### Mathematical Biosciences and Engineering

2015, Issue 3: 565-584. doi: 10.3934/mbe.2015.12.565

# Network-level reproduction number and extinction threshold for vector-borne diseases

• Received: 01 June 2014 Accepted: 29 June 2018 Published: 01 January 2015
• MSC : Primary: 92B30; Secondary: 62P10, 92B05.

• The basic reproduction number of deterministic models is an essential quantity to predict whether an epidemic will spread or not. Thresholds for disease extinction contribute crucial knowledge of disease control, elimination, and mitigation of infectious diseases. Relationships between basic reproduction numbers of two deterministic network-based ordinary differential equation vector-host models, and extinction thresholds of corresponding stochastic continuous-time Markov chain models are derived under some assumptions. Numerical simulation results for malaria and Rift Valley fever transmission on heterogeneous networks are in agreement with analytical results without any assumptions, reinforcing that the relationships may always exist and proposing a mathematical problem for proving existence of the relationships in general. Moreover, numerical simulations show that the basic reproduction number does not monotonically increase or decrease with the extinction threshold. Consistent trends of extinction probability observed through numerical simulations provide novel insights into mitigation strategies to increase the disease extinction probability. Research findings may improve understandings of thresholds for disease persistence in order to control vector-borne diseases.

Citation: Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases[J]. Mathematical Biosciences and Engineering, 2015, 12(3): 565-584. doi: 10.3934/mbe.2015.12.565

### Related Papers:

• The basic reproduction number of deterministic models is an essential quantity to predict whether an epidemic will spread or not. Thresholds for disease extinction contribute crucial knowledge of disease control, elimination, and mitigation of infectious diseases. Relationships between basic reproduction numbers of two deterministic network-based ordinary differential equation vector-host models, and extinction thresholds of corresponding stochastic continuous-time Markov chain models are derived under some assumptions. Numerical simulation results for malaria and Rift Valley fever transmission on heterogeneous networks are in agreement with analytical results without any assumptions, reinforcing that the relationships may always exist and proposing a mathematical problem for proving existence of the relationships in general. Moreover, numerical simulations show that the basic reproduction number does not monotonically increase or decrease with the extinction threshold. Consistent trends of extinction probability observed through numerical simulations provide novel insights into mitigation strategies to increase the disease extinction probability. Research findings may improve understandings of thresholds for disease persistence in order to control vector-borne diseases.

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