Analysis of a Chagas disease model with <i>Trypanosoma cruzi</i> dixenous development delays and predation transmission

In this paper, we develop and investigate an eco-epidemiological Chagas disease model with two delays and a Holling-Ⅱ predation mechanism with an aim to examine the impact of Chagas-paraiste dixenous development delays on the transmission of Chagas disease. A couple of thresholds of vector reproduction ratio $ \mathcal{R}_v $ and disease reproduction ratios $ \mathcal{R}_0 $ are analytically derived to jointly determine the dynamics of the proposed model. For the two-delay model, we find that the dynamics depends on the relation between the half-saturation constant of Holling-Ⅱ response ($ n $), relevant to predation transmission, and vector amount ($ \frac{1}{\sigma} $) at which the Ricker-type reproduction rate of vectors attains its maximum regulated by a factor ($ 1-\frac{\mu_v}{r_v} $), relevant to reproduction of vector species. To be specific, if $ n<\frac{1}{\sigma}(1-\frac{\mu_v}{r_v}) $, the dynamics are complex, in which both backward bifurcation and bistability are observed; if the relation is opposite, the dynamics are rather simple and forward bifurcation and global stability are accordingly shown, where the properties of the relevant characteristic equations are examined and the feasible Lyapunov functionals are constructed. Our theoretical findings and biological implications are subsequently inspected by numerical simulations. Moreover, it reveals that the Chagas-parasite development delay within the bodies of triatomines displays a significant role in the risk of Chagas disease transmission, while the impact of the delay within the bodies of vertebrate hosts is marginal.