Using Markov Chains to Predict the End of Lockdown in Shanghai
I. Introduction
Since the first confirmed COVID-19 case on March 1st, 2022, another wave of the COVID-19 pandemic hit Shanghai, China’s most populous city. Following China’s zero-COVID strategy, the leading industrial and manufacturing center embarked on an unprecedented, almost draconian lockdown. Amidst lockdown, most people are mandated to stay in their compounds, with only medical staff commuting between hospitals and residences for NAT testing. As a result, Shanghai’s unemployment rate rose to 6.1% in April, the highest level since the 6.2% peak in February 2020. Retail sales also shrank by 11.1% and industrial production by 2.9% in comparison to last year.
II. Methodology
To predict the end of the lockdown in Shanghai, we follow the zero-COVID strategy, modeling the number of COVID-19 cases to find when cases reach zero using Markov chains. A Markov chain is a stochastic model that describes the sequence of possible events in which the probability of each event depends on the previous event. As such, the construction of a Markov chain includes a stochastic matrix that represents the probability of each event occurring and an initial state matrix that describes the starting state.
In the case of the COVID-19 pandemic in Shanghai, we use data from the daily official reports of the Shanghai Municipal Health Commission from March 1st to May 15th, 2022 to build these matrices. For the purpose of this study, we also assume that the probability of each event remains constant throughout the lockdown. Our initial state matrix is a 2-by-1 matrix that includes the number of Shanghai inhabitants who are not diagnosed with SARS-CoV-2 and those who are diagnosed, which we set as 0 to represent pre-lockdown Shanghai.
Furthermore, for the stochastic matrix, we find the proportion of people who are not diagnosed with SARS-CoV-2 (No Covid to No Covid: ~0.9997), the proportion of people who recover (Yes Covid to No Covid: ~0.001), the proportion of people who are newly diagnosed with SARS-CoV-2 (No Covid to Yes Covid: ~0.0003), and the proportion of people who were previously diagnosed with SARS-CoV-2 and did not yet recover (Yes Covid to Yes Covid: ~0.999). For the calculation of these proportions, we use data on the total cumulative number of cases (all new cases subtracted by the number of recovered cases), the total number of new cases (symptomatic and asymptomatic), and the number of recovered patients. From these data, we calculate a daily SARS-CoV-2 infection rate and a daily recovery rate, whose average is the total proportion of people who are newly diagnosed with SARS-CoV-2 and the proportion of people who recover, respectively. Because the columns of the stochastic matrix must each have a sum of 1, we subtract each of the two proportions from 1 to find the proportion of people who are not diagnosed with SARS-CoV-2 (1 — No Covid to Yes Covid = No Covid to No Covid) and the proportion of people who were previously diagnosed with SARS-CoV-2 and did not yet recover (1 — Yes Covid to No Covid = Yes Covid to Yes Covid).
To find the steady-state vector, we solve the following system for x1 and x2:
We find that x1 = 0.799106 and x2 = 0.200894. According to the Shanghai Statistical Bulletin of the National Economic and Social Development, the total population of Shanghai in 2021 is around 24,894,300. This means that, in the long term, around 5,001,116 people will be diagnosed with SARS-CoV-2.
III. Conclusion
According to our Markov chain model, the number of COVID-19 cases in Shanghai will not approach zero if the proportion of people who are newly diagnosed with SARS-CoV-2 and the proportion of people who recover remain constant. As such, this has two possible implications. The rate of people getting diagnosed with SARS-CoV-2 must increase and the rate of recovery must decrease for the steady-state proportion of people diagnosed with SARS-CoV-2 to approach zero. This study also raises questions about the feasibility of the zero-COVID strategy given the current pandemic situation and the strictness of lockdown placed upon the economic center of China.