Nonlinear Dynamics

The novel coronavirus disease (namely COVID-19) has taken attention because of its deadliness across the globe, causing a massive death as well as critical situation around the world. It is an infectious disease which is caused by newly discovered coronavirus. Our study demonstrates with a nonlinear model of this devastating COVID-19 which narrates transmission from human-to-human in the society. Pontryagins Maximum principle has also been applied in order to obtain optimal control strategies where the maintenance of social distancing is the major control. The target of this study is to find out the most fruitful control measures of averting coronavirus infection and eventually, curtailed of the COVID-19 transmission among people. The model is investigated analytically by using most familiar necessary conditions of Pontryagins maximum principle. Furthermore, numerical simulations have been performed to illustrate the analytical results. The analysis reveals that implementation of educational campaign, social distancing and developing human immune system are the major factors which can be able to plunge the scenario of becoming infected.

An explicit solution of an initial value problem for the Susceptible-Vaccinated-Infectious-Recovered (SVIR) epidemic model is obtained, and various properties of the explicit solution are investigated. It is shown that the parametric form of the explicit solution satisfies some linear differential system including a positive solution of an integral equation. In this paper integral equations play an important role in establishing the explicit solution of the SVIR epidemic model, in particular, the number of infected individuals can be represented in a simple form by using a positive solution of an integral equation. Uniqueness of positive solutions of the SVIR epidemic model is also investigated, and it is shown that the explicit solution is a unique solution in the class of positive solutions.

In December 2019, a new virus belonging to the coronavirus strain has been discovered in Wuhan, China, this virus has attracted world-wide attention and it spread rapidly in the world, reaching nearly 216 countries in the world in November 2020. In this chapter, we study the fractional incommensurate SIQR (susceptible, infections,quarantined and removed) COVID-19 model with nonlinear saturated incidence rate using Atangana-Baleanu fractional derivatives. The existence and uniqueness of the solutions for the fractional model is proved using fixed point theorem, the model are shown to have two equilibrium point (disease-free and an endemic equilibrium). Some numerical simulations using Euler method are also carried out to support our theoretical results. We estimated the value of the fractional orders and the parameters of the proposed model using the least squares method.. Further, the sensitivity analysis of the parameter is performed as a result, our incommensurate model gives a good approximation to real data of COVID-19.

Newly proposed SIQR model defines exponent{lambda} of exponential function expressing daily number of isolated persons as linear equation of isolation rate q and social distancing ratio x. In order to dynamically analyze the process of COVID-19 epidemic in seven countries by means of regression analyses of{lambda} , increasing rate of cumulative isolated persons(cases), IRCC, is proposed as practical index for the isolation rate q. IRCC is correlated with q in the form of q=C {middle dot} IRCC, where C is a normalizing coefficient. At first, C is formulated in two modes, one is simple and the other complex, under the constraint conditions by definition 0[≤]x, q[≤]1, which give allowable narrow path of C between upper and lower boundaries. Then, the dynamic locus of q-x relation is analyzed for each of seven countries including Japan and the United States using formulated isolation rate q, and characteristic q-x behavior for each country is derived. At the same time, it is shown that specific path selection of C gives almost same linear loci of q-x relation derived by mathematical sequential method imitating a bipedal walk. In addition, increasing rates of cumulative PCR tests, IRCT, for six countries are discussed in relation with IRCC, and are shown that IRCT contributes to the promotion of the isolation rate via IRCC.

An ordinary differential dynamical model is developed to describe the transmission of infectious diseases, considering heterogeneities of region, age, vaccination, and immigration and real-time vaccination simultaneously using a tenderized formulation. Numerical experiments are performed in Xiamen city, China, with the whole population partitioned into 6 regions x 4 age groups x 4 vaccination status groups, showing the numerical stability of the developed model. The heterogeneity consideration makes our model adequate to evaluate specific interventions accurately within specific sub-populations and carry them out. Author summaryNot applicable.

