Global Stability of Two-Group Epidemic Models with Distributed Delays and Random Perturbation

and Applied Analysis 3 I ′ k kEk − ( d k γk ) Ik, Rk γkIk − d kRk, k 1, 2, . . . , n. 2.1 HereΛk represents influx of individuals into the kth group, d k , d E k , d k , and d k represent death rates of S, E, I, and R populations in the kth group, respectively, k represents the rate of becoming infectious after a latent period in the k-th group, and γk represents the recovery rate of infectious individuals in the k-th group. All parameter values are assumed to be nonnegative and Λk, d k , d E k > 0 for all k. Note that ( ∂ ∂t ∂ ∂r ) ik t, r − ( d k γk ) ik t, r , ik t, 0 kEk t , 2.2 whose solution is ik t, r ik t − r, 0 e− d k γk r kEk t − r e− d k γk r . 2.3 Substituting 2.3 into 2 , we obtain Sk Λk − n ∑ j 1 βkjSk t ∫∞ r 0 hj r jEj t − r e− d I j γj dr − d kSk, E′ k n ∑ j 1 βkjSk t ∫∞ r 0 hj r jEj t − r e− d I j γj dr − ( d k k ) Ek, I ′ k kEk − ( d k γk ) Ik, Rk γkIk − d kRk, k 1, 2, . . . , n. 2.4 Since the variables Ik and Rk do not appear in the first two equations of 2.4 , Li et al. consider the following reduced system with distributed time delays and general kernel functions 1 : Sk Λk − n ∑ j 1 βkjSk t ∫∞ r 0 fj r Ej t − r dr − d kSk, E′ k n ∑ j 1 βkjSk t ∫∞ r 0 fj r Ej t − r dr − ( d k k ) Ek. 2.5 Here the kernel function fk r 0 is continuous and 1 ∫∞ r 0 fk r dr hk > 0. System 2.5 can be interpreted as a multigroup model for an infectious disease whose latent period 4 Abstract and Applied Analysis r in hosts has a general probability density function 1/hk fk r dr, for the k-th group. Let S0 k Λk/d k , hk ∫∞ r 0 fk r dr. The next-generation matrix for system 2.5 is M0 ( βkjS 0 k hk d k k )