A nonlinear population model of an infectious disease transmission is proposed, and the method of functional homotopic mapping is used to explore the law of the infectious disease transmission. A pair of functional homotopic mappings is constructed and the corresponding linear system of the model is discussed. Under some circumstances, the zero point is a stable node, that is, the original epidemic transmission system is a stable node at zero point. Therefore, when the time variable $t$ of the system model tends to infinity, it tends to zero solution. Better measures can be taken to control the epidemic disease. Finally, an example illustrates the correctness of the method used. The functional homotopic mapping method can take corresponding measures to control it. The expression of the solution obtained can be further analyzed. It is therefore possible to continue further discussions of the various properties of other related physical quantities.