Self-organization of oscillation in an epidemic model for COVID-19

On the basis of a compartment model, the infection curve is investigated when the net rate of change λ=(dI/dt)/I of the number of infected individuals ?? is given by an ellipse (??/??₀ )²+[(??−??₀ )/Δ]²=1 in the λ−?? plane which is supported in [??_ℓ,??_ℎ] where ??_ℎ=??₀+Δ, ??_ℓ=??₀−Δ. With a≡(??_ℎ−??_ℓ)/(??_ℎ+??_ℓ )=Δ/??₀ , it is rigorously shown that (1) when a<1 or ??_ℓ>0, oscillation of the infection curve is self-organized and the period ?? of the oscillation is given by T??₀=??(??_ℎ−??_ℓ)/(??_ℎ??_ℓ)^1/2, (2) when a=1 or ??_ℓ=0, the infection curve shows a critical behavior where it decays obeying a power law function with exponent −2 in the long time limit after a peak, and (3) when a>1 or ??_ℓ<0, the infection curve decays exponentially in the long time limit after a peak and the relaxation time τ is given by τ??₀=(??_ℎ−??_ℓ)/2(−??_ℎ??_ℓ)^1/2. The present result indicates that the pandemic can be controlled by a measure which keeps ??_ℓ<0.