Travelling waves for an epidemic model with non-smooth treatment rates

This is the post-print version of the final published paper that is available from the link below. Copyright @ 2010 IOP Publishing Ltd and SISSA.We consider a susceptible–infected–removed (SIR) epidemic model with two types of nonlinear treatment rates: (i) piecewise linear treatment rate with saturation effect, (ii) piecewise constant treatment rate with a jump (Heaviside function). For case (i), we compute travelling front solutions whose profiles are heteroclinic orbits which connect either the disease-free state to an infective state or two endemic states with each other. For case (ii), it is shown that the profile has the following properties: the number of susceptibles is monotonically increasing and the number of infectives approaches zero at infinity, while their product converges to a constant. Numerical simulations are performed for all these cases. Abnormal behaviour like travelling waves with non-monotonic profile or oscillations is observed

Rayleigh processes, real trees, and root growth with re-grafting

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N tends to infinity of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous--Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N tends to infinity of the Aldous--Broder chain. A key technical ingredient in this work is the use of a pointed Gromov--Hausdorff distance to metrize the space of rooted compact real trees.Comment: 48 Pages. Minor revision of version of Feb 2004. To appear in Probability Theory and Related Field

Numerical Solution of Quantum-Mechanical Pair Equations

We discuss and illustrate the numerical solution of the differential equation satisfied by the first‐order pair functions of Sinanoğlu. An expansion of the pair function in spherical harmonics and the use of finite difference methods convert the differential equation into a set of simultaneous equations. Large systems of such equations can be solved economically. The method is simple and straightforward, and we have applied it to the first‐order pair function for helium with 1 / r_(12) as the perturbation. The results are accurate and encouraging, and since the method is numerical they are indicative of its potential for obtaining atomic‐pair functions in general

Subtree prune and regraft: a reversible real tree-valued Markov process

We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree prune and regraft (SPR) Markov chains that appear in phylogenetic analysis. A key technical ingredient in this work is the use of a novel Gromov--Hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure. Also, the investigation of the Dirichlet form hinges on a new path decomposition of the Brownian excursion.Comment: Published at http://dx.doi.org/10.1214/009117906000000034 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Qualitative assessment of the role of public health education program on HIV transmission dynamics

© The author 2010. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.This paper presents a nonlinear deterministic model for assessing the impact of public health education campaign on curtailing the spread of the HIV pandemic in a population. Rigorous qualitative analysis of the model reveals that it exhibits the phenomenon of backward bifurcation (BB), where a stable disease-free equilibrium coexists with a stable endemic equilibrium when a certain threshold quantity, known as the 'effective reproduction number' (Reff), is less than unity. The epidemiological implication of BB is that a public health education campaign could fail to effectively control HIV, even when the classical requirement of having the associated reproduction number less than unity is satisfied. Furthermore, an explicit threshold value is derived above which such an education campaign could lead to detrimental outcome (increase disease burden), and below which it would have positive population-level impact (reduce disease burden in the community). It is shown that the BB phenomenon is caused by imperfect efficacy of the public health education program. The model is used to assess the potential impact of some targeted public health education campaigns using data from numerous countries.Kano State Government of Nigeria (N.H.); Natural Science and Engineering Research Council and Mathematics of Information Technology and Complex Systems of Canada (A.B.G.

Terrestrial Planet Formation in a protoplanetary disk with a local mass depletion: A successful scenario for the formation of Mars

Models of terrestrial planet formation for our solar system have been successful in producing planets with masses and orbits similar to those of Venus and Earth. However, these models have generally failed to produce Mars-sized objects around 1.5 AU. The body that is usually formed around Mars' semimajor axis is, in general, much more massive than Mars. Only when Jupiter and Saturn are assumed to have initially very eccentric orbits (e $\sim$ 0.1), which seems fairly unlikely for the solar system, or alternately, if the protoplanetary disk is truncated at 1.0 AU, simulations have been able to produce Mars-like bodies in the correct location. In this paper, we examine an alternative scenario for the formation of Mars in which a local depletion in the density of the protosolar nebula results in a non-uniform formation of planetary embryos and ultimately the formation of Mars-sized planets around 1.5 AU. We have carried out extensive numerical simulations of the formation of terrestrial planets in such a disk for different scales of the local density depletion, and for different orbital configurations of the giant planets. Our simulations point to the possibility of the formation of Mars-sized bodies around 1.5 AU, specifically when the scale of the disk local mass-depletion is moderately high (50-75%) and Jupiter and Saturn are initially in their current orbits. In these systems, Mars-analogs are formed from the protoplanetary materials that originate in the regions of disk interior or exterior to the local mass-depletion. Results also indicate that Earth-sized planets can form around 1 AU with a substantial amount of water accreted via primitive water-rich planetesimals and planetary embryos. We present the results of our study and discuss their implications for the formation of terrestrial planets in our solar system.Comment: Accepted for publication in The Astrophysical Journa

More Schooling, More Children: Compulsory Schooling Reforms and Fertility in Europe

We study the relationship between education and fertility, exploiting compulsory schooling reforms in Europe as source of exogenous variation in education. Using data from 8 European countries, we assess the causal effect of education on the number of biological kids and the incidence of childlessness. We find that more education causes a substantial decrease in childlessness and an increase in the average number of children per woman. Our findings are robust to a number of falsification checks and we can provide complementary empirical evidence on the mechanisms leading to these surprising results.