The Application of PageRank model in urban epidemic prevention and control

The ongoing COVID-19 epidemic has had a great impact on social activities and the economy. The usage technical analysis tools to provide a more accurate and efficient reference for epidemic control measures is of great significance. This paper analyzes the characteristics and deficiencies of the existing technical methods, such as regression model, simulation calculation, differential equation and so on. By analyzing past outbreak cases and comparing the epidemic prevention measures of different cities, we discuss the importance of early and timely prevention in controlling the epidemic, and the importance of analyzing and formulating plans in advance. We then make the key observation that the spread of the virus is related to the topology of the urban network. This paper further proposes an epidemic analysis model of the optimized PageRank model, and gives a ranking algorithm for virus transmission risk levels based on road nodes, forming a visual risk warning level map, and applies the algorithm to the epidemic analysis of Yuegezhuang area in Beijing. Finally, more in-depth research directions and suggestions for prevention and control measures are put forward. © 2023 SPIE.

Long-term prediction of the COVID-19 epidemics induced by Omicron-virus in China based on a novel non-autonomous delayed SIR model

Since the outbreak of COVID-19, the severe acute respiratory syndrome coronavirus 2 genome is still mutating. Omicron, a recently emerging virus with a shorter incubation period, faster transmission speed, and stronger immune escape ability, is soaring worldwide and becoming the mainstream virus in the COVID-19 pandemic. It is especially critical for the governments, healthcare systems, and economic sectors to have an accurate estimate of the trend of this disaster. By using different mathematical approaches, including the classical susceptible-infected-recovered (SIR) model and its extensions, many investigators have tried to predict the outbreaks of COVID-19. In this study, we employed a novel model which is based upon the well-known susceptible-infected-removed (SIR) model with the time-delay and time-varying coefficients in our previous works. We aim to predict the evolution of the epidemics effectively in nine cities and provinces of China, including A City, B City, C City, D City, E City, F City, G City, H City and I Province. The results show it is effective to model the spread of the large-scale and sporadic COVID-19 induced by Omicron virus by the novel non-autonomous delayed SIR compartment model. The significance of this study is that it can provide the management department of epidemic control with theoretical references and subsequent evaluation of the prevention, control measures, and effects.

Modeling the influence of the information domain on countermeasure effectiveness in case of COVID-19

A common way to model an epidemic — restricted to contagion aspects only — is a modification of the Kermack-McKendrick SIR Epidemic model (SIR model) with differential equations. (Mis-)Information about epidemics may influence the behavior of the people and thus the course of epidemics as well. We have thus coupled an extended SIR model of the COVID-19 pandemic with a compartment model of the (mis-)information-based attitude of the population towards epidemic countermeasures. The resulting combined model is checked concerning basic plausibility properties like positivity and boundedness. It is calibrated using COVID-19 data from RKI and attitude data provided by the COVID-19 Snapshot Monitoring (COSMO) study. The values of parameters without corresponding observation data have been determined using an L2-fit under mild additional assumptions. The predictions of the calibrated model are essentially in accordance with observations. An uncertainty analysis of the model shows, that our results are in principle stable under measurement errors. We also assessed the scale, at which specific parameters can influence the evolution of epidemics. Another result of the paper is that in a multi-domain epidemic model, the notion of controlled reproduction number has to be redefined when being used as an indicator of the future evolution of epidemics.

A mathematical model and simulation scenarios for T and B cells immune response to severe acute respiratory syndrome-coronavirus-2

Severe acute respiratory syndrome coronavirus is a type 2 highly contagious, and transmissible among humans;the natural human immune response to severe acute respiratory syndrome-coronavirus-2 combines cell-mediated immunity (lymphocyte) and antibody production. In the present study, we analyzed the dynamic effects of adaptive immune system cell activation in the human host. The methodology consisted of modeling using a system of ordinary differential equations;for this model, the equilibrium free of viral infection was obtained, and its local stability was determined. Analysis of the model revealed that lymphocyte activation leads to total pathogen elimination by specific recognition of viral antigens;the model dynamics are driven by the interaction between respiratory epithelial cells, viral infection, and activation of helper T, cytotoxic T, and B lymphocytes. Numerical simulations showed that the model solutions match the dynamics involved in the role of lymphocytes in preventing new infections and stopping the viral spread;these results reinforce the understanding of the cellular immune mechanisms and processes of the organism against severe acute respiratory syndrome-coronavirus-2 infection, allowing the understanding of biophysical processes that occur in living systems, dealing with the exchange of information at the cellular level.

