Existence of periodic orbits for periodic eco-epidemic models with disease in the prey

For an eco-epidemic model with disease in the prey and periodic coefficients it is conjectured in [Xingge Niu, Tailei Zhang, Zhidong Teng, The asymptotic behavior of a nonautonomous eco-epidemic model with disease in the prey, Applied Mathematical Modelling 35, 457-470 (2011)] that, when the infected prey is permanent, there is a positive periodic orbit that is globally asymptotically stable in the interior of $(\R^+)^3$. In this paper we prove the existence part of the conjecture.