Fig. 2

3D surface and contour plots showing the influence of various parameter pairs on \(R_0\) in the proposed HBV model. Subfigures (a) and (b) illustrate the joint impact of the male infection clearance probability \(q_m\) and the male-to-female contact rate \(c_{mf}\). Subfigures (c) and (d) analyze how the natural mortality rate \(\mu _0\) and non-sexual transmission rate \(\beta\) shape \(R_0\). Subfigures (e) and (f) explore the sensitivity of \(R_0\) to the interaction between male-to-female contact rate \(c_{mf}\) and male-to-female transmission probability \(\beta _{mf}\). These visualizations help identify key drivers in transmission dynamics and potential intervention targets.