A Continuous Markov-Chain Model for the Simulation of COVID-19 Epidemic Dynamics

Abstract

To address the urgent need to accurately predict the spreading trend of the COVID-19 epidemic, a continuous Markov-chain model was, for the first time, developed in this work to predict the spread of COVID-19 infection. A probability matrix of infection was first developed in this model based upon the contact frequency of individuals within the population, the individual's characteristics, and other factors that can effectively reflect the epidemic's temporal and spatial variation characteristics. The Markov-chain model was then extended to incorporate both the mutation effect of COVID-19 and the decaying effect of antibodies. The developed comprehensive Markov-chain model that integrates the aforementioned factors was finally tested by real data to predict the trend of the COVID-19 epidemic. The result shows that our model can effectively avoid the prediction dilemma that may exist with traditional ordinary differential equations model, such as the susceptible-infectious-recovered (SIR) model. Meanwhile, it can forecast the epidemic distribution and predict the epidemic hotspots geographically at different times. It is also demonstrated in our result that the influence of the population's spatial and geographic distribution in a herd infection event is needed in the model for a better prediction of the epidemic trend. At the same time, our result indicates that no simple derivative relationship exists between the threshold of herd immunity and the virus basic reproduction number R₀. The threshold of herd immunity achieved through natural immunity is significantly higher than 1 - 1/R₀. These not only explain the theoretical misconceptions of herd immunity thresholds in herd immunity theory but also provide a guidance for predicting the optimal vaccination coverage. In addition, our model can predict the temporal and spatial distribution of infections in different epidemic waves. It is implied from our model that it is challenging to eradicate COVID-19 in the short term for a large population size and a wide spatial distribution. It is predicted that COVID-19 is likely to coexist with humans for a long time and that it will exhibit multipoint epidemic effects at a later stage. The statistical evidence is consistent with our prediction and strongly supports our modeling results.

Keywords: COVID-19; Markov-chain model; herd immunity threshold; mutation; reproduction number.

Figure 1

An illustration of a constraint-based Markov-chain model with a distance threshold (A) and an illustration of a constraint-free Markov-chain model (B).

Figure 2

Epidemic trend predicted by three different models.

Figure 3

Herd immunity threshold predicted by three different approaches.

Figure 4

Predicted infection probability at different vaccination coverage percentages using a simplified Markov-chain model.

Figure 5

Epidemic trend predicted by the Markov-chain model with the consideration of multiple factors (A) and the predicted epidemic distribution at time point B (B), at time point C (C), at time point D (D), at time point E (E), at time point F (F), at time point G (G), at time point H (H), and at time point I (I).

Figure 6

An implication of our prediction for the real-world scenario: an interpretation of the epidemic in the United States.