Mathematical Analysis of the Transmission Dynamics of Skin Cancer Caused by UV Radiation

Abstract

Nowadays, skin cancer is a worldwide panic. It is related to ultraviolet radiation. In this paper, we have formulated a SIRS type mathematical model to show the effects of ultraviolet radiation on skin cancer. At first, we have showed the boundedness and positivity of the model solutions to verify the model’s existence and uniqueness. The boundedness and positivity gave the solutions of our model bounded and positive, which was very important for real-world situation because in real world, population cannot be negative. Then, we have popped out all the equilibrium points of our model and verified the stability of the equilibrium points. This stability test expressed some physical situation of our model. The disease-free equilibrium point is locally asymptotically stable if $R_0<1$ and if $R_0>1$ , then it is unstable. Again, the endemic equilibrium point is stable, if $R_0>1$ and unstable if $R_0<1$ . In order to understand the dynamical behavior of the model’s equilibrium points, we examined the phase portrait. We also have observed the sensitivity of the model parameters. After this, we have investigated the different scenarios of bifurcations of the model’s parameters. At the set of Hopf bifurcation parameters when infection rate due to UV rays is less than ${\alpha }_1=0.01$ , proper control may eradicate the existence of disease. From transcritical bifurcation, we can say when recovery rate greater than 1.9, then the disease of skin cancer can be eliminated and when recovery rate less than 1.9 then the disease of skin cancer cannot be eradicated. Finally, numerical analysis is done to justify our analytical findings.