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Register a new userTwo critical times for the SIR model
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We consider the SIR model and study the first time the number of infected individuals begins to decrease and the first time this population is below a given threshold. We interpret these times as functions of the initial susceptible and infected populations and characterize them as solutions of a certain partial differential equation. This allows us to obtain integral representations of these times and in turn to estimate them precisely for large populations.
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We prove some refinements of concentration compactness principle for Sobolev space $W^{1,n}$ on a smooth compact Riemannian manifold of dimension $n$. As an application, we extend Aubins theorem for functions on $mathbb{S}^{n}$ with zero first order moments of the area element to higher order moments case. Our arguments are very flexible and can be easily modified for functions satisfying various boundary conditions or belonging to higher order Sobolev spaces.
We consider the Nelson model on some static space-times and investigate the problem of absence of a ground state. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. We investigate the absence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass $m(x)$ tends to $0$ at spatial infinity. Using path space techniques, we show that if $m(x)leq C |x|^{-mu}$ at infinity for some $C>0$ and $mu>1$ then the Nelson Hamiltonian has no ground state.
With respect to generalized trigonometric functions, since the discovery of double-angle formula for a special case by Edmunds, Gurka and Lang in 2012, no double-angle formulas have been found. In this paper, we will establish new double-angle formulas of generalized trigonometric functions in two special cases.
Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the $p$-Laplacian, which is known as a typical nonlinear differential operator. Compared to GTFs with one parameter, there are few applications of GTFs with two parameters to differential equations. We will apply GTFs with two parameters to studies on the inviscid primitive equations of oceanic and atmospheric dynamics, new formulas of Gaussian hypergeometric functions, and the $L^q$-Lyapunov inequality for the one-dimensional $p$-Laplacian.
We consider an anisotropic first-order ODE aggregation model and its approximation by a second-order relaxation system. The relaxation model contains a small parameter $varepsilon$, which can be interpreted as inertia or response time. We examine rigorously the limit $varepsilon to 0$ of solutions to the relaxation system. Of major interest is how discontinuous (in velocities) solutions to the first-order model are captured in the zero-inertia limit. We find that near such discontinuities, solutions to the second-order model perform fast transitions within a time layer of size $mathcal{O}(varepsilon^{2/3})$. We validate this scale with numerical simulations.
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