This paper studies disease transmission on small-world networks, focusing on the conditions under which epidemics occur. It examines both site and bond percolation models, where the probability of infection or transmission is varied. The models show epidemic behavior when the infection or transmission probability exceeds the threshold for site or bond percolation on the network. Exact solutions for the percolation threshold are derived for various cases, and numerical simulations confirm these results. Small-world networks, characterized by a small number of "degrees of separation," allow for faster disease spread compared to regular lattices. The paper introduces a variant of the Watts-Strogatz model, where a low-dimensional lattice with shortcuts is used to simulate disease propagation. It also considers models where transmission is less than 100% efficient, modeled as bond percolation. For site percolation, the percolation threshold $ p_c $ is determined by the fraction of susceptible individuals. The threshold is found to depend on the number of shortcuts $ \phi $ and the susceptibility $ p $. For $ k = 1 $, the threshold is given by $ p_c = \frac{\sqrt{4\phi^2 + 12\phi + 1} - 2\phi - 1}{4\phi} $. For general $ k $, the threshold is a root of a polynomial of order $ k + 1 $. For bond percolation, the threshold $ p_c $ is determined by the transmissibility of the disease. For $ k = 1 $, the threshold is the same as for site percolation. For $ k > 1 $, the threshold is more complex and requires solving a system of equations. The paper also considers a combined model where both site and bond percolation occur. The percolation threshold is determined by the product of the site and bond probabilities. Numerical simulations confirm the analytic results, showing that epidemics occur when the system is above the percolation threshold. The simulations also demonstrate that the spread of disease is faster in more susceptible populations. The results show that the percolation threshold on small-world networks can be accurately determined using analytic methods, and that the behavior of these networks is crucial in understanding the spread of diseases. The paper concludes that the method used can be extended to other lattice structures, providing a general framework for studying disease transmission on complex networks.This paper studies disease transmission on small-world networks, focusing on the conditions under which epidemics occur. It examines both site and bond percolation models, where the probability of infection or transmission is varied. The models show epidemic behavior when the infection or transmission probability exceeds the threshold for site or bond percolation on the network. Exact solutions for the percolation threshold are derived for various cases, and numerical simulations confirm these results. Small-world networks, characterized by a small number of "degrees of separation," allow for faster disease spread compared to regular lattices. The paper introduces a variant of the Watts-Strogatz model, where a low-dimensional lattice with shortcuts is used to simulate disease propagation. It also considers models where transmission is less than 100% efficient, modeled as bond percolation. For site percolation, the percolation threshold $ p_c $ is determined by the fraction of susceptible individuals. The threshold is found to depend on the number of shortcuts $ \phi $ and the susceptibility $ p $. For $ k = 1 $, the threshold is given by $ p_c = \frac{\sqrt{4\phi^2 + 12\phi + 1} - 2\phi - 1}{4\phi} $. For general $ k $, the threshold is a root of a polynomial of order $ k + 1 $. For bond percolation, the threshold $ p_c $ is determined by the transmissibility of the disease. For $ k = 1 $, the threshold is the same as for site percolation. For $ k > 1 $, the threshold is more complex and requires solving a system of equations. The paper also considers a combined model where both site and bond percolation occur. The percolation threshold is determined by the product of the site and bond probabilities. Numerical simulations confirm the analytic results, showing that epidemics occur when the system is above the percolation threshold. The simulations also demonstrate that the spread of disease is faster in more susceptible populations. The results show that the percolation threshold on small-world networks can be accurately determined using analytic methods, and that the behavior of these networks is crucial in understanding the spread of diseases. The paper concludes that the method used can be extended to other lattice structures, providing a general framework for studying disease transmission on complex networks.