Chaos, Solitons & Fractals

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.

A mathematical model for the transmission dynamics of Coronavirus diseases (COVID-19) is proposed by incorporating self-protection behavior changes in the population. The disease-free equilibrium point is computed and its stability analysis is studied. The basic reproduction number(R0) of the model is computed and the disease-free equilibrium point is locally and globally stable for R0 < 1 and unstable for R0 > 1. Based on the available data the unknown model parameters are estimated using a combination of least square and Bayesian estimation methods for different countries. Using forward sensitivity index the model parameters is carried out to determine and identify the key factors for the spread of disease dynamics. From country to country the sensitive parameters for the spread of the virus varies. It is found out that the reproduction number depends mostly on the infection rates, the threshold value of the force of infection for a population, the recovery rates, and the virus decay rate in the environment. It is also demonstrated that control of the effective transmission rate (recommended human behavioral change towards self-protective measures) is essential to stop the spreading of the virus. Numerical simulations also show that the viruss transmission dynamics depend mostly on those sensitive parameters.

The world has now paid a lot of attention to the outbreak of novel coronavirus (COVID-19). This virus mainly transmitted between humans through directly respiratory droplets and close contacts. However, there is currently some evidence where it has been claimed that it may be indirectly transmitted. In this work, we study the mode of transmission of COVID-19 epidemic system based on the susceptible-infected-recovered (SIR) model. We have calculated the basic reproduction number R0 by next-generation matrix method. We observed that if R0 < 1, then disease-free equilibrium point is locally as well as globally asymptotically stable but when R0 > 1, the endemic equilibrium point exists and is globally stable. Finally, some numerical simulation is presented to validate our results.

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.

We propose an SEIARD mathematical model to investigate the current outbreak of coronavirus disease (COVID-19) in Mexico. Our model incorporates the asymptomatic infected individuals, who represent the majority of the infected population (with symptoms or not) and could play an important role in spreading the virus without any knowledge. We calculate the basic reproduction number (R0) via the next-generation matrix method and estimate the per day infection, death and recovery rates. The local stability of the disease free equilibrium is established in terms of R0. A sensibility analysis is performed to determine the relative importance of the model parameters to the disease transmission. We calibrate the parameters of the SEIARD model to the reported number of infected cases and fatalities for several states in Mexico by minimizing the sum of squared errors and attempt to forecast the evolution of the outbreak until August 2020.

A novel coronavirus, designated as COVID-19, emerged in Wuhan, China, at the end of 2019. In this paper, a mathematical model is proposed to analyze the dynamic behavior of COVID-19. Based on inter-city networked coupling effects, a fractional-order SEIHDR system with the real-data from 23 January to 18 March, 2020 of COVID19 is discussed. Meanwhile, hospitalized individuals and the mortality rates of three types of individuals (exposed, infected and hospitalized) are firstly taken into account in the proposed model. And infectivity of individuals during incubation is also considered in this paper. By applying least squares method and predictor-correctors scheme, the numerical solutions of the proposed system in the absence of the inter-city network and with the inter-city network are stimulated by using the real-data from 23 January to 18 - m March, 2020 where m is equal to the number of prediction days. Compared with integer-order system ( = 0), the fractional-order model without network is validated to have a better fitting of the data on Beijing, Shanghai, Wuhan, Huanggang and other cities. In contrast to the case without network, the results indicate that the inter-city network system may be not a significant case to virus spreading for China because of the lock down and quarantine measures, however, it may have an impact on cities that have not adopted city closure. Meanwhile, the proposed model better fits the data from 24 February to 31, March in Italy, and the peak number of confirmed people is also predicted by this fraction-order model. Furthermore, the existence and uniqueness of a bounded solution under the initial condition are considered in the proposed system. Afterwards, the basic reproduction number R0 is analyzed and it is found to hold a threshold: the disease-free equilibrium point is locally asymptotically stable when R0 [&le;] 1, which provides a theoretical basis for whether COVID-19 will become a pandemic in the future.

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.

In this paper, a generalized fractional-order SEIR model is proposed, denoted by SEIQRP model, which has a basic guiding significance for the prediction of the possible outbreak of infectious diseases like COVID-19 and other insect diseases in the future. Firstly, some qualitative properties of the model are analyzed. The basic reproduction number R0 is derived. When R0 < 1, the disease-free equilibrium point is unique and locally asymptotically stable. When 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 United States is predicted. Considering the influence of the individual behavior and government mitigation measurement, a modified SEIQRP model is proposed, defined as SEIQRPD model. According to the real data of the United States, 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 United States in the next two weeks is investigated, and the peak of isolated cases are 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 developed a deterministic mathematical model of the pandemic COVID-19 transmission in Ethiopia, which allows transmission by exposed humans. We proposed an SEIR model using system of ordinary differential equations. First the major qualitative analysis, like the disease free equilibruim point, endemic equilibruim point, basic reproduction number, stability analysis of equilibrium points and sensitivity analysis was rigorously analysed. Second, we introduced time dependent controls to the basic model and extended to an optimal control model of the disease. We then analysed using Pontryagins Maximum Principle to derive necessary conditions for the optimal control of the pandemic. The numerical simulation indicated that, an integrated strategy effective in controling the epidemic and the gvernment must apply all control strategies in combating COVID-19 at short period of time.

