Mathematical analysis of a stochastic model for spread of Coronavirus

Introducing a novel mean-reverting Ornstein-Uhlenbeck process based stochastic epidemic model

The major objective of this paper is to examine a novel mean-reverting Ornstein-Uhlenbeck process-based stochastic SIRD model for transmission the epidemic disease that is a great crisis in numerous societies. For this purpose, the deterministic model is further converted into the stochastic form by allowing the infection rate satisfies the mean-reverting Ornstein-Uhlenbeck process to account the uncertainties involved in epidemic spread. At first using Lyapunov functions, the solution's uniqueness and positivity will be demonstrated. Subsequently, the stochastic epidemic threshold [Formula: see text] that controls the disease's extinction and persistence in the mean is identified analytically. It has been established that when [Formula: see text] the disease will extinguish, whereas if [Formula: see text] the disease is persistent. At last, several numerical simulations are presented to demonstrate the findings of the hypothetical investigation results. These simulations served to vividly illustrate and validate the implications derived from the hypothetical analysis.

Dynamical behaviors of a Lotka-Volterra competition system with the Ornstein-Uhlenbeck process

The competitive relationship is one of the important studies in population ecology. In this paper, we investigate the dynamical behaviors of a two-species Lotka-Volterra competition system in which intrinsic rates of increase are governed by the Ornstein-Uhlenbeck process. First, we prove the existence and uniqueness of the global solution of the model. Second, the extinction of populations is discussed. Moreover, a sufficient condition for the existence of the stationary distribution in the system is obtained, and, further, the formulas for the mean and the covariance of the probability density function of the corresponding linearized system near the equilibrium point are obtained. Finally, numerical simulations are applied to verify the theoretical results.

Conceptual analysis of the combined effects of vaccination, therapeutic actions, and human subjection to physical constraint in reducing the prevalence of COVID-19 using the homotopy perturbation method

Background

The COVID-19 pandemic has put the world's survival in jeopardy. Although the virus has been contained in certain parts of the world after causing so much grief, the risk of it emerging in the future should not be overlooked because its existence cannot be shown to be completely eradicated.

Results

This study investigates the impact of vaccination, therapeutic actions, and compliance rate of individuals to physical limitations in a newly developed SEIQR mathematical model of COVID-19. A qualitative investigation was conducted on the mathematical model, which included validating its positivity, existence, uniqueness, and boundedness. The disease-free and endemic equilibria were found, and the basic reproduction number was derived and utilized to examine the mathematical model's local and global stability. The mathematical model's sensitivity index was calculated equally, and the homotopy perturbation method was utilized to derive the estimated result of each compartment of the model. Numerical simulation carried out using Maple 18 software reveals that the COVID-19 virus's prevalence might be lowered if the actions proposed in this study are applied.

Conclusion

It is the collective responsibility of all individuals to fight for the survival of the human race against COVID-19. We urged that all persons, including the government, researchers, and health-care personnel, use the findings of this research to remove the presence of the dangerous COVID-19 virus.

Dynamical analysis of a stochastic delayed epidemic model with lévy jumps and regime switching

In this paper a delayed stochastic SLVIQR epidemic model, which can be applied for modeling the new coronavirus COVID-19 after a calibration, is derived. Model is constructed by assuming that transmission rate satisfies the mean-reverting Ornstain-Uhlenbeck process and, besides a standard Brownian motion, another two driving processes are considered: a stationary Poisson point process and a continuous finite-state Markov chain. For the constructed model, the existence and uniqueness of positive global solution is proven. Also, sufficient conditions under which the disease would lead to extinction or be persistent in the mean are established and it is shown that constructed model has a richer dynamic analysis compared to existing models. In addition, numerical simulations are given to illustrate the theoretical results.

From pandemic to a new normal: Strategies to optimise governmental interventions in Indonesia based on an SVEIQHR-type mathematical model

There are five different forms of intervention presently realised by the Indonesian government in an effort to end the COVID-19 pandemic: vaccinations, social restrictions, tracings, testings, and treatments. In this paper, we construct a mathematical model of type SVEIQHR (susceptible-vaccinated-exposed-infected-quarantined-hospitalised-recovered) for the disease's spread in the country, which incorporates as parameters the rates of the above interventions, as well as the vaccine's efficacy. We determine the model's equilibria and basic reproduction number. Using the model, we formulate strategies by which the interventions should be realised in order to optimise their impact. The results show that, in a disease-free state, when the number of new cases rises, the best strategy is to implement social restrictions, whereas in an endemic state, if a near-lockdown policy is undesirable, carrying out vaccinations is the best strategy; however, efforts should be aimed not primarily towards increasing the vaccination rate, but towards the use of high-efficacy vaccines.

