Effect of population migration and punctuated lockdown on the spread of infectious diseases

TL;DR

The paper develops a SEIRS epidemic diffusion model with population migration across $n$ cities to study post-lockdown spread under saturated incidence and waning immunity. It derives the full multi-city ODE system and computes the reproduction number $R_0$ for two-city and three-city scenarios using the next-generation matrix approach, showing how migration rates $m_{ij}$ modulate transmission. Numerical simulations reveal that higher migration generally increases $R_0$ and peak infections, with punctuated lockdown producing damped oscillations and delayed resurgences as recovered individuals become susceptible again. The results emphasize mobility management and intermittent lockdown strategies to prevent healthcare overload in the absence of vaccines, along with monitoring of recovered individuals to limit resurgence.

Abstract

One of the critical measures to control infectious diseases is a lockdown. Once past the lockdown stage in many parts of the world, the crucial question now concerns the effects of relaxing the lockdown and finding the best ways to implement further lockdown(s), if required, to control the spread. With the relaxation of lockdown, people migrate to different cities and enhance the spread of the disease. This work presents the population migration model for n-cities and applies the model for migration between two and three cities. The reproduction number is calculated, and the effect of the migration rate is analyzed. A punctuated lockdown is implemented to simulate a protocol of repeated lockdowns that limits the resurgence of infections. A damped oscillatory behavior is observed with multiple peaks over a period.

Figures (9)

• Figure 1: Variation of $R_0$ with $m_{12}$ for $m_{21} = 0.05$.
• Figure 2: Evolution of infected and recovered population for migration rate $m_{12}= 0.01$ and $m_{21}=0.02$
• Figure 3: Evolution of infected and recovered population for migration rate $m_{12}= 0.05$ and $m_{21}=0.1$
• Figure 4: Evolution of infected population for both cities for various migration rates.
• Figure 5: Evolution of infected population for both cities for various (reverse)migration rates.