Multiscale analysis for a vector-borne epidemic model

Abstract

ABSTRACT. Traditional studies about disease dynamics have focused on global stability issues, due to their epidemiological importance. We study a classical SIR-SI model for arboviruses in two different directions: we begin by describing an alternative proof of previously known global stability results by using only a Lyapunov approach. In the sequel, we take a different view and we argue that vectors and hosts can have very distinctive intrinsic time-scales, and that such distinctiveness extends to the disease dynamics. Under these hypothesis, we show that two asymptotic regimes naturally appear: the fast host dynamics and the fast vector dynamics. The former regime yields, at leading order, a SIR model for the hosts, but with a rational incidence rate. In this case, the vector disappears from the model, and the dynamics is similar to a directly contagious disease. The latter yields a SI model for the vectors, with the hosts disappearing from the model. Numerical results show the performance of the approximation, and a rigorous proof validates the reduced models.

Time-scale separation and centre manifold analysis describing vector-borne disease dynamics

“…This modelling approach is motivated by the fact that the life spans of the mosquito and of the human host differ by orders of magnitude; hence, the mosquitos act as fast vectors between humans, but the immune status of the human hosts, susceptibility to some strains and immunity to others, leads to the time pattern of human disease as observed, for example, in clinical notification data. Rather recently, a first study on a simple susceptible-infected-recovered (SIR) model for the humans and a susceptibleinfected (SI) model for the mosquitos has investigated the separation of time scales explicitly [17]. Mathematical modelling efforts in dengue fever go back as far as the 1970s [9].…”

State estimators for some epidemiological systems

“…We then considered the reduced Bailey-Dietz model studied in Souza (2014), which is a special case of the generalised SIR model, and constructed the proposed observer for this reduced system. This observer was applied with real data coming from the district of Jacarepaguá in the city of Rio de Janeiro to estimate the susceptible population in this area and, as indicated by the analysis in Section 4, the estimate converges very fast towards the population state -this can also be seen graphically in Figure 6(B), where simulations with three different arbitrary initial conditions are displayed, and the corresponding estimates become visually indistinguishable after a short time.…”

“…Souza (2014), but see also a more informal derivation on Keeling and Rohani (2008). For convenience, we recall that it is given by (20)…”

“…As an example, we can mention the fast-vector regime of the Bailey-Dietz model discussed in Souza (2014) -but see also the discussion in Keeling and Rohani (2008) -given by…”

“…The classical Bailey-Dietz (Bailey, 1975;Dietz, 1975) model in adimensional variables-cf. Souza (2014); Esteva and Vargas (1998)-is given byẋ…”