To date, over 130 million people on infected with COVID-19. It causes more 2.8 millions deaths. This paper introduces a symptomatic-asymptomatic-recoverer-dead differential equation model (SARDDE). It gives the conditions of the asymptotical stability on the disease-free equilibrium of SARDDE. It proposes the necessary conditions of disease spreading for the SARDDE. Based on the reported data of the first and the second COVID-19 epidemics in Beijing and simulations, it determines the parameters of SARDDE, respectively. Numerical simulations of SARDDE describe well the outcomes of current symptomatic and asymptomatic individuals, recovered symptomatic and asymptomatic individuals, and died individuals, respectively. The numerical simulations suggest that both symptomatic and asymptomatic individuals cause lesser asymptomatic spread than symptomatic spread; blocking rate of about 90% cannot prevent the spread of the COVID19 epidemic in Beijing; the strict prevention and control strategies implemented by Beijing government is not only very effective but also completely necessary. The numerical simulations suggest also that using the data from the beginning to the day after about two weeks at the turning point can estimate well or approximately the following outcomes of the two COVID-19 academics, respectively. It is expected that the research can provide better understanding, explaining, and dominating for epidemic spreads, prevention and control measures.

In order to quantitatively characterize the epidemic of COVID-19, useful relations among parameters describing an epidemic in general are derived based on the Kermack-McKendrick model. The first relation is 1/{tau}grow =1/{tau}trans-1/{tau}inf, where{tau} grow is the time constant of the exponential growth of an epidemic,{tau} trans is the time for a pathogen to be transmitted from one patient to uninfected person, and the infectious time{tau} inf is the time during which the pathogen keeps its power of transmission. The second relation p({infty}) {approx} 1-exp(-(R0-1)/0.60) is the relation between p({infty}), the final size of the disaster defined by the ratio of the total infected people to the population of the society, and the basic reproduction number, R0, which is the number of persons infected by the transmission of the pathogen from one infected person during the infectious time. The third relation 1/{tau}end = 1/{tau}inf-(1-p({infty}))/{tau}trans gives the decay time constant{tau} end at the ending stage of the epidemic. Derived relations are applied to influenza in Japan in 2019 for characterizing the epidemic.

This study shows that the disease free equilibrium (E0) for COVID-19 coronavirus does not satisfy the criteria for a locally or globally asymptotic stability. This implies that as a pandemic as declared by WHO (2020) the COVID-19 coronavirus does not have a curative vaccine yet and precautionary measures are advised through quarantine and observatory procedures. Also, the Basic Reproductive number (R0 < 1) by Equation (33) shows that there is a chance of decline of secondary infections when the ratio between the incidence rate in the population and the total number of infected population quarantined with observatory procedure. The effort to evaluate the disease equilibrium shows that unless there is a dedicated effort from government, decision makers and stakeholders, the world would hardly be reed of the COVID-19 coronavirus and further spread is eminent and the rate of infection will continue to increase despite the increased rate of recovery because of the absence of vaccine at the moment.

In this project, we study a class of fractional order generalized SEIR epidemic models. Based on the public data from Jan. 22th to May 15th, 2020, we reliably estimate key epidemic parameters and make predictions on the peak point and possible ending time for the target region. We analyze the current management strategy and predict the future implementation of different management strategies. Numerical simulations which support our analysis are also given.

When a Susceptible-Infective-Recovered (SIR) model with a constant contact rate is used to describe the dynamics of directly transmitted infections, oscillations, which decay exponentially with time, are obtained. Due to damped oscillations, intermittent vaccination schemes can be designed in order to reduce or even eliminate the infection. A simple intermittent vaccination can be described by a series of pulses, i.e., a proportion of susceptible individuals is vaccinated intermittently at every fixed period of time. Analysis of the model is done by numerical simulations in order to determine the trajectories in the phase space. It is observed that as the proportion of vaccinated individuals increases, closed orbits with multiple cycles appear, even irregular trajectories arise occasionally. These results can be understood by comparing with bifurcations occurring in a discrete logistic model describing a single population. Further, bifurcations occurring in epidemiological models that use periodic functions to mimic seasonal variations in the disease transmission are discussed.

