scholarly journals A Mathematical Model of Epidemics—A Tutorial for Students

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1174 ◽  
Author(s):  
Yutaka Okabe ◽  
Akira Shudo
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Analytical Solutions ◽  
Sir Model ◽  
Exact Analytical Solutions ◽  
The Mathematical Model ◽  
Epidemic Diseases

This is a tutorial for the mathematical model of the spread of epidemic diseases. Beginning with the basic mathematics, we introduce the susceptible-infected-recovered (SIR) model. Subsequently, we present the numerical and exact analytical solutions of the SIR model. The analytical solution is emphasized. Additionally, we treat the generalization of the SIR model including births and natural deaths.

Torus-Like Balloons

2020 ◽  
Vol 56 ◽  
pp. 59-65
Author(s):  
Ivïalo M. Mladenov ◽  
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Elliptic Integrals ◽  
Deployable Structures ◽  
The Future ◽  
The Mathematical Model

The concepts for inflatable deployable structures have been under development and evaluation for many years. Strangely enough only the Mylar balloon up to now has been described adequately. Here we provide the mathematical model and its analytical solution for the torus-like balloons. Their characteristics and shapes are described explicitly in terms of elliptic integrals. The obtained results are commented shortly and the possible directions for the related studies in the future are outlined.

scholarly journals Approximation of potential-driven flow dynamics in large-scale self-similar tree networks

2011 ◽  
Vol 467 (2134) ◽  
pp. 2810-2824 ◽  
Author(s):  
Jason Mayes ◽  
Mihir Sen
Keyword(s):  
Mathematical Model ◽  
Analytical Solutions ◽  
Large Scale ◽  
Differential Algebraic Equations ◽  
Flow Dynamics ◽  
Algebraic Equations ◽  
Tree Networks ◽  
Self Similar ◽  
Large Scale Flow ◽  
The Mathematical Model

Dynamic analysis of large-scale flow networks is made difficult by the large system of differential-algebraic equations resulting from its modelling. To simplify analysis, the mathematical model must be sufficiently reduced in complexity. For self-similar tree networks, this reduction can be made using the network’s structure in way that can allow simple, analytical solutions. For very large, but finite, networks, analytical solutions are more difficult to obtain. In the infinite limit, however, analysis is sometimes greatly simplified. It is shown that approximating large finite networks as infinite not only simplifies the analysis, but also provides an excellent approximate solution.

scholarly journals Explicit Form of Exact Analytical Solution for Calculating Ground Displacement and Stress Induced by Shallow Tunneling and Its Application

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Caixia Guo ◽  
Kaihang Han ◽  
Heng Kong ◽  
Leilei Shi
Keyword(s):  
Analytical Solution ◽  
Explicit Form ◽  
Analytical Solutions ◽  
Pile Foundation ◽  
Pile Foundations ◽  
Physical Plane ◽  
Exact Analytical Solutions ◽  
Implicit Form ◽  
Existing Structure ◽  
Ground Displacements

In urban environment, it is often unavoidable for shallow tunnels to be constructed adjacent to existing pile foundations. To obtain the ground displacements and stresses induced by shallow tunneling and existing pile foundation loads, the key procedure involves superimposing the analytical solution for shallow tunneling in green-field with the analytical solution for existing structure loads. In green-field, the complex variable method provides exact analytical solutions of ground displacements and stresses caused by shallow tunneling. However, the exact analytical solutions are not directly expressed as explicit functions of the coordinates (x, y) in the physical plane (called implicit form of exact analytical solutions), whereas the displacements and stresses induced by existing structure loads are explicit functions of the coordinates (x, y) in the physical plane, which makes it difficult to superpose the displacements and stresses induced by existing structure loads. In this paper, explicit form of exact analytical solutions of displacements and stresses induced by shallow tunneling in green-field is obtained by using the inverse conformal transformation and the Cauchy–Riemann equations. Comparison with implicit form of exact analytical solutions shows that the explicit form of exact analytical solutions is intuitional and easily used by engineers, and moreover, the calculation amount is much smaller than that for the implicit form of exact analytical solutions. Then, an application involving superimposing the explicit form of exact analytical solutions with Mindlin’s solution is implemented to analyze the secondary stress field and the related potential plastic zone caused by shallow tunneling adjacent to pile foundations. Moreover, the influences of pile foundation parameters on the ranges and shapes of the potential plastic zones induced by nearby tunneling are analyzed.

