibility of more infection in susceptible individuals. A dynamics in the mean fluctuation of the force of infection is investigated with different parameter regimes. A significant correlation is established between the force of infection and corresponding Shannon entropy under the same parameters, which provides evidence that infection reaches a significant proportion of the susceptible.

Inspired by the experimental and numerical findings, we study the dynamic instabilities of two coupled nonlinear delay differential equations that are used to describe the coherent oscillations between the top and bottom boundary layers in turbulent Rayleigh-Bénard convection. By introducing two sensitivity parameters for the instabilities of the top and bottom boundary layers, we find three different types of solutions, namely in-phase single-period oscillations, multi-period oscillations and chaos. The chaos solution contains rare but large amplitude fluctuations. The statistical properties of these fluctuations are consistent with those observed in the experiment for the massive eruption of thermal plumes, which causes random reversals of the large-scale circulation in turbulent Rayleigh-Bénard convection. Our study thus provides new insights into the origin of rare massive eruptions and sudden changes of large-scale flow pattern that are often observed in convection systems of geophysical and astrophysical scales.

This paper studies the rumor propagation model with heterogeneous networks in a multilingual environment. Firstly, a rumor propagation model with two language spreaders, in which the immunologic mechanism is considered in the ignorant, is proposed on heterogeneous networks. Secondly, the basic reproduction number and the dynamic behaviors are analyzed by using the next-generation matrix method and Lyapunov stability theory, respectively. Moreover, two control strategies are designed to effectively suppress the spread of the rumor. The one is continuous control strategy. By applying real-time control to the spreaders, the rumor spreading time can be greatly reduced and the rumor can die out in a short time. The other is event-triggered impulsive control strategy, which can effectively reduce the consumption of resources and ensure the extinction of the rumor. Finally, the correctness of theoretical analysis and the feasibility of control methods are verified by numerical simulations.

In this paper, the nonsmooth dynamics of two contacting rigid bodies is analysed in the presence of dry friction. In three dimensions, slipping can occur in continuously many directions. Then, the Coulomb friction model leads to a system of differential equations, which has a codimension-2 discontinuity set in the phase space. The new theory of extended Filippov systems is applied to analyse the dynamics of a rigid body moving on a fixed rigid plane to explore the possible transitions between the slipping and rolling behaviour. The paper focuses on finding the so-called limit directions of the slipping equations at the discontinuity. This leads to a complete qualitative description of the possible scenarios of the dynamics in the vicinity of the discontinuity. It is shown that the new approach consistently extends the information provided from the static friction force of the rolling behaviour. The methods are demonstrated on an application example.

We have presented in the current analytic research the generating formulae and results of direct mathematical modelling of non-classical trends for COVID-19’s evolution in world which, nevertheless, can be divided into two types: (1) the general trends for European countries such as Germany presented by the curve of modified sigmoid-type with up-inclination of the upper limit of saturation (at the end of first wave of pandemic) as well as for other cases of key countries that suffered from pandemic such as USA, India, Brazil, Russia (we conclude that the same type of coronavirus pandemic is valid for most of the countries in world with similar scenarios of the same type for general trends); (2) non-classical general trends for Middle East countries (such as Iran), with the appropriate bulge on graphical plots at the beginning of first wave of pandemic. We expect that the second wave of pandemic will pass its peak at the end of December 2020 for various countries. Moreover, the second wave of pandemic will have come to end at first decade of January 2021 in Germany and Iran (but at the end of January 2021 in India as well), so we should restrict ourselves in modelling the first and second waves of pandemic within this time period for these countries. Thus, the model of first approximation is considered here which allows to understand the mean-time trends of COVID-19 evolution for the first + second waves of pandemic for USA, Brazil and Russia, or predict the approximated time period of the upcoming third wave of pandemic in cases of India, Germany and Iran.

In this study, a new SIVS epidemic model for human papillomavirus (HPV) is proposed. The global dynamics of the proposed model are analyzed under pulse vaccination for the susceptible unvaccinated females and males. The threshold value for the disease-free periodic solution is obtained using the comparison theory for ordinary differential equations. It is demonstrated that the disease-free periodic solution is globally stable if the reproduction number is less than unity under some defined parameters. Moreover, we found the critical value of the pulse vaccination for susceptible females needed to control the HPV. The uniform persistence of the disease for some parameter values is also analyzed. The numerical simulations conducted agreed with the theoretical findings. It is found out using numerical simulation that the pulse vaccination has a good impact on reducing the disease.

Approximate analytical methods, such as the multiple scales (MS) and direct normal form (DNF) techniques, have been used extensively for investigating nonlinear mechanical structures, due to their ability to offer insight into the system dynamics. A comparison of their accuracy has not previously been undertaken, so is addressed in this paper. This is achieved by computing the backbone curves of two systems: the single-degree-of-freedom Duffing oscillator and a non-symmetric, two-degree-of-freedom oscillator. The DNF method includes an inherent detuning, which can be physically interpreted as a series expansion about the natural frequencies of the underlying linear system and has previously been shown to increase its accuracy. In contrast, there is no such inbuilt detuning for MS, although one may be, and usually is, included. This paper investigates the use of the DNF detuning as the chosen detuning in the MS method as a way of equating the two techniques, demonstrating that the two can be made to give identical results up to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ^2$$\end{document}ε2 order. For the examples considered here, the resulting predictions are more accurate than those provided by the standard MS technique. Wolfram Mathematica scripts implementing these methods have been provided to be used in conjunction with this paper to illustrate their practicality. Electronic supplementary material The online version of this article (10.1007/s11071-018-4534-1) contains supplementary material, which is available to authorized users.

