Global warming and socioeconomic conditions: Comment on “Modeling the impact of global warming on vector-borne infections” by Eduardo Massad, Francisco Antonio Bezerra Coutinho, Luiz Fernandes Lopez and Daniel Rodrigues da Silva

Abstract

The review by Massad et al. dealt with global warming and vector-borne infections [1], in which they did not only present a mathematical model, but explained step by step all terms that appeared in the equations. The non-autonomous model was applied to malaria and dengue infections, obtaining the time dependent threshold for transmission R(t), which was studied by the sensitivity analysis. The authors have made numerous important contributions in this area. Two features of the model deserve some comments. The model considered a time-delay only in the passage from susceptible to infectious mosquitoes. The extrinsic incubation period depends strongly on temperature [2], as well as the transition time from egg up to adult mosquito [3], while the intrinsic incubation period (negligible compared to long life of humans) apparently does not depend on temperature. Neither the incubation period (humans are lead directly from susceptible to infectious stage) nor the transition time of aquatic phase were considered in the model, but the authors privileged extrinsic incubation period in their model through the time-delay. This choice can be understood as aiming to retrieve the formulae proposed by Macdonald for R0 [4]. With respect to the seasonal variation, this effect was allowed only to ‘mosquitoes production from eggs’. The authors considered a constant fraction per time ps describing the surviving on aquatic phase and becoming adult. The introduction of seasonality in the model is not clearly justified, because this is not related to any entomological parameters, as well as the step function θ seems artificial. Instead of interpreting the forcing parameters c and d as describing the durations of summer and winter seasons, the forcing term should be linked with the temperature dependent transition rate from eggs to adult (redefining ps ). Allowing to the temperature be dependent on time, T (t), the parameters c and d can be fitted from a time-series data [5], e.g., T (t)= c+ d sin( 2π 365 t + φ), and the function θ arises naturally through the lower and upper bounds of temperature at which transition from aquatic phase to adult can occur [3]. Therefore, all entomological parameters used by the authors can be set depending on the function T (t) (in some extent they did it in the last model): rM , τI and μM , which are, respectively, intrinsic oviposition rate, extrinsic incubation period and mortality rate of mosquitoes.