The two blue arrows denotes within-group infections (R 11 and R 22 ) and the black arrows denote inter-group infection cycle (R 12 R 21 ). The term R 11 R 22 subtracted in (3) accounts for the redundancy that resulted when adding R 11 and R 22 . Similar results for the basic reproduction number of a multi-group model were also obtained in [26, 27] .