The effect of the needle exchange program on the spread of some sexually transmitted diseases

Eliza Bánhegyi; Attila Dénes; János Karsai; László Székely; Eliza Bánhegyi; Attila Dénes; János Karsai; László Székely

doi:10.3934/mbe.2019225

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 4506-4525. doi: 10.3934/mbe.2019225

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The effect of the needle exchange program on the spread of some sexually transmitted diseases

1.
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., H-6720 Szeged, Hungary
2.
Department of Mathematics, Institute of Environmental Engineering Systems, Szent István University, Páter Károly utca 1., H-2100 Gödöllõ, Hungary

Received: 31 December 2018 Accepted: 09 May 2019 Published: 21 May 2019

In this paper we consider a model for the spread of a sexually transmitted disease considering sexual transmission and spread via infected needles among intravenous drug users. Besides the transmission among drug users, we also consider sexual contacts between intravenous drug users and non-drug users. Furthermore, the needles are considered as a vector population. For several European countries, a sharp increase of sexually transmitted diseases was reported and several others are rated as endangered based on the number of syringes given out per intravenous drug users per year. The main purpose of the paper is to investigate the dynamics of this model including the effect of needle exchange and study the risk of an increased transmission among non-drug users, induced by the reduction of the needle exchange program. Following the determination of the basic reproduction number $\mathcal{R}_0$ it is shown that all solutions tend to the unique disease-free equilibrium if $\mathcal{R}_0$ < 1. We also prove that the disease persists in the human population if $\mathcal{R}_0$ > 1. Our numerical simulations, based on real life and hypothetical data for HIV, suggest that a decrease in the rate of the distribution and discharge rate of new needles might imply that the considered disease is becoming endemic in the considered human population of drug users and non-drug users. A variant of our model with time- variable needle distribition parameter is fitted to recent HIV data from Hungary to give a forecast for the number of infected in the following years.
- compartmental model,
- sexually transmitted diseases,
- needle exchange program,
- global stability,
- uniform persistence
Citation: Eliza Bánhegyi, Attila Dénes, János Karsai, László Székely. The effect of the needle exchange program on the spread of some sexually transmitted diseases[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4506-4525. doi: 10.3934/mbe.2019225

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Abstract

In this paper we consider a model for the spread of a sexually transmitted disease considering sexual transmission and spread via infected needles among intravenous drug users. Besides the transmission among drug users, we also consider sexual contacts between intravenous drug users and non-drug users. Furthermore, the needles are considered as a vector population. For several European countries, a sharp increase of sexually transmitted diseases was reported and several others are rated as endangered based on the number of syringes given out per intravenous drug users per year. The main purpose of the paper is to investigate the dynamics of this model including the effect of needle exchange and study the risk of an increased transmission among non-drug users, induced by the reduction of the needle exchange program. Following the determination of the basic reproduction number $\mathcal{R}_0$ it is shown that all solutions tend to the unique disease-free equilibrium if $\mathcal{R}_0$ < 1. We also prove that the disease persists in the human population if $\mathcal{R}_0$ > 1. Our numerical simulations, based on real life and hypothetical data for HIV, suggest that a decrease in the rate of the distribution and discharge rate of new needles might imply that the considered disease is becoming endemic in the considered human population of drug users and non-drug users. A variant of our model with time- variable needle distribition parameter is fitted to recent HIV data from Hungary to give a forecast for the number of infected in the following years.

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