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[EN] The classic nonlinear Kermack-McKendrick model based upon a system of differential equations has been widely applied to model the rise and fall of global pandemic and also seasonal epidemic by introducing a forced harmonic infectivity which would change throughout the year. These methods work well in their respective domains of applicability, and for certain diseases, but they fail when both seasonality and high infectivity are combined. In this paper we consider a Susceptible-Infected-Recovered, or SIR, model with two latent states to model the propagation and evolutionary history of varicella in humans. We show that infectivity can be calculated from real data and we find a nonstandard seasonal variation that cannot be fitted with a single harmonic. Moreover, we show that infectivity for the present strains of the virus has raised following a sigmoid function in a period of several centuries. This could allow the design of vaccination strategies and the study of the epidemiology of varicella and herpes zoster. (C) 2016 Elsevier B.V. All rights reserved.

The last several years have brought an avalanche of data from various high throughput, genome‐wide analyses of yeast and other model organisms. Still, scientists struggle to comprehend the complex behavior of biological systems. One method that has been available for decades but now is more necessary than ever is the mathematical modeling of biological systems. Unfortunately, a chasm of terminology and techniques has separated most biologists from mathematical modelers. This chapter hopes to bridge that gap for metabolic models by delineating the general process used to develop a system of differential equations that describes a biochemical pathway. This modeling process can be generally applied to many biological phenomena. In addition, the specific approach of Biochemical Systems Theory (BST) is demonstrated for the nitty‐gritty details of the model equations. These methods are demonstrated using the core section of ceramide metabolism in yeast.

This chapter discusses applications to inelastic deformation and damage evolution in composites. The transformation field analysis (TFA) method evaluates averages of local and overall strain and stress fields caused in elastic heterogeneous solids, such as composite materials, laminates, and polycrystals, by piecewise uniform distributions of transformation strains or eigenstrains; these is caused by thermal changes, phase transformations, moisture absorption, or inelastic deformation. Also, equivalent eigenstrains applied in modeling of local cracks, voids, and other damage modes can be included in the analysis. Moreover, the method provides a system of differential equations describing evolution of the local transformation fields under varying overall loads, according to local constitutive relations prescribed in the constituents. The chapter discusses applications to inelastic deformation and damage evolution in composites.

Publisher Summary This chapter discusses differential games. In control theory, a certain evolutionary process (typically given by a time-dependent system of differential equations) depends on a control variable. A certain cost is associated with the control variable and with the corresponding evolutionary process. The goal is to choose the best control—that is, the control for which the cost is minimized. A similar evolutionary process is used in differential games, but it depends on two (or more) controls. Each player is in charge of one of the controls. Each player wishes to minimize its own cost (the cost functions of the different players may be related). However, at each time t , a player must decide on its control without knowing what the other (usually opposing) players intend to do. The strategy of each player at time t depends only on whatever information the player was able to gain by observing the evolving process up to time t only.

Publisher Summary This chapter discusses integrable combinations applicable to systems of ordinary differential equations. Integrable combinations yield one or more ordinary differential equations that can be integrated exactly. The idea behind integrable combinations is that by combining pieces of a system of differential equations, a combination of the dependent variables can be determined explicitly in terms of the independent variable. Integration of the system of ordinary differential equations, dxi/dt = fi(t, x1, x2,.,xn), for i = 1, 2,…,n, is often accomplished by choosing integrable combinations. An integrable combination is a differential equation which is derived from a system of differential equations and is readily integrable. The chapter also highlights that each linearly independent integrable combination yields a first integral of the original system.

Publisher Summary This chapter presents some numerical studies of an area-preserving mapping. The equatorial plane is a surface of section for the flow governed by the Hamiltonian. In a typical orbit, the particle crosses the equatorial plane in the northerly direction; it penetrates to certain latitude and then gets reflected back, penetrating to certain latitude below the equator. To determine the qualitative nature of the section map Σ for those orbits that penetrate close to the singularity, one must first determine the nature of the flow near the singularity r = 0. Near the singularity, for z > 0, the flow on each constant energy surface is topologically equivalent to the orbits of the system of differential equations. The mapping M 1 is a model for the flow when the particle starts in the equatorial plane, goes north, and then comes back, while the mapping M 2 is a model for the flow when the particle starts in the equatorial plane, goes south, and then returns to the equatorial plane.

Publisher Summary This chapter discusses the Lefschetz fixed-point formula. The Lefschetz fixed-point formula is known for quite general spaces. The theory of structural stability for n-dimensional manifolds has made remarkable progress in the last 15 years. According to a claim made by Lefschetz, the coefficients are only known approximately in a system of differential equations arising out of a practical problem: various errors are inevitably involved in their determination. Lefschetz was, of course, talking about the definition of structural stability for differential equations, but the same may be said for diffeomorphisms.