During epidemic outbreaks, there are various types of information about epidemic prevention disseminated simultaneously among the population. Meanwhile, the mass media also scrambles to report the information related to the epidemic. Inspired by these phenomena, we devise a model to discuss the dynamical characteristics of the co-evolution spreading of multiple information and epidemic under the influence of mass media. We construct the co-evolution model under the framework of two-layered networks and gain the dynamical equations and epidemic critical point with the help of the micro-Markov chain approach. The expression of epidemic critical point show that the positive and negative information have a direct impact on the epidemic critical point. Moreover, the mass media can indirectly affect the epidemic size and epidemic critical point through their interference with the dissemination of epidemic-relevant information. Though extensive numerical experiments, we examine the accuracy of the dynamical equations and expression of the epidemic critical point, showing that the dynamical characteristics of co-evolution spreading can be well described by the dynamic equations and the epidemic critical point is able to be accurately calculated by the derived expression. The experimental results demonstrate that accelerating positive information dissemination and enhancing the propaganda intensity of mass media can efficaciously restrain the epidemic spreading. Interestingly, the way to accelerate the dissemination of negative information can also alleviate the epidemic to a certain extent when the positive information hardly spreads. Current results can provide some useful clues for epidemic prevention and control on the basis of epidemic-relevant information dissemination.

COVID-19 has spread around the world since December 2019, creating one of the greatest pandemics ever witnessed. According to the current reports, this is a situation when people need to be more careful and take the precaution measures more seriously, unless the condition may become even worse. Maintaining social distances and proper hygiene, staying at isolation or adopting the self-quarantine method are some of the common practices that people should use to avoid the infection. And the growing information regarding COVID-19 and its symptoms help the people to take proper precautions. In this present study, we consider an SEIRS epidemiological model on COVID-19 transmission which accounts for the effect of an individual's behavioural response due to the information regarding proper precautions. Our results indicate that if people respond to the growing information regarding awareness at a higher rate and start to take the protective measures, then the infected population decreases significantly. The disease fatality can be controlled only if a large proportion of individuals become immune, either by natural immunity or by a proper vaccine. In order to apply the latter option, we need to wait until a safe and proper vaccine is developed and it is a time-taking process. Hence, in the latter part of the work, an optimal control problem is considered by implementing control strategies to reduce the disease burden. Numerical figures show that the control denoting behavioural response works with higher intensity immediately after implementation and then gradually decreases with time. Further, the control policy denoting hospitalisation of infected individuals works with its maximum intensity for quite a long time period following a sudden decrease. As, the implementation of the control strategies reduce the infected population and increase the recovered population, so, it may help to reduce the disease transmission at this current epidemic situation.

Wuhan shutdown was implemented on January 23 and the first level response to public health emergencies (FLRPHE) was launched over the country, and then China got the outbreak of COVID-19 under control. A mathematical model is established to study the transmission of COVID-19 in Wuhan. This research investigates the spread of COVID-19 in Wuhan and assesses the effectiveness of control measures including the Wuhan city travel ban and FLRPHE. Based on the dynamical analysis and data fitting, the transmission of COVID-19 in Wuhan is estimated and the effects of control measures including Wuhan city travel ban and FLRPHE are investigated. According to the assumptions, the basic reproduction number for COVID-19 estimated that for Wuhan equal to 7.53 and there are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4.718\times 10^4$$\end{document}4.718×104 infectious people in Wuhan as of January 23. The interventions including the Wuhan city travel ban and FLRPHE reduce the size of peak and the cumulative number of confirmed cases of COVID-19 in Wuhan by 99%. The extraordinary efforts implemented by China effectively contain the transmission of COVID-19 and protect public health in China.

Since the outbreak of coronavirus disease in 2019 (COVID-19), the disease has rapidly spread to the world, and the cumulative number of cases is now more than 2.3 million. We aim to study the spread mechanism of rumors on social network platform during the spread of COVID-19 and consider education as a control measure of the spread of rumors. Firstly, a novel epidemic-like model is established to characterize the spread of rumor, which depends on the nonautonomous partial differential equation. Furthermore, the registration time of network users is abstracted as 'age,' and the spreading principle of rumors is described from two dimensions of age and time. Specifically, the susceptible users are divided into higher-educators class and lower-educators class, in which the higher-educators class will be immune to rumors with a higher probability and the lower-educators class is more likely to accept and spread the rumors. Secondly, the existence and uniqueness of the solution is discussed and the stability of steady-state solution of the model is obtained. Additionally, an interesting conclusion is that the education level of the crowd is an essential factor affecting the final scale of the spread of rumors. Finally, some control strategies are presented to effectively restrain the rumor propagation, and numerical simulations are carried out to verify the main theoretical results.