Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-16T07:26:09.125Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Screening in Structured SIR Epidemics

Published online by Cambridge University Press:  06 June 2012

B. Ainseba  and
M. Iannelli
B. Ainseba
Affiliation:
Bordeaux University, Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Case 26, Université Bordeaux Ségalen, 3 Place de la Victoire, 33076 Bordeaux Cedex, France
M. Iannelli*
Affiliation:
Department of mathematics, University of Trento, 38050 Povo, Trento, Italy
*
⋆ Corresponding author. E-mail: [email protected]
Get access

Abstract

We present a model for describing the spread of an infectious disease with publicscreening measures to control the spread. We want to address the problem of determining anoptimal screening strategy for a disease characterized by appreciable duration of theinfectiveness period and by variability of the transmission risk. The specific disease wehave in mind is the HIV infection. However the model will apply to a disease for whichclass-age structure is significant and should not be disregarded.

Keywords

PDE in connection with biologypopulation dynamicsepidemiologyoptimal control problems involving partial differential equations
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 3: Epidemiology , 2012 , pp. 12 - 27
Copyright
© EDP Sciences, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Almeder, C., Feichtinger, G., Sanderson, W. C., Veliov, V. M.. Prevention and medication of HIV/AIDS : the case of Botswana, Central European J. Oper. Res. 15 (2007), 47–61. CrossRefGoogle Scholar
S. Anita. Analysis and Control of Age-Dependent Population Dynamics, Kluwer, Boston, MA, 2000.
Anita, S., Iannelli, M., Kim, M.Y., Park, E.J.. Optimal harvesting for periodic age-dependent population dynamics, SIAM J. Appl. Math. 58 (1999), 1648–1666. Google Scholar
Armbruster, B., Brandeau, M. L.. Optimal mix of screening and contact tracing for endemic diseases, Math. Biosci. 209 (2007), 386–402. CrossRefGoogle ScholarPubMed
Ball, F., Becker, N. G.. Control of transmission with two types of infection, Math. Biosci. 200 (2006), 170–187. CrossRefGoogle Scholar
V. Barbu, M. Iannelli. Controlling the SIS Epidemics, Proceedings of the Conference on Mathematical Models in Medical and Health Sciences, Nashville, Tennessee, edited by M. A. Horn, G. Simonett, G. F. Webb, 1998.
Barbu, V., Iannelli, M.. Optimal control of population dynamics, J. Optim. Theory Appl. 102 (1999), 1–14. CrossRefGoogle Scholar
Behncke, H.. Optimal control of deterministic epidemics, Optim. Control Appl. Meth. 21 (2000), 269–285. CrossRefGoogle Scholar
Blayneh, K. W., Gumel, A. B., Lenhart, S., Clayton, T.. Backward Bifurcation and Optimal Control in Transmission Dynamics of West Nile Virus, Bulletin of Mathematical Biology, 72 (2010), 1006–1028. CrossRefGoogle ScholarPubMed
Castillo-Chavez, C., Feng, Z.. Global stability of an age structured model for TB and its applications to optimal vaccination strategies, Math. Biosci. 151 (1998), 135–154. CrossRefGoogle Scholar
Feichtinger, G., Veliov, V. M., Tsachev, T.. Maximum principle for age and duration structured systems : a tool for optimal prevention and treatment of HIV, Mathematical Population Studies, 11 (2004), 3–28. CrossRefGoogle Scholar
Feng, Z., Thieme, H. R.. Recurrent outbreaks of childhood diseases revisited : the impact of isolation, Math. Biosci. 128 (1995), 93–130. CrossRefGoogle ScholarPubMed
Fister, K. R., Lenhart, S.. Optimal control of a competitive system with age-structure, J. Math. Anal. Appl. 291 (2004), 526–537. CrossRefGoogle Scholar
Hadeler, K.P., Muller, J.. Optimal harvesting and optimal vaccination, Math. Biosci. 206 (2007), 249–272. CrossRefGoogle ScholarPubMed
Hansen, E., Day, T.. Optimal control of epidemics with limited resources, J. Math. Biol. 62 (2011), 423–451. CrossRefGoogle ScholarPubMed
H. W. Hethcote, J. A. York. Gonorrhea Transmission and Control, Lectures Notes in Biomathematics 56, Springer Verlag 1984.
Hethcote, H. W.. Optimal ages of vaccination for measles, Math. Biosci. 89 (1988), 29–52. CrossRefGoogle Scholar
Hethcote, H. W., Waltman, P.. Optimal vaccination schedules in a deterministic epidemic model, Math. Biosci. 18 (1973), 365–381. CrossRefGoogle Scholar
Hritonenko, N., Yatsenko, Y.. The structure of optimal time- and age-dependent harvesting in the Lotka McKendrik population model, Math. Biosci. 208 (2007), 48–62. CrossRefGoogle Scholar
Hyman, J. M., Li, J., Stanley, E. A.. Modeling the impact of random screening and contact tracing in reducing the spread of HIV, Math. Biosci. 181 (2003), 17–54. CrossRefGoogle ScholarPubMed
Jung, E., Lenhart, S., Feng, Z.. Optimal Control of Treatments in a Two Strain Tuberculosis Model, Discrete and Continuous Dynamical Systems 2 (2002), 473–482. Google Scholar
Kassa, S. M., Ouhinou, A.. Epidemiological models with prevalence dependent endogenous self-protection measure, Math. Biosci. 229 (2011), 41–49. CrossRefGoogle ScholarPubMed
Liu, X., Stechlinski, P.. Pulse and constant control schemes for epidemic models with seasonality, Nonlinear Analysis : RWA 12 (2011), 931–946. CrossRefGoogle Scholar
Milner, F.A. and Zhao, R.. A new mathematical model of syphilis, Math. Model. Nat. Phenom. 5 (2010), 96–108. CrossRefGoogle Scholar
Morton, R., Wickwire, K. H.. On the optimal control of a deterministic epidemic, Adv. Appl. Prob. 6 (1974), 622–635. CrossRefGoogle Scholar
Mubayi, A., Kribs Zaleta, C., Martcheva, M., Castillo-Chavez, C.. A cost based comparison of quarantine strategies for new emerging diseases, Math. Biosc. Eng. 7 (2010), 687–717. CrossRefGoogle Scholar
Muller, J., Kretzschmar, M., Dietz, K.. Contact tracing in stochastic and deterministic epidemic models, Math. Biosci. 164 (2000), 39–64. CrossRefGoogle ScholarPubMed
Ndeffo Mbah, M. L., Gilligan, C. A.. Optimization of control strategies for epidemics in heterogeneous populations with symmetric and asymmetric transmission, J. Theor. Biology 262 (2010), 757–763. CrossRefGoogle Scholar
Sanders, J. L.. Quantitative guidelines for communicable disease control programs, Biometrics, 27 (1971), 883–893. CrossRefGoogle Scholar
Terry, A. J.. Pulse Vaccination Strategies in a Metapopulation SIR model, Math. Biosc. Eng. 7 (2010), 455–477. CrossRefGoogle Scholar
Wickwire, K. H.. A note on the optimal control of carrier-borne epidemics, J. Appl. Prob. 12 (1975), 565–568. CrossRefGoogle Scholar
Wickwire, K. H.. Optimal isolation policies for deterministic and stochastic epidemics, Math. Biosci. 26 (1975), 325–346. CrossRefGoogle Scholar
Wickwire, K. H.. Mathematical models for the control of pests and infectious diseases : a survey, Theor. Pop. Biol. 11 (1977), 182–238. CrossRefGoogle ScholarPubMed