A fractional mathematical model with nonlinear partial differential equations for transmission dynamics of severe acute respiratory syndrome coronavirus 2 infection.

This study presents a fractional mathematical model based on nonlinear Partial Differential Equations (PDEs) of fractional variable-order derivatives for the host populations experiencing transmission and evolution of the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) pandemic. Five host population groups have been considered, the Susceptible, Exposed, Infected, Recovered, and Deceased (SEIRD). The new model, not introduced before in its current formulation, is governed by nonlinear PDEs with fractional variable-order derivatives. As a result, the proposed model is not compared with other models or real scenarios. The advantage of the proposed fractional partial derivatives of variable orders is that they can model the rate of change of subpopulation for the proposed model. As an efficient tool to obtain the solution of the proposed model, a modified analytical technique based on the homotopy and Adomian decomposition methods is introduced. Then again, the present study is general and is applicable to a host population in any country.

A multiconsistent computational methodology to resolve a diffusive epidemiological system with effects of migration, vaccination and quarantine.

BACKGROUND: We provide a compartmental model for the transmission of some contagious illnesses in a population. The model is based on partial differential equations, and takes into account seven sub-populations which are, concretely, susceptible, exposed, infected (asymptomatic or symptomatic), quarantined, recovered and vaccinated individuals along with migration. The goal is to propose and analyze an efficient computer method which resembles the dynamical properties of the epidemiological model. MATERIALS AND METHODS: A non-local approach is utilized for finding approximate solutions for the mathematical model. To that end, a non-standard finite-difference technique is introduced. The finite-difference scheme is a linearly implicit model which may be rewritten using a suitable matrix. Under suitable circumstances, the matrices representing the methodology are M-matrices. RESULTS: Analytically, the local asymptotic stability of the constant solutions is investigated and the next generation matrix technique is employed to calculate the reproduction number. Computationally, the dynamical consistency of the method and the numerical efficiency are investigated rigorously. The method is thoroughly examined for its convergence, stability, and consistency. CONCLUSIONS: The theoretical analysis of the method shows that it is able to maintain the positivity of its solutions and identify equilibria. The method's local asymptotic stability properties are similar to those of the continuous system. The analysis concludes that the numerical model is convergent, stable and consistent, with linear order of convergence in the temporal domain and quadratic order of convergence in the spatial variables. A computer implementation is used to confirm the mathematical properties, and it confirms the ability in our scheme to preserve positivity, and identify equilibrium solutions and their local asymptotic stability.

Identification of time delays in COVID-19 data

COVID-19 data released by public health authorities is subject to inherent time delays. Such delays have many causes, including delays in data reporting and the natural incubation period of the disease. We develop and introduce a numerical procedure to recover the distribution of these delays from data.We extend a previously-introduced compartmental model with a nonlinear, distributed-delay term with a general distribution, obtaining an integrodifferential equation. We show this model can be approximated by a weighted-sum of constant time-delay terms, yielding a linear problem for the distribution weights. Standard optimization can then be used to recover the weights, approximating the distribution of the time delays. We demonstrate the viability of the approach against data from Italy and Austria.We find that the delay-distributions for both Italy and Austria follow a Gaussian-like profile, with a mean of around 11 to 14 days. However, we note that the delay does not appear constant across all data types, with infection, recovery, and mortality data showing slightly different trends, suggesting the presence of independent delays in each of these processes. We also found that the recovered delay-distribution is not sensitive to the discretization resolution.These results establish the validity of the introduced procedure for the identification of time-delays in COVID-19 data. Our methods are not limited to COVID-19, and may be applied to other types of epidemiological data, or indeed any dynamical system with time-delay effects.

Transient zonal modeling of SARS-CoV-2 airborne infection risk for incomplete mixing air distribution methods

A widely used analytical model to quantitatively assess airborne infection risk is the Wells-Riley model based on the assumption of complete air mixing in a single zone. This study aimed to extend the Wells-Riley model so that the infection risk can be calculated in spaces where complete mixing is not present. This is done by evaluating the time-dependent distribution of infectious quanta in each zone and by solving the coupled system of differential equations based on the zonal quanta concentrations. In conclusion, this study shows that using the Wells-Riley model based on the assumption of completely mixing air may overestimate the long-range airborne infection risk compared to some high-efficiency ventilation systems such as displacement ventilation, but also underestimate the infection risk in a room heated with warm air supplied from the ceiling. © 2022 17th International Conference on Indoor Air Quality and Climate, INDOOR AIR 2022. All rights reserved.

Uncertain hypothesis test for uncertain differential equations

Uncertain hypothesis test is a statistical tool that uses uncertainty theory to determine whether some hypotheses are correct or not based on observed data. As an application of uncertain hypothesis test, this paper proposes a method to test whether an uncertain differential equation fits the observed data or not. In order to demonstrate the test method, some numerical examples are provided. Finally, both uncertain currency model and stochastic currency model are used to model US Dollar to Chinese Yuan (USD–CNY) exchange rates. As a result, it is shown that the uncertain currency model fits the exchange rates well, but the stochastic currency model does not.

On a reaction–diffusion system modelling infectious diseases without lifetime immunity

In this paper, we study a mathematical model for an infectious disease caused by a virus such as Cholera without lifetime immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction–diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behaviour of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving a priori estimates. The analysis developed in this paper can be employed to study other epidemic models in biological, ecological and health sciences.

THREE-DIMENSIONAL ANALYTICAL SOLUTION OF THE ADVECTION-DIFFUSION EQUATION FOR AIR POLLUTION DISPERSION

We develop a new analytical solution of a three-dimensional atmospheric pollutant dispersion. The main idea is to subdivide vertically the planetary boundary layer into sub-layers, where the wind speed and eddy diffusivity assume average values for each sub-layer. Basically, the model is assessed and validated using data obtained from the Copenhagen diffusion and Prairie Grass experiments. Our findings show that there is a good agreement between the predicted and observed crosswind-integrated concentrations. Moreover, the calculated statistical indices are within the range of acceptable model performance.

The Analytic Solutions of the Fractional-Order Model for the Spatial Epidemiology of the COVID-19 Infection

This paper provides a mathematical fractional-order model that accounts for the mindset of patients in the transmission of COVID-19 disease, the continuous inflow of foreigners into the country, immunization of population subjects, and temporary loss of immunity by recovered individuals. The analytic solutions, which are given as series solutions, are derived using the fractional power series method (FPSM) and the residual power series method (RPSM). In comparison, the series solution for the number of susceptible members, using the FPSM, is proportional to the series solution, using the RPSM for the first two terms, with a proportional constant of ψΓnα+1, where ψ is the natural birth rate of the baby into the susceptible population, Γ is the gamma function, n is the nth term of the series, and α is the fractional order as the initial number of susceptible individuals approaches the population size of Ghana. However, the variation in the two series solutions of the number of members who are susceptible to the COVID-19 disease begins at the third term and continues through the remaining terms. This is brought on by the nonlinear function present in the equation for the susceptible subgroup. The similar finding is made in the series solution of the number of exposed individuals. The series solutions for the number of deviant people, the number of nondeviant people, the number of people quarantined, and the number of people recovered using the FPSM are unquestionably almost identical to the series solutions for same subgroups using the RPSM, with the exception that these series solutions have initial conditions of the subgroup of the population size. It is observed that, in this paper, the series solutions of the nonlinear system of fractional partial differential equations (PDEs) provided by the RPSM are more in line with the field data than the series solutions provided by the FPSM.

