American Institute of Mathematical Sciences

Advanced
2009, 6(1): 135-143. doi: 10.3934/mbe.2009.6.135

Modeling HIV outbreaks: The male to female prevalence ratio in the core population

Brandy Rapatski 1, and  James Yorke 2,

1. 

Department of Mathematics, Richard Stockton College of New Jersey, Pomona, NJ 08240, United States
2. 

Departments of Mathematics & Physics, University of Maryland, College Park, MD 20742, United States

Received  October 2007 Revised  September 2008 Published  December 2008

What affects the ratio of infected men to infected women in the core population in a heterosexual HIV epidemic? Hethcote & Yorke [5] introduced the term "core" initially to loosely describe the collection of individuals having the most unprotected sex partners. We study the early epidemic during the exponential growth phase and focus on the core group because most infected people were infected by people in the core. We argue that in the early outbreak phase of an epidemic, there is an identity, which we call the "outbreak equation." It relates three ratios that describe the core men versus the core women, namely, the ratio $E$ of numbers of all core men to all core women, the ratio $C$ of numbers of infected core men to core women, and the ratio $M$ of the infectiousness of a typical core man to that of a typical core woman. Then the relationship between the ratios is $E=MC^2$ in the early outbreak phase. We investigate two very different scenarios, one in which there are two times as many core men as core women ($E=2$) and the other in which core men equal core women ($E=1$). In the first case, the HIV epidemic grows at a much faster rate. We conclude that if the female core group was larger, that is, if more women in the total population were promiscuous (or if fewer men were promiscuous) then the HIV epidemic would grow more slowly.
Keywords: effective contact rate, infectiousness, human immunodeficiency virus, core population, infectiousness multiplier.
Mathematics Subject Classification: Primary: 92D30.
Citation: Brandy Rapatski, James Yorke. Modeling HIV outbreaks: The male to female prevalence ratio in the core population. Mathematical Biosciences & Engineering, 2009, 6 (1) : 135-143. doi: 10.3934/mbe.2009.6.135
[1]

Luis F. Gordillo, Stephen A. Marion, Priscilla E. Greenwood. The effect of patterns of infectiousness on epidemic size. Mathematical Biosciences & Engineering, 2008, 5 (3) : 429-435. doi: 10.3934/mbe.2008.5.429
[2]

Bertrand Lods. Variational characterizations of the effective multiplication factor of a nuclear reactor core. Kinetic & Related Models, 2009, 2 (2) : 307-331. doi: 10.3934/krm.2009.2.307
[3]

Chunlai Mu, Jun Zhou, Yuhuan Li. Fast rate of dead core for fast diffusion equation with strong absorption. Communications on Pure & Applied Analysis, 2010, 9 (2) : 397-411. doi: 10.3934/cpaa.2010.9.397
[4]

Song Liang, Yuan Lou. On the dependence of population size upon random dispersal rate. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2771-2788. doi: 10.3934/dcdsb.2012.17.2771
[5]

Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053
[6]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907
[7]

Marius Cocou. A dynamic viscoelastic problem with friction and rate-depending contact interactions. Evolution Equations & Control Theory, 2020, 9 (4) : 981-993. doi: 10.3934/eect.2020060
[8]

Najat Ziyadi. A male-female mathematical model of human papillomavirus (HPV) in African American population. Mathematical Biosciences & Engineering, 2017, 14 (1) : 339-358. doi: 10.3934/mbe.2017022
[9]

Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561
[10]

Tzy-Wei Hwang, Feng-Bin Wang. Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 147-161. doi: 10.3934/dcdsb.2013.18.147
[11]

Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483
[12]

Suqi Ma, Qishao Lu, Shuli Mei. Dynamics of a logistic population model with maturation delay and nonlinear birth rate. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 735-752. doi: 10.3934/dcdsb.2005.5.735
[13]

Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233
[14]

Xun-Yang Wang, Khalid Hattaf, Hai-Feng Huo, Hong Xiang. Stability analysis of a delayed social epidemics model with general contact rate and its optimal control. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1267-1285. doi: 10.3934/jimo.2016.12.1267
[15]

Islam A. Moneim, David Greenhalgh. Use Of A Periodic Vaccination Strategy To Control The Spread Of Epidemics With Seasonally Varying Contact Rate. Mathematical Biosciences & Engineering, 2005, 2 (3) : 591-611. doi: 10.3934/mbe.2005.2.591
[16]

Xing Liang, Lei Zhang. The optimal distribution of resources and rate of migration maximizing the population size in logistic model with identical migration. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020280
[17]

Caiping Liu, Heungwing Lee. Lagrange multiplier rules for approximate solutions in vector optimization. Journal of Industrial & Management Optimization, 2012, 8 (3) : 749-764. doi: 10.3934/jimo.2012.8.749
[18]

Santanu Sarkar. Analysis of Hidden Number Problem with Hidden Multiplier. Advances in Mathematics of Communications, 2017, 11 (4) : 805-811. doi: 10.3934/amc.2017059
[19]

Renato Soeiro, Abdelrahim Mousa, Tânia R. Oliveira, Alberto A. Pinto. Dynamics of human decisions. Journal of Dynamics & Games, 2014, 1 (1) : 121-151. doi: 10.3934/jdg.2014.1.121
[20]

Marek Bodnar, Urszula Foryś. Time Delay In Necrotic Core Formation. Mathematical Biosciences & Engineering, 2005, 2 (3) : 461-472. doi: 10.3934/mbe.2005.2.461

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences