ASSESSING BIOLOGICAL CONTROL METHOD ONTHE PROGRESSION OF ANAPLASMOSIS DISEASE INDOMINANT CATTLE SPECIES IN THE MATABELELAND NORTH PROVINCE.

The paper defines the model with predation removing both susceptible and infected ticks at rate h·WP (system (1.1) shows −(μT + h·WP) in both tick equations), and then claims an effective reproduction number R1 = sqrt(b0 b1 βB βT / ( μB μT (μB+Λ)(μT+h·WP) )) governing local stability of the disease-free equilibrium E1, asserting stability when R1 < 1 and instability when R1 > 1. However, for the DFE with WP treated as a fixed parameter, ST* = b1/(μT + h·WP), not b1/μT. The paper’s Jacobian at E1 uses b1/μT (e.g., the −b1βT/μT entries), indicating that ST* was incorrectly evaluated at μT rather than μT + h·WP. This propagates into the next-generation calculation, dropping one factor of (μT + h·WP) and yielding an incorrect threshold. The correct threshold is R_true = sqrt( b0 b1 βB βT / ( μB (μB+Λ) (μT + h·WP)^2 ) ). Consequently, R1 can exceed 1 even when R_true < 1, so the paper’s instability claim based on R1 fails in general. Additional inconsistencies include a sign error in Lemma 1 (it states E1 is locally asymptotically stable when R1 > 1, then later uses the conventional R1 < 1) and misprints in the DFE listing. These issues are visible in the model equations (1.1), the invariant set and equilibrium discussion, the stated R1 (1.6), and the Jacobian (1.7) with b1/μT substitutions.