An age-of-infection model with both symptomatic and asymptomatic infections.

We formulate a general age-of-infection epidemic model with two pathways: the symptomatic infections and the asymptomatic infections. We then calculate the basic reproduction number [Formula: see text] and establish the final size relation. It is shown that the ratio of accumulated counts of symptomatic patients and asymptomatic patients is determined by the symptomatic ratio f which is defined as the probability of eventually becoming symptomatic after being infected. We also formulate and study a general age-of-infection model with disease deaths and with two infection pathways. The final size relation is investigated, and the upper and lower bounds for final epidemic size are given. Several numerical simulations are performed to verify the analytical results.

Post-Pandemic Sector-Based Investment Model Using Generalized Liouville–Caputo Type

In this article, Euler's technique was employed to solve the novel post-pandemic sector-based investment mathematical model. The solution was established within the framework of the new generalized Caputo-type fractional derivative for the system under consideration that serves as an example of the investment model. The mathematical investment model consists of a system of four fractional-order nonlinear differential equations of the generalized Liouville–Caputo type. Moreover, the existence and uniqueness of solutions for the above fractional order model under pandemic situations were investigated using the well-known Schauder and Banach fixed-point theorem technique. The stability analysis in the context of Ulam—Hyers and generalized Ulam—Hyers criteria was also discussed. Using the investment model under consideration, a new analysis was conducted. Figures that depict the behavior of the classes of the projected model were used to discuss the obtained results. The demonstrated results of the employed technique are extremely emphatic and simple to apply to the system of non-linear equations. When a generalized Liouville–Caputo fractional derivative parameter (ρ) is changed, the results are asymmetric. The current work can attest to the novel generalized Caputo-type fractional operator's suitability for use in mathematical epidemiology and real-world problems towards the future pandemic circumstances.

Incorporation of awareness programs into a model of the spread of HIV/AIDS among people who inject drugs

Mathematical modeling techniques have been used extensively during the human immunodeficiency virus (HIV) epidemic. Drug injection causes increased HIV spread in most countries globally. The media is crucial in spreading health awareness by changing mixing behavior. The published studies show some of the ways that differential equation models can be employed to explain how media awareness programs influence the spread and containment of disease (Greenhalgh et al. Appl Math Comput. 2015;251:539–563). Here we build a differential equation model which shows how disease awareness programs can alter the HIV prevalence in a group of people who inject drugs (PWIDs). This builds on previous work by Greenhalgh and Hay (1997) and Liang et al. (2016). We have constructed a mathematical model to describe the improved model that reduces the spread of the diseases through the effect of awareness of disease on sharing needles and syringes among the PWID population. The model supposes that PWIDs clean their needles before use rather than after. We carry out a steady state analysis and examine local stability. Our discussion has been focused on two ways of studying the influence of awareness of infection levels in epidemic modeling. The key biological parameter of our model is the basic reproductive number R0$$ {R}_0 $$. R0$$ {R}_0 $$ is a crucial number which determines the behavior of the infection. We find that if R0$$ {R}_0 $$ is less than one then the disease-free steady state is the unique steady state and moreover whatever the initial fraction of infected individuals then the disease will die out as time becomes large. If R0$$ {R}_0 $$ exceeds one there is the disease-free steady state and a unique steady state with disease present. We also showed that the disease-free steady state is locally asymptotically stable if R0$$ {R}_0 $$ is less than one, neutrally stable if R0$$ {R}_0 $$ is equal to one and unstable if R0$$ {R}_0 $$ exceeds one. In the last case, when R0$$ {R}_0 $$ is greater than one the endemic steady state was locally asymptotically stable. Our analytical results are confirmed by using simulation with realistic parameter values. In nontechnical terms, the number R0$$ {R}_0 $$ is a critical value describing how the disease will spread. If R0$$ {R}_0 $$ is less than or equal to one then the disease will always die out but if R0$$ {R}_0 $$ exceeds one and disease is present the disease will sustain itself and moreover the numbers of PWIDs with disease will tend to a unique nonzero value.

