Turing instability and pattern formation induced by noise in the modified SIR model

In the process of the spread of infectious diseases, it is often affected by demographic noise and stochastic disturbance. But the relationship between the noise and the dynamical mechanism is always in a state of searching. This paper mainly presents the combined effect of noise and diffusion on pattern formation. A SIR model with a well-mixed patch is proposed to explain the bizarre, periodic phenomenon of the epidemic. The Langevin and power spectral equations are derived through the Fourier transform and field theory techniques. And the Turing instability regions are analyzed through the dispersion equation and the power spectrum to show the role of diffusion and noise in the types of pattern formation. It is found that the noise could further extend the regions of Turing instability and make irregular patterns occur. Meanwhile, our results could quantitatively illustrate the spread of Covid-19.