In this study, an improved Susceptible-Infected-Recovered (SIR) epidemic diffusion model for cholera is extended by including migration for susceptible people. This model is applied to a metapopulation that consists of two isolated cities where just susceptible individuals can migrate between the cities. The disease-free equilibrium, the endemic equilibrium points, and the basic reproductive number with unequal migration rates are analyzed for this metapopulation. Firstly, the study showed that the basic reproductive number depends on the migration rates between the cities. Then, showed that when the epidemic SIR system is stable, then the infected cases for cholera outbreak can reach zero in one city, but the infected cases in the other city still can stay positive. Finally, discussed three scenarios that depend on population sizes and migration rates of susceptible people between the cities and showed how important the migration rates are in the diffusion of the cholera outbreak by visualizing these three scenarios. Mathematics Subject ClassificationPrimary: 92B05; Secondary: 92D40

A time delay epidemic model is presented for the spread of the Coronavirus 2019 (COVID-19) in China. The time delay effects affect infected individuals. Monte Carlo simulation is performed to estimate the transmission and recovery rates. The basic reproduction number is estimated in terms of the average infected ratio. This ratio can be used to monitor the policy performance of disease control during the spread of the disease.

In this work, an attempt is made to analyse the dynamics of COVID-19 outbreak mathematically using a modified SEIR model with additional compartments and a nonlinear incidence rate with the help of bifurcation theory. Existence of a forward bifurcation point is presented by deriving conditions in terms of parameters for the existence of disease free and endemic equilibrium points. The significance of having two additional compartments, viz., protective and hospital quarantine compartments, is then illustrated via numerical simulations. From the analysis and results, it is observed that, by properly selecting transfer functions to place exposed and infected individuals in protective and hospital quarantine compartments, respectively, and with apt governmental action, it is possible to contain the COVID-19 spread effectively. Finally, the capability of the proposed model in predicting/representing the COVID-19 dynamics is presented by comparing with real-time data.

We evaluate the effectiveness of COVID-19 control strategies of 25 countries which have endured more than four weeks of community infections. With an extended SEIR model that allows infections in both the exposed and infected states, the key epidemic parameters are estimated from each countrys data, which facilitate the evaluation and cross-country comparison. It is found quicker control measures significantly reduce the average reproduction numbers and shorten the time length to infection peaks. If the swift control measures of Korea and China were implemented, average reductions of 88% in the confirmed cases and 80% in deaths would had been attained for the other 23 countries from start to April 10. Effects of earlier or delayed interventions in the US and the UK are experimented which show at least 75% (29%) less infections and deaths can be attained for the US (the UK) under a Five-Day Earlier experiment. The impacts of two removal regimes (Korea and Italy) on the total infection and death tolls on the other countries are compared with the natural forecast ones, which suggest there are still ample opportunity for countries to reduce the final death numbers by improving the removal process.

The goal of this study is to establish a general framework for predicting the so-called critical "Turning Period" in an infectious disease epidemic such as the COVID-19 outbreak in China early this year. This framework enabled a timely prediction of the turning period when applied to Wuhan COVID-19 epidemic and informed the relevant authority for taking appropriate and timely actions to control the epidemic. It is expected to provide insightful information on turning period for the worlds current battle against the COVID-19 pandemic. The underlying mathematical model in our framework is the individual Susceptible-Exposed-Infective-Removed (iSEIR) model, which is a set of differential equations extending the classic SEIR model. We used the observed daily cases of COVID-19 in Wuhan from February 6 to 10, 2020 as the input to the iSEIR model and were able to generate the trajectory of COVID-19 cases dynamics for the following days at midnight of February 10 based on the updated model, from which we predicted that the turning period of CIVID-19 outbreak in Wuhan would arrive within one week after February 14. This prediction turned to be timely and accurate, providing adequate time for the government, hospitals, essential industry sectors and services to meet peak demands and to prepare aftermath planning. Our study also supports the observed effectiveness on flatting the epidemic curve by decisively imposing the "Lockdown and Isolation Control Program" in Wuhan since January 23, 2020. The Wuhan experience provides an exemplary lesson for the whole world to learn in combating COVID-19.