scholarly journals On the Sodium Concentration Diffusion with Three-Dimensional Extracellular Stimulation

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Luisa Consiglieri ◽  
Ana Rute Domingos
Keyword(s):  
Mathematical Model ◽  
Boundary Condition ◽  
Analytical Solution ◽  
Parabolic Equations ◽  
Three Dimensional ◽  
Sodium Concentration ◽  
Dynamical Boundary Condition ◽  
Sodium Diffusion ◽  
The Mathematical Model ◽  
Concentration Diffusion

We deal with the transmembrane sodium diffusion in a nerve. We study a mathematical model of a nerve fibre in response to an imposed extracellular stimulus. The presented model is constituted by a diffusion-drift vectorial equation in a bidomain, that is, two parabolic equations defined in each of the intra- and extra-regions. This system of partial differential equations can be understood as a reduced three-dimensional Poisson-Nernst-Planck model of the sodium concentration. The representation of the membrane includes a jump boundary condition describing the mechanisms involved in the excitation-contraction couple. Our first novelty comes from this general dynamical boundary condition. The second one is the three-dimensional behaviour of the extracellular stimulus. An analytical solution to the mathematical model is proposed depending on the morphology of the excitation.

scholarly journals Analysis of the non-stationary mathematical model of a simple hydraulic network with a centrifugal pump

Vestnik IGEU ◽  
2020 ◽  
pp. 64-70
Author(s):  
V.A. Naumov
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Centrifugal Pump ◽  
Similarity Criteria ◽  
Dimensionless Form ◽  
Maximum Volume ◽  
Stationary Flows ◽  
Variable Level ◽  
Hydraulic Network ◽  
The Mathematical Model

Simple hydraulic networks with a centrifugal pump are not only part of complex networks, but are also widely used in Autonomous water supply and Sewerage systems. The mathematical model of simple networks taking into account the variable level of liquid in reservoirs includes the well-known Bernoulli equation for non-stationary flows. Published works on this problem do not take into account the non-stationary nature of the flow due to the variable liquid level. The conditions for using the quasi-stationary model are not discussed. Similarity criteria for the issue were not found. The purpose of the study is to analyze the non-stationary mathematical model of the object, including the definition of criteria for similarity of the problem and their impact on the solution. The well-known equations of fluid quantity balance and Bernoulli for non-stationary flows with smoothly changing characteristics were used as a mathematical model of a simple hydraulic network. The pressure characteristic of a centrifugal pump is approximated by a well-established dependence in the form of a square three-member. The system of differential equations was reduced to a dimensionless form. Analytical and numerical methods were used to solve the problem. The analysis of the mathematical model of pumping liquid by a centrifugal pump in a hydraulic network with a variable level was carried out. The dimensionless form of the system of equations allowed us to determine three similarity criteria for the problem, including the analog of the Struhal number Str. The analytical solution to the Cauchy problem is found in the quasi-stationary formulation (Str = 0). The solution of the problem in the full statement is obtained by the numerical method. The results of the study of the influence of similarity criteria on the solution are presented. The dimensionless flow rate of the liquid decreases with increasing Str values. In this case, the maximum volume of liquid and the time to reach it increases. Increasing the values of the other two criteria leads to an increase in both the flow rate and the maximum volume of the liquid. The analytical solution in the quasi-rational formulation can be used only for Str < 0,1. The results obtained can be used in the design of Autonomous Water supply and Sewerage systems. Further research for the non-self-similar area of hydraulic resistance and for variable fluid viscosity is promising.

scholarly journals Analytical solution and adaptation to the parameter estimation of the SIR model

2020 ◽  
pp. 40-43
Author(s):  
S. M. Ivanov
Keyword(s):  
Parameter Estimation ◽  
Graduate Students ◽  
Analytical Solution ◽  
General Solution ◽  
Population Size ◽  
Analytical Solutions ◽  
Sir Model ◽  
Mathematical Epidemiology ◽  
Linear Regressions ◽  
Inverse Proportionality

The article deals with analytical solution and adaptation to the parameter estimation of the SIR model of the epidemic. By a special replacement of the exponential function by inverse proportionality, the approximate general solution of the SIR model is found. It is spoken in detail about the process of integration of ordinary differential equations of the SIR model. The equality of the sum of the obtained analytical solutions and population size is checked. The obtained solutions are simple and understandable. To parametrically estimate the SIR model, its general solution is adapted to paired linear regressions. The article is of interest for students, graduate students and scientists involved in mathematical epidemiology.

