Nonlinear Dynamics

We propose an epidemic model in which all diseases are transmitted by direct transmission (infectives to susceptibles) and indirect transmission which occurs through shedding of virus by infectives and acquisition by susceptibles. The model takes into account the effect of latency period and time needed for a susceptible to become infective by indirect contact. Under the certain assumptions, the basic reproduction number of the model is identified from the direct and indirect basic reproduction numbers. The main goal is to analyse the asymptotic behaviours, global stability, bifurcation and to detecte the most sensitive parameters. Numerical simulations are carried out to illustrate the theoretical part.

In this paper, we analyze the real-time infection data of COVID-19 epidemic for 21 nations up to June 30, 2020. For most of these nations, the total number of infected individuals exhibits a succession of exponential growth and power-law growth before the flattening of the curve. In particular, we find a universal [Formula] growth before they reach saturation. However, at present, India, which has I(t) ~ t2, and Russia and Brazil, which have I(t) ~ t, are yet to flatten their curves. Thus, the polynomials of the I(t) curves provide valuable information on the stage of the epidemic evolution, thus on the life cycle of COVID-19 pandemic. Besides these detailed analyses, we compare the predictions of an extended SEIR model and a delay differential equation-based model with the reported infection data and observed good agreement among them, including the [Formula] behaviour. We argue that the power laws in the epidemic curves may be due to lockdowns.

The SIR model of the epidemic spread is used for consideration the problem of the competition of two viruses having different contagiousness. It is shown how the more contagious strain replaces over time the less contagious one. In particular the results can be applied to the current situation when the omicron strain appeared in population affected by the delta strain. PACS number(s)02.50.-r, 05.60.-k, 82.39.-k, 87.19.Xx

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.

The SEIR model is one of modified models of SIR, especially taken into account of exposed people. SEIR equations can be solved numerically, but it is hard to obtain analytically. Here, we propose some approximate solutions of SEIR equations, one of which is related with the logistic formula in Biology. As the second aim, the SEIR model is applied to the 7th-wave of COVID-19 in Japan. The basic reproduction number () in the SEIR model is estimated for the Omicron wave. We make use of data of the removed number R(t) rather than that of the infective number, because the latter seems to be ambiguous. This analysis gives = 10 with{gamma} = 1 and {sigma} = 0.5.

In this article, we formulate and analyse a mathematical model for the co-infection of Hepatitis B virus and COVID-19. We incorporate into our framework Hepatitis B virus prevention, COVID-19 prevention, COVID-19 vaccination, and environmental factors so as to investigate their effect on transmission dynamics. First, we derive the basic reproduction number for HBV only, COVID-19 only, and co-infection stochastic models using the next generation matrix method. Next, we establish the conditions for stability in the stochastic sense for HBV only, COVID-19 only sub-models, and the co-infection model. Furthermore, we devote our attention to finding sufficient conditions for extinction and persistence. Finally, by using the Euler-Murayama scheme, we illustrate the dynamics of the co-infection, COVID-19, HBV and the effect of some parameters on disease transmission dynamics by means of numerical simulations.

The data on SARS-CoV-2 (COVID-19) in South Africa shows seasonal transmission patterns to date, with the peaks having occurred in winter and summer since the out-breaks began. The transmission dynamics have mainly been driven by variations in environmental factors and virus evolution, and the two are at the center of driving the different waves of the disease. It is thus important to understand the role of seasonality in the transmission dynamics of COVID-19. In this paper a compartmental model with a time dependent transmission rate is formulated and the stabilities of the steady states analysed. We note that if R0 < 1, the disease-free equilibrium is globally asymptotically stable, and the disease completely dies out and when R0 > 1, the system admits a positive periodic solution, and the disease is uniformly or periodically persistent. The model is fitted to data on new cases in South Africa for the first four waves. The model results clearly indicate the need to consider seasonality in the transmission dynamics of COVID-19 and its importance in modelling fluctuations in the data for new cases. The potential impact of seasonality in the transmission patterns of COVID-19 and the public health implications are discussed.