Computational and Mathematical Methods in Medicine Prediction of COVID-19 in BRICS Countries: An Integrated Deep Learning Model of CEEMDAN-R-ILSTM-Elman

Since the outbreak of COVID-19, BRICS countries have experienced different epidemic spread due to different health conditions, social isolation measures, vaccination rates, and other factors. A descriptive analysis is conducted for the spread of the epidemic in the BRICS countries. Considering the nonlinear and nonstationary characteristics of COVID-19 data, a principle of decomposition-reconstruction(R)-prediction-integration is proposed. Correspondingly, this paper constructs an integrated deep learning prediction model of CEEMDAN-R-ILSTM-Elman. Specifically, the prediction model is integrated by complete ensemble empirical mode decomposition (CEEMDAN), improved long-term and short-term memory network (ILSTM), and Elman neural network. First, the data is decomposed by adopting CEEMDAN. Then, by calculating the permutation entropy and average period, the decomposed eigenmode component IMFs are reconstructed into four sequences of high, medium, low level, and trend term. Thus, ILSTM and Elman algorithms are used for component sequence prediction, whose results are integrated as the final results. The ILSTM is established based on the LSTM model and the improved beetle antennae search algorithm (IBAS). The ILSTM mainly considers that the prediction accuracy of LSTM model is vulnerable to the influence of parameter selection. The IBAS with adaptive step size is used to automatically optimize the super parameters of LSTM model and to improve the modeling efficiency and prediction accuracy. Experimental results indicate that compared with other benchmark models, CEEMDAN-R-ILSTM-Elman integrated model predicts the number of newly confirmed cases of COVID-19 in BRICS countries with higher accuracy and lower error. Strict social policies have a greater impact on the infection rate and mortality rate of the epidemic. During July-August 2021, epidemic spread in BRICS countries will slow down, and the overall situation is still quite severe.

Orthonormal Bernoulli Polynomials for Solving a Class of Two Dimensional Stochastic Volterra–Fredholm Integral Equations

Analyzing the mathematical models involving Itô integral, in particular in science and engineering has received much attention, and the reason for this issue is the randomness and lack of access to the exact answer of this type of models. For this purpose, in this paper, the approximate solution of two dimensional (2-D) stochastic Volterra–Fredholm integral equations (SVFIEs) based on the operational matrix method and orthonormal Bernoulli polynomials (OBP) is investigated. Some results and convergence analysis are also presented. Finally, by presenting three examples and reviewing the results and numerical comparisons, we showed that the proposed method has an excellent performance.

Impact of Infective Immigrants on COVID-19 Dynamics

The COVID-19 epidemic is an unprecedented and major social and economic challenge worldwide due to the various restrictions. Inflow of infective immigrants have not been given prominence in several mathematical and epidemiological models. To investigate the impact of imported infection on the number of deaths, cumulative infected and cumulative asymptomatic, we formulate a mathematical model with infective immigrants and considering vaccination. The basic reproduction number of the special case of the model without immigration of infective people is derived. We varied two key factors that affect the transmission of COVID-19, namely the immigration and vaccination rates. In addition, we considered two different SARS-CoV-2 transmissibilities in order to account for new more contagious variants such as Omicron. Numerical simulations using initial conditions approximating the situation in the US when the vaccination program was starting show that increasing the vaccination rate significantly improves the outcomes regarding the number of deaths, cumulative infected and cumulative asymptomatic. Other factors are the natural recovery rates of infected and asymptomatic individuals, the waning rate of the vaccine and the vaccination rate. When the immigration rate is increased significantly, the number of deaths, cumulative infected and cumulative asymptomatic increase. Consequently, accounting for the level of inflow of infective immigrants may help health policy/decision-makers to formulate policies for public health prevention programs, especially with respect to the implementation of the stringent preventive lock down measure.

