  The reproduction number of deterministic models is an essential quantity to
predict whether an epidemic will spread or die out. Thresholds for disease
extinction contribute crucial knowledge on disease control, elimination, and
mitigation of infectious diseases. Relationships between the basic reproduction
numbers of two network-based ordinary differential equation vector-host models,
and extinction thresholds of corresponding 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 the relationships
may always exist and proposing a mathematical problem of proving their
existences in general. Moreover, numerical simulations show that the
reproduction number is not monotonically increasing or decreasing with the
extinction threshold. Key parameters in predicting uncertainty of extinction
thresholds are identified using Latin Hypercube Sampling/Partial Rank
Correlation Coefficient. 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.