To date, over 178 million people on infected with COVID-19. It causes more 3.8 millions deaths. Based on a previous symptomatic-asymptomatic-recoverer-dead differential equation model (SARDDE) and the clinic data of the first COVID-19 epidemic in Shanghai, this paper determines the parameters of SARDDE. Numerical simulations of SARDDE describe well the outcomes of current symptomatic individuals, recovered symptomatic individuals, and died individuals, respectively. The numerical simulations suggest that both symptomatic and asymptomatic individuals cause lesser asymptomatic spread than symptomatic spread; blocking rate of about 95.5% cannot prevent the spread of the COVID19 epidemic in Shanghai. The strict prevention and control strategies implemented by Shanghai government is not only very effective but also completely necessary. The numerical simulations suggest also that using the data from the beginning to the day after about 19 days at the turning point can estimate well the following outcomes of the COVID-19 academic. It is expected that the research can provide better understanding, explaining, and dominating for epidemic spreads, prevention and control measures.

We derive a novel hybrid approach, a combination of neural networks and inverse problem, in order to forecast COVID-19 cases, and more generally any infectious disease. For this purpose, we extract a second order nonlinear differential equation for the total confirmed cases from a SIR-like model. That differential equation is the key factor of the present study. The neural network and inverse problems are used to compute the trial functions for total cases and the model parameters, respectively. The number of suspected and infected individuals can be found using the trial function of total confirmed cases. We divide the time domain into two parts, training interval (first 365/395 days) and test interval (first 366 to 395/ 396 to 450 days), and train the neural networks on the preassigned training zones. To examine the efficiency and effectiveness, we apply the proposed method to Canada, and use the Canadian publicly available database to estimate the parameters of the trial function involved with total cases. The trial functions of model parameters show that the basic reproduction number was closed to unity over a wide range, the first from 100 to 365 days of the current pandemic in Canada. The proposed prediction models, based on influence of previous time and social economic policy, show excellent agreement with the data. The test results revel that the single path prediction can forecast a period of 30 days, and forecasting using previous social and economical situation can forecast a range of 55 days.

COVID-19 pandemic has impacted people all across the world. As a result, there has been a collective effort to monitor, predict, and control the spread of this disease. Among this effort is the development of mathematical models that could capture accurately the available data and simulate closely the futuristic scenarios. In this paper, a fractional-order memory-dependent model for simulating the spread of COVID-19 is proposed. In this model, the impact of governmental action and public perception are incorporated as part of the time-varying transmission rate. The model simulation is performed using the two-step generalized exponential time-differencing method and tested for data from Wuhan, China. The mean-square errors demonstrate the merit of the fractional-order model and provide a good estimate of the optimal order.

The coronavirus disease 2019 (COVID-19) has been damaging our daily life after declaration of pandemic. Therefore, we have started studying on the characteristics of Susceptible-Infectious-Recovered (SIR) model to know about the truth of infectious disease and our future. After detailed studies on the characteristics of the SIR model for the various parameter dependencies with respect to such as the outing restriction (lockdown) ratio and vaccination rate, we have finally noticed that the second term (isolation term) in the differential equation of the number of the infected is quite similar to the "helium ash particle loss term" in deuterium-tritium (D-T) nuclear fusion. Based on this analogy, we have found that isolation of the infected is not actively controlled in the SIR model. Then we introduce the isolation control time parameter q and have studied its effect on this pandemic. Required isolation time to terminate the COVID-19 can be estimated by this proposed method. To show this isolation control effect, we choose Tokyo for the model calculation because of high population density. We determine the reproduction number and the isolation ratio in the initial uncontrolled phase, and then the future number of the infected is estimated under various conditions. If the confirmed case can be isolated in 3[~]8 days by widely performed testing, this pandemic could be suppressed without awaiting vaccination. If the mild outing restriction and vaccination are taken together, the isolation control time can be longer. We consider this isolation time control might be the only solution to overcome the pandemic when vaccine is not available.

We show phase-wise growth of COVID 19 pandemic and explain it by comparing real time data with Discrete Generalized Growth model and Discrete Generalized Richard Model. The comparison of COVID 19 is made for China, Italy, Japan and the USA. The mathematical techniques makes it possible to calculate the rate of exponential growth of active cases, estimates the size of the outbreak, and measures the deviation from the exponential growth indicating slowing down effect. The phase-wise pandemic evolution following the real time data of active cases defines the impact-point when the preventive steps, taken to eradicate the pandemic, becomes effective. The study is important to devise the measures to handle emerging threat of similar COVID-19 outbreaks in other countries, especially in the absence of a medicine.