A proposed fractional dynamic system and Monte Carlo-based back analysis for simulating the spreading profile of COVID-19

This paper presents a dynamic system for estimating the spreading profile of COVID-19 in Thailand, taking into account the effects of vaccination and social distancing. For this purpose, a compartmental network is built in which the population is divided into nine mutually exclusive nodes, including susceptible, insusceptible, exposed, infected, vaccinated, recovered, quarantined, hospitalized, and dead. The weight of edges denotes the interaction between the nodes, modeled by a series of conversion rates. Next, the compartmental network and corresponding rates are incorporated into a system of fractional partial differential equations to define the model governing the problem concerned. The fractional degree corresponding to each compartment is considered the node weight in the proposed network. Next, a Monte Carlo-based optimization method is proposed to fit the fractional compartmental network to the actual COVID-19 data of Thailand collected from the World Health Organization. Further, a sensitivity analysis is conducted on the node weights, i.e., fractional orders, to reveal their effect on the accuracy of the fit and model predictions. The results show that the flexibility of the model to adapt to the observed data is markedly improved by lowering the order of the differential equations from unity to a fractional order. The final results show that, assuming the current pandemic situation, the number of infected, recovered, and dead cases in Thailand will, respectively, reach 4300, 4.5 × 10 6 , and 36,000 by the end of 2021.

Investigation of the Stochastic Modeling of COVID-19 with Environmental Noise from the Analytical and Numerical Point of View

In this article, we propose a novel mathematical model for the spread of COVID-19 involving environmental white noise. The new stochastic model was studied for the existence and persistence of the disease, as well as the extinction of the disease. We noticed that the existence and extinction of the disease are dependent on R0 (the reproduction number). Then, a numerical scheme was developed for the computational analysis of the model; with the existing values of the parameters in the literature, we obtained the related simulations, which gave us more realistic numerical data for the future prediction. The mentioned stochastic model was analyzed for different values of σ1,σ2 and β1,β2, and both the stochastic and the deterministic models were compared for the future prediction of the spread of COVID-19.

Network Autoregressive Model for the Prediction of COVID-19 Considering the Disease Interaction in Neighboring Countries

Predicting the way diseases spread in different societies has been thus far documented as one of the most important tools for control strategies and policy-making during a pandemic. This study is to propose a network autoregressive (NAR) model to forecast the number of total currently infected cases with coronavirus disease 2019 (COVID-19) in Iran until the end of December 2021 in view of the disease interactions within the neighboring countries in the region. For this purpose, the COVID-19 data were initially collected for seven regional nations, including Iran, Turkey, Iraq, Azerbaijan, Armenia, Afghanistan, and Pakistan. Thenceforth, a network was established over these countries, and the correlation of the disease data was calculated. Upon introducing the main structure of the NAR model, a mathematical platform was subsequently provided to further incorporate the correlation matrix into the prediction process. In addition, the maximum likelihood estimation (MLE) was utilized to determine the model parameters and optimize the forecasting accuracy. Thereafter, the number of infected cases up to December 2021 in Iran was predicted by importing the correlation matrix into the NAR model formed to observe the impact of the disease interactions in the neighboring countries. In addition, the autoregressive integrated moving average (ARIMA) was used as a benchmark to compare and validate the NAR model outcomes. The results reveal that COVID-19 data in Iran have passed the fifth peak and continue on a downward trend to bring the number of total currently infected cases below 480,000 by the end of 2021. Additionally, 20%, 50%, 80% and 95% quantiles are provided along with the point estimation to model the uncertainty in the forecast.

A Mathematical Model of COVID-19 with Vaccination and Treatment

We formulate and theoretically analyze a mathematical model of COVID-19 transmission mechanism incorporating vital dynamics of the disease and two key therapeutic measures-vaccination of susceptible individuals and recovery/treatment of infected individuals. Both the disease-free and endemic equilibrium are globally asymptotically stable when the effective reproduction number R 0(v) is, respectively, less or greater than unity. The derived critical vaccination threshold is dependent on the vaccine efficacy for disease eradication whenever R 0(v) > 1, even if vaccine coverage is high. Pontryagin's maximum principle is applied to establish the existence of the optimal control problem and to derive the necessary conditions to optimally mitigate the spread of the disease. The model is fitted with cumulative daily Senegal data, with a basic reproduction number R 0 = 1.31 at the onset of the epidemic. Simulation results suggest that despite the effectiveness of COVID-19 vaccination and treatment to mitigate the spread of COVID-19, when R 0(v) > 1, additional efforts such as nonpharmaceutical public health interventions should continue to be implemented. Using partial rank correlation coefficients and Latin hypercube sampling, sensitivity analysis is carried out to determine the relative importance of model parameters to disease transmission. Results shown graphically could help to inform the process of prioritizing public health intervention measures to be implemented and which model parameter to focus on in order to mitigate the spread of the disease. The effective contact rate b, the vaccine efficacy ε, the vaccination rate v, the fraction of exposed individuals who develop symptoms, and, respectively, the exit rates from the exposed and the asymptomatic classes σ and ϕ are the most impactful parameters.