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.

Sequel to [10], who studied the dynamics of COVID-19 using an SEIRUS model. We consider an SEIRS model capturing saturated incidence with treatment response. In this theoretical model, we assumed that the treatment response is proportional to the number of infected as long as the incidence cases are within the capacity of the healthcare system, after which the value becomes constant, when the number of confirmed cases exceed the carrying capacity of the available medical facilities. Thus, we obtain the reproduction number stating that when R0 < 1, the disease free equilibrium is globally asymptotically stable. Also, we studied the existence of the local and global stability of the disease free and endemic equilibria and found that the kind of treatment response and inhibitory measures deployed in tackling the COVID-19 pandemic determines whether the disease will die out or become endemic.

In this study, we investigate the stochastic dynamics of an extended SIS epidemic model in densely populated environments within a Markov jump process framework. We solve the master equation in closed form and obtain exact solutions of the time-dependent distribution of the number of infected individuals, the quasi-stationary distribution, the extinction time distribution of the epidemic, and the distribution of the first-passage time at which the number of infections reaches a certain threshold. The approximated quasi-stationary distribution and mean extinction time are also derived using the large deviation theory. Interestingly, we find that the first nonzero eigenvalue of the generator matrix of the Markovian model characterizes the extinction rate of the epidemic, while the second nonzero eigenvalue characterizes its outbreak rate. We also examine the stochastic bifurcation for our model based on the time evolution of the probability distribution and the bifurcation threshold of the basic reproduction number for the stochastic SIS model is shown to be large than that for its deterministic counterpart. Finally, we demonstrate that analyzing the first-passage time distribution can offer early warning for interventions and optimize the allocation of emergency beds.

Equations for infection spread in a closed population are found in discrete approximation, corresponding to the published statistical data, and in continuous time in the form of delay differential equations. We consider the epidemic as dependent upon four key parameters: the size of population involved, the mean number of dangerous contacts of one infected person per day, the probability to transmit infection due to such contact and the mean duration of disease. In the simplest case of free-running epidemic in an infinite population, the number of infected rises exponentially day by day. Here we show the model for epidemic process in a closed population, constrained by isolation, treatment and so on. The four parameters introduced here have the clear sense and are in association with the well-known concept of reproduction number in the continuous susceptible--infectious--removed, susceptible--exposed--infectious--removed (SIR, SEIR) models. We derive the initial rate of infection spread from the published statistical data for the initial stage of epidemic, when the quarantine measures were absent. On this basis, we can found the corresponding basic reproduction number mentioned above. Our approach allows evaluating the influence of quarantine measures on free pandemic process that leads to the time-dependent rate of infection and suppression of infection. We found a good correspondence of the theory and reliable statistical data. The initially formulated discrete model, describing epidemic course day by day is transferred to differential form. The conditions for saturation of epidemic are found by solving the delay differential equations. They differ essentially from ones in SIR model due to finite delay, typical for COVID-19. The proposed model opens up the possibility to predict the optimal level of social quarantine measures. The model is quite flexible and it can be extended to more complex cases.

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.

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.

The transmission and spread of infectious disease like Covid-19 occurs through horizontal and vertical mode. The causative pathogens for such kind of disease may be bacterium, protozoa, virus or toxin. The infectious diseases like AIDS, SARS, MARS, Polio Plague, Bubonic Plague and Covid-19 have destroyed the social and economic structure of world population. The world scientific community adopts different mechanisms to model and analyse the population dynamics of infectious disease outbreaks. Mathematical Modelling is the most effective tool to take the informed decision about the containment, control and eradication of the pandemic. The main focus of Government and public health authorities is to design the strategy in destabilising the spread and impact of the infections. A series of models-SIR, SEIR, SEIRD, SEAIHCRD, SAUQAR has been under study to combat the Covid-19 since its inception. An effort has been made to design the model based on reproduction number, endemic equilibrium and disease-free equilibrium to curtail the impact of Covid-19 through stability analysis methods-Hurwitz stability criteria, Lyapunov Method and Linear Stability Analysis.

