Fig. 4

Contour plots of \(R_{0}\). a, b the \((\delta ,\omega )\) planes with various stray dogs’ food resources \(K_{d2}\). The contour plots measure the effectiveness of invalid sheep vaccination rate \((\delta )\) and vaccination rate \((\omega )\) on reducing \(R_{0}\). c–f the \((\gamma ,\theta )\) planes respond to different \(K_{d2}\) as well as \(\delta\). The contour plots depict the effectiveness of domestic dog’s recovery rate \((\gamma )\) and stray dog’s disposing rate \((\theta )\) on decreasing \(R_{0}\). Other parameters are listed in the Table 1