scholarly journals Growth of N-dimensional spherical bubble within viscous, superheated liquid: Analytical solution

2019 ◽  
pp. 380-380
Author(s):  
Gamiel Shalaby ◽  
Ali Abu-Bakr
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Variable Viscosity ◽  
Superheated Liquid ◽  
Spherical Bubble ◽  
The Mathematical Model

In this paper, we present the study of the bevaiour of spherical bubble in N-dimensions fluid. The fluid is a mixture of vapour and superheated liquid. The mathematical model is formulated in N-dimensions fluid on the basis of continuity and momentum equations, and solved its analytically. The variable viscosity is taken in an account problem. The obtained results show that the radius of bubble increases with the decreasing of the value of N-dimensions.

AI OPTIMIZATION OF THE EXOTHERMIC REACTION OF ETHYLENE OXIDE WITH WATER

2021 ◽  
pp. 2150033
Author(s):  
Khaled A. Al-Utaibi ◽  
Ayesha sohail ◽  
Andleeb Zafar ◽  
Rana Talha ◽  
Sadiq M. Sait
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Ethylene Oxide ◽  
Ethylene Glycol ◽  
Exothermic Reaction ◽  
Plug Flow ◽  
Material Balance ◽  
Chemical Species ◽  
Future Research ◽  
The Mathematical Model

A computational framework, for the numerical approximation of the exothermic reaction of ethylene oxide (EO) with water, to form ethylene glycol is presented in this paper. Ethylene Glycol also known as Mono-ethylene Glycol (MEG), is a diol with a boiling of 198[Formula: see text]C and conventionally produced through hydrolysis of ethylene oxide which is obtained through the oxidation of ethylene. It is used as an excellent automobile coolant as the 1:1 ratio mixture of MEG with Water boils at 129[Formula: see text]C and freezes at [Formula: see text]C. Other than its use as an antifreeze, it is also used as a reagent during the production of polyester fibers, pharmaceutics, cosmetics, hydraulic fluids, printing inks, explosives, polyesters and paint solvents. The mathematical model presented here, consists of an energy balance and a material balance system, described in an axisymmetric coordinate system. The optimized resulting values using the artificial intelligence approach are summarized in this paper. We derive an analytical solution. The analytical solution for the mathematical model equations is in general not possible for this model but it may be possible to derive an analytical solution to this mathematical model if we consider the equation for the conservation of material (chemical species) as a formulation for plug flow and isothermal conditions. Noteworthy findings are reported in this paper for future research.

scholarly journals A Mathematical Model of Gas and Water Flow in a Swelling Geomaterial—Part 1. Verification with Analytical Solution

Minerals ◽  
2019 ◽  
Vol 10 (1) ◽  
pp. 30 ◽  
Author(s):  
Elias Ernest Dagher ◽  
Julio Ángel Infante Sedano ◽  
Thanh Son Nguyen
Keyword(s):  
Mathematical Model ◽  
Numerical Model ◽  
Analytical Solution ◽  
Analytical Solutions ◽  
Gas Flow ◽  
Numerical Models ◽  
Model Verification ◽  
Model Complexity ◽  
Gas Migration

Gas generation and migration are important processes that must be considered in a safety case for a deep geological repository (DGR) for the long-term containment of radioactive waste. Expansive soils, such as bentonite-based materials, are widely considered as sealing materials. Understanding their long-term performance as barriers to mitigate gas migration is vital in the design and long-term safety assessment of a DGR. Development and the application of numerical models are key to understanding the processes involved in gas migration. This study builds upon the authors’ previous work for developing a hydro-mechanical mathematical model for migration of gas through a low-permeable geomaterial based on the theoretical framework of poromechanics through the contribution of model verification. The study first derives analytical solutions for a 1D steady-state gas flow and 1D transient gas flow problem. Using the finite element method, the model is used to simulate 1D flow through a confined cylindrical sample of near-saturated low-permeable soil under a constant volume boundary stress condition. Verification of the numerical model is performed by comparing the pore-gas pressure evolution and stress evolution to that of the results of the analytical solution. The results of the numerical model closely matched those of the analytical solutions. Future studies will attempt to improve upon the model complexity and investigate processes and material characteristics that can enhance gas migration in a nearly saturated swelling geomaterial.

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