Continuous and pulsed epidemiological models for onchocerciasis with implications for eradication strategy

Glenn Ledder; Donna Sylvester; Rachelle R. Bouchat; Johann A. Thiel; Glenn Ledder; Donna Sylvester; Rachelle R. Bouchat; Johann A. Thiel

doi:10.3934/mbe.2018038

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 4: 841-862. doi: 10.3934/mbe.2018038

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Continuous and pulsed epidemiological models for onchocerciasis with implications for eradication strategy

1.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA
2.
Mathematics Department, Seattle University, 901 12th Ave, Seattle, WA 98122-1090, USA
3.
Department of Mathematics, Indiana University of Pennsylvania, Indiana, PA 15705, USA
4.
Department of Mathematics, New York City College of Technology -CUNY, Brooklyn, NY 11201, USA

Received: 01 March 2017 Accepted: 03 January 2018 Published: 01 August 2018
MSC : Primary: 92C60

Onchocerciasis is an endemic disease in parts of sub-Saharan Africa. Complex mathematical models are being used to assess the likely efficacy of efforts to eradicate the disease; however, their predictions have not always been borne out in practice. In this paper, we represent the immunological aspects of the disease with a single empirical parameter in order to reduce the model complexity. Asymptotic approximation allows us to reduce the vector-borne epidemiological model to a model of an infectious disease with nonlinear incidence. We then consider two versions, one with continuous treatment and a more realistic one where treatment occurs only at intervals. Thorough mathematical analysis of these models yields equilibrium solutions for the continuous case, periodic solutions for the pulsed case, and conditions for the existence of endemic disease equilibria in both cases, thereby leading to simple model criteria for eradication. The analytical results and numerical experiments show that the continuous treatment version is an excellent approximation for the pulsed version and that the current onchocerciasis eradication strategy is inadequate for regions where the incidence is highest and unacceptably slow even when the long-term behavior is the disease-free state.
- Epidemiology,
- dynamical models,
- asymptotics
Citation: Glenn Ledder, Donna Sylvester, Rachelle R. Bouchat, Johann A. Thiel. Continuous and pulsed epidemiological models for onchocerciasis with implications for eradication strategy[J]. Mathematical Biosciences and Engineering, 2018, 15(4): 841-862. doi: 10.3934/mbe.2018038

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Abstract

Onchocerciasis is an endemic disease in parts of sub-Saharan Africa. Complex mathematical models are being used to assess the likely efficacy of efforts to eradicate the disease; however, their predictions have not always been borne out in practice. In this paper, we represent the immunological aspects of the disease with a single empirical parameter in order to reduce the model complexity. Asymptotic approximation allows us to reduce the vector-borne epidemiological model to a model of an infectious disease with nonlinear incidence. We then consider two versions, one with continuous treatment and a more realistic one where treatment occurs only at intervals. Thorough mathematical analysis of these models yields equilibrium solutions for the continuous case, periodic solutions for the pulsed case, and conditions for the existence of endemic disease equilibria in both cases, thereby leading to simple model criteria for eradication. The analytical results and numerical experiments show that the continuous treatment version is an excellent approximation for the pulsed version and that the current onchocerciasis eradication strategy is inadequate for regions where the incidence is highest and unacceptably slow even when the long-term behavior is the disease-free state.

References

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