The simplest approximation for the first stages of the infection spread is considered. The specific feature of the COVID-19 characterized by its long latent period is taken into account. Exponential increase of numbers of infected people is determined by the half period of the maximal latent time for the COVID-19. The averaging over latent period leads to additional increase of the infected numbers. PACS number(s)02.50.-r, 05.60.-k, 82.39.-k, 87.19.Xx

In this paper we develop and analyse a malaria model with seasonality of mosquito life-history traits: periodic-mosquitoes per capita birth rate, -mosquitoes death rate, -probability of mosquito to human disease transmission, -probability of human to mosquito disease transmission and -mosquitoes biting rate. All these parameters are assumed to be time dependent leading to a nonautonomous differential equation systems. We provide a global analysis of the model depending on two thresholds parameters [Formula] and [Formula] (with [Formula]). When [Formula], then the disease-free stationary state is locally asymptotically stable. In the presence of the human disease-induced mortality, the global stability of the disease-free stationary state is guarantied when [Formula]. On the contrary, if [Formula], the disease persists in the host population in the long term and the model admits at least one positive periodic solution. Moreover, by a numerical simulation, we show that a subcritical (backward) bifurcation is possible at [Formula]. Finally, the simulation results are in accordance with the seasonal variation of the reported cases of a malaria-epidemic region in Mpumalanga province in South Africa.

The standard growth model of epidemic evolution such as the Richards generalized logistic function is remarkably successful because it agrees with almost all previous epidemic data. Yet, it fails to explain intervention measures for mitigations of the ongoing coronavirus 2019 disease (COVID-19) pandemic. It also fails to replicate an endemic phase that occurs in many countries epidemic curves (time series data of daily new cases). These discrepancies demonstrate that new epidemic laws are required to understand, predict and mitigate the COVID-19 pandemic. Here we show that almost all COVID-19 evolution can be modeled by three innovative epidemic laws. Specifically, based on the world COVID-19 data, we first divide an epidemic curve into three phases: an exponential growth phase, an exponential decay phase, and a constant endemic phase. We next integrate the growth and the decay phases into the first epidemic law with interventions as a model parameter. This law is completely opposite to the Richards generalized logistic function in terms of intervention measures. We then combine the first epidemic law with the endemic phase to form the second epidemic law, which makes the curve of cumulative cases increase linearly as time tends to infinity. The third epidemic law states if an epidemic is composed of multiple epidemic waves, the superposition principle applies. These laws were confirmed by the COVID-19 data from 18 countries including undeveloped, developing and developed countries. Finally, we pave the way for future research to incorporate the proposed theory into the classic SIR model. We anticipate that the results from this research can provide a scientific base for governments to mitigate the COVID-19 and other epidemic disasters.

We investigate time scale separation in the vector borne disease model SIRUV, as previously described in the literature [1], and recently reanalyzed with the singular perturbation technique [2]. We focus on the analysis with a single small parameter, the birth and death rate {micro}, whereas all other model parameters are much larger and describe fast transitions. The scaling of the endemic stationary state, the Jacobian matrix around it and its eigenvalues with this small parameter {micro} is calculated and the center manifold analysis performed with the method described in [3] which goes back to earlier work [4, 5], namely a transformation of the Jacobian matrix to block structure in zeroth order in the parameter {micro} is used and then a family of center manifolds with {micro} larger than zero is obtained.

Population is separated into five compartments for COVID-19; susceptible individuals (S), pre-symptomatic patients (P), asymptomatic patients (A), quarantined patients (Q) and recovered and/or dead patients (R). The time evolution of each compartment is described by a set of ordinary differential equations. Numerical solution to the set of differential equations shows that quarantining pre-symptomatic and asymptomatic patients is effective in controlling the pandemic. It is also shown that the ratio of non-symptomatic patients to the daily confirmed new cases can be as large as 20 and that the fraction of untraceable cases in new cases can be as large as 80%, depending on the policies for social distancing and PCR test.