In this work, a co-infection model for human papillomavirus (HPV) and Chlamydia trachomatis with cost-effectiveness optimal control analysis is developed and analyzed. The disease-free equilibrium of the co-infection model is shown not to be globally asymptotically stable, when the associated reproduction number is less unity. It is proven that the model undergoes the phenomenon of backward bifurcation when the associated reproduction number is less than unity. It is also shown that HPV re-infection ({varepsilon}p = 0) induced the phenomenon of backward bifurcation. Numerical simulations of the optimal control model showed that: (i) focusing on HPV intervention strategy alone (HPV prevention and screening), in the absence of Chlamydia trachomatis control, leads to a positive population level impact on the total number of individuals singly infected with Chlamydia trachomatis, (ii) Concentrating on Chlamydia trachomatis intervention controls alone (Chlamydia trachomatis prevention and treatment), in the absence of HPV intervention strategies, a positive population level impact is observed on the total number of individuals singly infected with HPV. Moreover, the strategy that combines and implements HPV and Chlamydia trachomatis prevention controls is the most cost-effective of all the control strategies in combating the co-infections of HPV and Chlamydia trachomatis.

Climate changes are affecting the control of many vector-borne diseases, particularly in Africa. In this work, a dengue fever model with protected travellers is formulated. Caputo-Fabrizio operator is utilized to obtain some qualitative information about the disease. The basic properties and the reproduction number is studied. The two steady states are determined and the local stability of the states are found to be asymptotically stable. The fixed pointed theory is made use to obtain the existence and uniqueness of solutions of the model. The numerical simulation suggests that the fractional-order affects the dynamics of dengue fever.

In this paper, we design a Nonlinear Observer (NLO) to estimate the effective reproduction number ([R]t) of infectious diseases. The NLO is designed from a discrete-time augmented Susceptible-Infectious-Removed (SIR) model. The observer gain is obtained by solving a Linear Matrix Inequality (LMI). The method is used to estimate[R] t in Jakarta using epidemiological data during COVID-19 pandemic. If the observer gain is tuned properly, this approach produces similar result compared to existing approach such as Extended Kalman filter (EKF).

Coronavirus disease 2019 (COVID-19) is a disease caused by Severe acute respiratory syndrome coronavirus 2 (SARS CoV-2). It was declared on March 11, 2020, by the World Health Organization as pandemic disease. The disease has neither approved medicine nor vaccine and has made government and scholars search for drastic measures in combating the pandemic. Regrettably, the spread of the virus and mortality due to COVID-19 has continued to increase daily. Hence, it is imperative to control the spread of the disease particularly using non-pharmacological strategies such as quarantine, isolation and public health education. This work studied the effect of these different control strategies as time-dependent interventions using mathematical modeling and optimal control approach to ascertain their contributions in the dynamic transmission of COVID-19. The model was proven to have an invariant region and was well-posed. The basic reproduction number was computed with and without interventions and was used to carry out the sensitivity analysis that identified the critical parameters contributing to the spread of COVID-19. The optimal control analysis was carried out using the Pontryagins maximum principle to figure out the optimal strategy necessary to curtail the disease. The findings of the optimal control analysis and numerical simulations revealed that time-dependent interventions reduced the number of exposed and infected individuals compared to time-independent interventions. These interventions were time-bound and best implemented within the first 100 days of the outbreak. Again, the combined implementation of only two of these interventions produced a good result in reducing infection in the population, while the combined implementation of all three interventions performed better, even though zero infection was not achieved in the population. This implied that multiple interventions need to be deployed early in order to the virus to the barest minimum.

As of 21st May 2020, there have been 4.89M confirmed cases worldwide and over 323,000 deaths of people who have tested positive for SARS-CoV-2. The outbreak of COVID-19, has not only caused widespread morbidity and mortality, but has also led to a catastrophic breakdown in the global economy and unprecedented social disruption. To lessen the global health consequences of COVID-19, sweeping COVID-19 lockdown and quarantine measures have been imposed within many nations. These measures have significantly impacted the worlds economy and in many cases has led to the loss of livelihood. Mathematical modeling of pandemics is of critical importance to understand the unfolding of transmission events and to formulate control measures. In this research letter, we have introduced a novel approach to forecasting epidemics like COVID-19. The proposed mathematical model stems from the fundamental principles of fluid dynamics, and can be utilized to make projections of the number of infected people. This unique mathematical model can be beneficial for predicting and designing potential strategies to mitigate the spread and impact of pandemics.

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.

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.

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.

Its spreading speed together with the risk of fatality might be the main characteristic that separates COVID-19 from other infectious diseases in our recent history. In this scenario, mathematical modeling for predicting the spread of the disease could have great value in containing the disease. Several very recent papers have contributed to this purpose. In this study we propose a birth-and-death model for predicting the number of COVID-19 active cases. It relation to the Susceptible-Infected-Recovered (SIR) model has been discussed. An explicit expression for the expected number of active cases helps us to identify a stationary point on the infection curve, where the infection ceases increasing. Parameters of the model are estimated by fitting the expressions for active and total reported cases simultaneously. We analyzed the movement of the stationary point and the basic reproduction number during the infection period up to the 20th of April 2020. These provide information about the disease progression path and therefore could be really useful in designing containment strategies.

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.

Covid-19 disease outspread and its subsequent control and inhibition strategies in various countries have been different which led to quite drastic difference in the outcome of the disease progression. In this paper we present an analytical and numerical study of Covid-19 disease spread and control by applying the modified SIR model of epidemic outbreak to explain the Covid spread from February-July 2020 in various countries. Two approaches are evident from the disease progression; one focused on disease eradication and inhibition, and the other is less restrictive dynamic response. Both the approaches are analytically modeled to determine the parameters that characterize the effectiveness of dealing with the disease progression. The model successfully explains the Covid-19 evolution and control of various countries over a vast span of four-five months. The study is highly beneficial to simply analytically and numerically model this complex problem of epidemic proliferation. It assists to easily determine the mathematical parameters for the Covid-19 control measures, helps in predicting the end of the epidemic, and most importantly conceiving the judicious way of unlock process to restore communication between cities, states and countries.