Publisher Summary This chapter discusses periodic solutions of perturbed Lotka–Volterra systems. The Lotka–Volterra system of differential equations is one of the early attempts in ecology to model a predator–prey relationship. It has been suggested that an appropriate model for a predator–prey systems should yield an unstable equilibrium point and a surrounding stable limit cycle. The system of equations is studied as a perturbation of the Lotka–Volterra system. The basic idea is to find conditions that make the critical point in the first quadrant an unstable spiral and then find conditions that guarantee that trajectories cross a certain closed curve containing the critical point from outside to inside.

The numerical solution of nonlinear partial differential equations is often complicated by large local gradients in the solution that evolve in time. As these gradients pass over a mesh point, the solution at that mesh point changes rapidly. When this happens, small time steps should be taken at these mesh points to accurately integrate the solution while elsewhere larger time steps would suffice. Thus, a moving front can result in a multirate system of differential equations at the mesh points and force a global small time step. If the mesh points are moved so the solution at the mesh points changes slowly, then then the same larger time step may be appropriate everywhere. Several strategies are described on how to best move the meshpoints to define a reference frame where the solution is slowly varying.

Publisher Summary This chapter discusses the concept of Laplace transforms. In applying the method of Laplace transforms to the initial value problem, three main steps are undertaken. The first step transforms a hard problem into a relatively easy problem. Then the easy problem is solved by finding X. Finally, X is inverted; that is, a solution x of the original problem is found from the solution of the transformed problem. This same procedure is followed in the solution of more complicated initial value problems. The chapter discusses the applications of Laplace transforms to differential equations. In establishing the existence of the Laplace transform of a function, it is necessary to show that a certain improper integral converges. The theory of Laplace transforms is applied to the solution of initial value problems. The chhapter discusses problems where the differential equation, or system of differential equations, is linear and has constant coefficients.

Publisher Summary This chapter describes the identification of compartmental systems. A linear time-invariant compartmental model, which is frequently used to describe a physiological process, can be represented by the system of differential equations. The chapter highlights a few properties of certain types of compartmental systems. In addition to the general case for the matrix A, the chapter focuses on two subclassifications of A: the mammillary system and the catenary system. It highlights two basic questions, one concerning the restriction of the eigenvalues of the compartmental matrix A to specified subsets of the complex plane and the other concerning a few properties of an impulse-response function in special cases. It also focuses on single input-observation compartmental systems.

Publisher Summary This chapter discusses a few new formulations related to pharmacokinetic systems analysis. It describes an n-compartmental process by a system of differential equations. The chapter presents a few new approaches along with some preliminary results in the analysis of pharmacokinetic systems. It discusses three approaches to improve the analysis of pharmacokinetic systems. Time delay effects play a significant role in the analysis and modeling of drug distribution. The rate coefficients are time dependent and not arbitrary constants. These two concepts are discussed in some detail in the chapter. The chapter also presents a parametric method of estimating the functional time-dependent form of the rate coefficients.

Publisher Summary This chapter describes a model for the statistical analysis of cell radiosensitivity during the cell cycle. It discusses the solutions of the system of differential equations which characterize this model when the rate constants k1 and k2 together with initial proportions of sensitive and resistant cells, S0 and R0, are treated as random variables with specified probability distributions. The analysis is performed under the assumption that the above random variables may be characterized by the uniform and beta probability distributions. In view of the range of variation of these variables and their inherent characteristics, these probability distributions seem quite relevant. The chapter describes the radiosensitivity model and presents the statistical characterization of this system.

This chapter focuses on the network analysis by polynomial matrices. For a closed network, F is a square polynomial matrix, and the characteristic equation is det F = 0. For a well-defined closed network, this determinant does not vanish identically, and its degree in p is, by definition, is the degree of the network. The characteristic equation remains invariant if F is premultiplied by an arbitrary unimodular matrix, so that det F can be called the network determinant. The closed network is represented as an n -port of constraints containing straps and ideal transformers whose ports are closed on a total of n separate positive RLC elements. The equations of the n -port of constraints are MI = 0; KV = 0 with MK = 0, where I and V are the vectors of port variables. The well-known condition MK = 0 merely expresses that the constraints are workless. The degree of a system of differential equations is the total number of arbitrary integration constants. In the detailed network equations, the only variables appearing under the differentiation sign are condenser voltages ( V C ) and inductor currents ( I L ). The degree is, thus, the number of linearly independent elements in the set ( V C , I L ).

Publisher Summary This chapter discusses the aspects of ordinary differential equations including theorems on existence, uniqueness of solutions, and the continuity and differentiability of solutions with respect to a parameter. If y' =f(x,y) is a system of differential equations where f satisfies a Lipschitz condition in y for a < x < b and all y ∈ En, then the solution y = y(x) is defined for a < x < b. The solutions of differential equations are continuous or differentiable with respect to a parameter if the initial conditions and f have the similar property. The chapter proves the differentiability of the solution in a parameter for the case where the parameter enters in the initial conditions.