Mathematical studies on the sterile insect technique for the Chikungunya disease and Aedes albopictus

Abstract

Chikungunya is an arthropod-borne disease caused by the Asian tiger mosquito, Aedes albopictus. It can be an important burden to public health and a great cause of morbidity and, sometimes, mortality. Understanding if and when disease control measures should be taken is key to curtail its spread. Dumont and Chiroleu (Math Biosc Eng 7(2):315-348, 2010) showed that the use of chemical control tools such as adulticide and larvicide, and mechanical control, which consists of reducing the breeding sites, would have been useful to control the explosive 2006 epidemic in Réunion Island. Despite this, chemical control tools cannot be of long-time use, because they can induce mosquito resistance, and are detrimental to the biodiversity. It is therefore necessary to develop and test new control tools that are more sustainable, with the same efficacy (if possible). Mathematical models of sterile insect technique (SIT) to prevent, reduce, eliminate or stop an epidemic of Chikungunya are formulated and analysed. In particular, we propose a new model that considers pulsed periodic releases, which leads to a hybrid dynamical system. This pulsed SIT model is coupled with the human population at different epidemiological states in order to assess its efficacy. Numerical simulations for the pulsed SIT, using an appropriate numerical scheme are provided. Analytical and numerical results indicate that pulsed SIT with small and frequent releases can be an alternative to chemical control tools, but only if it is used or applied early after the beginning of the epidemic or as a preventive tool.

Dynamically consistent nonstandard finite difference schemes for epidemiological models

“…The convergence of the nonstandard finite difference schemes (24) and (28) is not a problem due essentially to the asymptotic relation (23). For instance, the scheme (28) is of order 1 as a consequence of this asymptotic relation and Taylor expansion, which show that the local truncation error T k+1 of the scheme is given by…”

“…According to the approach of [44], formalized in [7], the two numerical methods (24) and (28) are non-standard finite difference schemes for the following reasons: (a) The standard denominator h of the discrete derivatives is replaced by the more complex function φ(h), which satisfies the requirement (23). (This denominator function is expected to capture the essential features of the dynamical system); (b) Nonlinear terms are approximated in a nonlocal way by using more than one point of the mesh.…”

“…It is straightforward to check that the NSFD scheme (24) has no extra fixed-points than those of (1). Let x * denote the DFE or EE of the system (1).…”

“…The nonstandard schemes (24) and (28) should, following the philosophy of [44], eliminate elementary instabilities in the first place. Since only the implicit scheme (24) is fully defined here, we restrict our analysis of elementary stability to it.…”

“…The Jacobian matrices are assumed to be diagonalizable, which is the usual case in applications. The denominator function that is needed in (24) can be taken to be…”