Time series forecasting of new cases and new deaths rate for COVID-19 using deep learning methods

The first known case of Coronavirus disease 2019 (COVID-19) was identified in December 2019. It has spread worldwide, leading to an ongoing pandemic, imposed restrictions and costs to many countries. Predicting the number of new cases and deaths during this period can be a useful step in predicting the costs and facilities required in the future. The purpose of this study is to predict new cases and deaths rate one, three and seven-day ahead during the next 100 days. The motivation for predicting every n days (instead of just every day) is the investigation of the possibility of computational cost reduction and still achieving reasonable performance. Such a scenario may be encountered in real-time forecasting of time series. Six different deep learning methods are examined on the data adopted from the WHO website. Three methods are LSTM, Convolutional LSTM, and GRU. The bidirectional extension is then considered for each method to forecast the rate of new cases and new deaths in Australia and Iran countries. This study is novel as it carries out a comprehensive evaluation of the aforementioned three deep learning methods and their bidirectional extensions to perform prediction on COVID-19 new cases and new death rate time series. To the best of our knowledge, this is the first time that Bi-GRU and Bi-Conv-LSTM models are used for prediction on COVID-19 new cases and new deaths time series. The evaluation of the methods is presented in the form of graphs and Friedman statistical test. The results show that the bidirectional models have lower errors than other models. A several error evaluation metrics are presented to compare all models, and finally, the superiority of bidirectional methods is determined. This research could be useful for organisations working against COVID-19 and determining their long-term plans.

A two diffusion stochastic model for the spread of the new corona virus SARS-CoV-2

We propose a refined version of the stochastic SEIR model for epidemic of the new corona virus SARS-Cov-2, causing the COVID-19 disease, taking into account the spread of the virus due to the regular infected individuals (transmission coefficient β ), hospitalized individuals (transmission coefficient l β , l > 0 ) and superspreaders (transmission coefficient β ' ). The model is constructed from the corresponding ordinary differential model by introducing two independent environmental white noises in transmission coefficients for above mentioned classes - one noise for infected and hospitalized individuals and the other for superspreaders. Therefore, the model is defined as a system of stochastic differential equations driven by two independent standard Brownian motions. Existence and uniqueness of the global positive solution is proven, and conditions under which extinction and persistence in mean hold are given. The theoretical results are illustrated via numerical simulations.

Time-variant reliability-based prediction of COVID-19 spread using extended SEIVR model and Monte Carlo sampling

A probabilistic method is proposed in this study to predict the spreading profile of the coronavirus disease 2019 (COVID-19) in the United State (US) via time-variant reliability analysis. To this end, an extended susceptible-exposed-infected-vaccinated-recovered (SEIVR) epidemic model is first established deterministically, considering the quarantine and vaccination effects, and then applied to the available COVID-19 data from US. Afterwards, the prediction results are described as a time-series of the number of people infected, recovered, and dead. Upon introducing the extended SEIVR model into a limit-state function and defining the model parameters including transmission, recovery, and mortality rates as random variables, the problem is transformed into a reliability model and analyzed by the Monte Carlo sampling. The findings are subsequently given in the form of exceedance probabilities (EPs) of the three main outputs, namely, the maximum number of infected cases, the total number of recovered cases, and the total number of fatal cases. Afterwards, by incorporating time into the formulation of the reliability problem, the EPs are calculated over time and presented as 3D probability graphs, illustrating the relationship between the number of cases affected (i.e., infected, recovered, or dead), exceedance probability, and time. The results for the US demonstrate that, by the end of 2021, the number of the infected (active) cases decreases to 0.8 million and number of cases recovered and fatalities increases to 41.3 million and 0.6 million, respectively.