Python optimization code for solving a mathematical modeling of COVID_19

In recent years, mathematical modeling has played a key role in many life applications such as computer science, physics, chemistry, and genetics. Actually, in this paper, our focus is on the classifications and the importance of mathematical programming and its applications in health problems especially the Mathematical Modeling of COVID_19. According to the era of the Corona pandemic, it has been using mathematical equations to employ mathematical programming in epidemics and the mechanism of spreading in urban areas. The solution of the problem is presented in two directions;the first was by graphic representation and the other by using computational software via the Python language. © 2023 Author(s).

Total Value Adjustment of Multi-Asset Derivatives under Multivariate CGMY Processes

Counterparty credit risk (CCR) is a significant risk factor that financial institutions have to consider in today's context, and the COVID-19 pandemic and military conflicts worldwide have heightened concerns about potential default risk. In this work, we investigate the changes in the value of financial derivatives due to counterparty default risk, i.e., total value adjustment (XVA). We perform the XVA for multi-asset option based on the multivariate Carr–Geman–Madan–Yor (CGMY) processes, which can be applied to a wider range of financial derivatives, such as basket options, rainbow options, and index options. For the numerical methods, we use the Monte Carlo method in combination with the alternating direction implicit method (MC-ADI) and the two-dimensional Fourier cosine expansion method (MC-CC) to find the risk exposure and make value adjustments for multi-asset derivatives.

A modified variable‐order fractional SIR model to predict the spread of COVID‐19 in India

The first case of COVID‐19 in India detected on January 30, 2020, after its emergence in Wuhan, China, in December 2019. The lockdown was imposed as anemergency measure by the Indian government to prevent the spread of COVID‐19 but gradually eased out due to its vast economic consequences. Just 15 days after the relaxation of lockdown restrictions, Delhi became India's worst city in terms of COVID‐19 cases. In this paper, we propose a variable‐order fractional SIR (susceptible, infected, removed) model at state‐level scale. We introduce a algorithm that uses the differential evolution algorithm in combination with Adam–Bashforth–Moulton method to learn the parameters in a system of variable‐order fractional SIR model. The model can predict the confirm COVID‐19 cases in India considering the effects of nationwide lockdown and the possible estimate of the number of infliction inactive cases after the removal of lockdown on June 1, 2020. A new parameter p is introduced in the classical SIR model representing the fraction of infected people that get tested and are thereby quarantined. The COVID‐19 trajectory in Delhi, as per our model, predicts the slowing down of the spread between January and February 2021, touching a peak of around 5 lakh confirmed cases. [ FROM AUTHOR] Copyright of Mathematical Methods in the Applied Sciences is the property of John Wiley & Sons, Inc. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full . (Copyright applies to all s.)

Analytical and Numerical Investigation of the SIR Mathematical Model

This is a theoretical study of the SIR model — a popular mathematical model of the propagation of infectious diseases. We construct a solution of the Cauchy problem for a system of two ordinary differential equations describing in integral form the concentration dynamics of infected and recovered individuals in an immune population. A qualitative analysis is carried out of the stationary system states using the Lyapunov function. An expression is obtained for the coordinates of the equilibrium points in terms of the Lambert W-function for arbitrary initial values. The application of the SIR model for the description of COVID-19 propagation dynamic is demonstrated. © 2023, Springer Science+Business Media, LLC, part of Springer Nature.

On the Aspects of Vitamin D and COVID-19 Infections and Modeling Time-Delay Body's Immune System with Time-Dependent Effects of Vitamin D and Probiotic

This chapter provides a summary of recent views on the aspects of vitamin D levels and the relationship between the prevalence rates of vitamin D deficiency and COVID-19 death toll of several countries in Europe and Asia. The chapter also discusses a new modified time-delay immune system model with time-dependent of the body's immune healthy cells, vitamin D, and probiotic. The model can be used to assess the timely progression of healthy immune cells with the effects of the levels of vitamin D and probiotics supplement. It also can help to predict when the infected cells and virus particles free state can ever be reached as time progresses with and without considering the vitamin D and probiotic supplements. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.