key: cord-0538040-8f62m7aa
authors: Li, Quan-Lin; Wang, Chengliang; Xu, Yiming; Zhang, Chi; Chang, Yanxia; Wu, Xiaole; Fan, Zhen-Ping; Liu, Zhi-Guo
title: Infections Forecasting and Intervention Effect Evaluation for COVID-19 via a Data-Driven Markov Process and Heterogeneous Simulation
date: 2021-01-07
journal: nan
DOI: nan
sha: 2a662c8b743755c9e21f35315c989511048c14eb
doc_id: 538040
cord_uid: 8f62m7aa

The Coronavirus Disease 2019 (COVID-19) pandemic has caused tremendous amount of deaths and a devastating impact on the economic development all over the world. Thus, it is paramount to control its further transmission, for which purpose it is necessary to find the mechanism of its transmission process and evaluate the effect of different control strategies. To deal with these issues, we describe the transmission of COVID-19 as an explosive Markov process with four parameters. The state transitions of the proposed Markov process can clearly disclose the terrible explosion and complex heterogeneity of COVID-19. Based on this, we further propose a simulation approach with heterogeneous infections. Experimentations show that our approach can closely track the real transmission process of COVID-19, disclose its transmission mechanism, and forecast the transmission under different non-drug intervention strategies. More importantly, our approach can helpfully develop effective strategies for controlling COVID-19 and appropriately compare their control effect in different countries/cities.

method to evaluate the basic reproductive number 14 . The transmission of COVID-19 was found to be seasonal 15, 16 . Also, the infection of COVID-19 becomes really terrible at a very low temperature in the winter 17 . In the beginning of 2021, the COVID-19 has become even more active and serious in many countries. Therefore, the prevention and control of COVID-19 in the world will continue facing huge challenges in 2021.

So far, we have found that the prevention and control of COVID-19 include four non-drug intervention strategies [18] [19] [20] [21] [22] [23] : Travel restrictions [24] [25] [26] [27] , early identification and isolation [28] [29] [30] [31] [32] [33] [34] [35] [36] , increasing social distance [37] [38] [39] [40] , and exposure restrictions 30, [41] [42] [43] [44] . It was indicated that we will have to live with COVID-19 in the decades ahead 45 . Thus, the non-drug intervention strategies can play an important role in the prevention and control of COVID-19 transmission due to the fact that no effective drug is found for treating COVID-19.

To describe and analyze the complex transmission of COVID-19, this paper is devoted to developing an approach for evaluating and predicting the transmission processes of COVID-19, which is modeled as a Markov process with explosive structure. Our model needs four basic parameters to capture the basic characteristics of the COVID-19 transmission process: the explosive growth of patients, and the batching infection of an infected person. These four parameters can be estimated based on public data (https://github.com /CSSEGISandData/COVID -19) . We also propose a simulation approach with heterogeneous structure based on the proposed Markov model and the four parameters. Experimentations on the cases of six countries/cities show that our approach works well in tracking the real transmission process of COVID-19.

We set up a novel block-structured Markov process to describe and analyze the Our Markov process of COVID-19 captures several key random factors in the transmission process of COVID-19. By applying the theory of Markov processes, the random factors can be determined based on the framework of exponential distributions and/or Poisson processes. Then, the random factors can be further simplified to four key parameters as follows: Fig. 1 . From Fig. 1 , we define the sets of states as follows: Using the different levels, the Markov process

is a QBD (quasi-birth and death) process whose infinitesimal generator is given by

where , is a matrix of suitable size based on the sizes of the above levels from Level to Level .

position of vector ω , while all the other elements are zero.

We define 

Note that the initially patient number k can be directly given from the actual data on the first day of our COVID-19 observation.

In order to take the value of the infection batch size, d, we can consider the patients as two groups with different batch sizes. Then, the overall batch size is the weighted average of these two groups. The weight values can be approximately designed according to the number of existing patients in the different groups. To better show this idea, let k represents the number of the overall patients, and i k represents that of group i (i =1,2). Then, the weight of group i can be determined Fig. 1 for more details.

The Markov process 

We approximately infer the two weights, i r , of the batch size of each group, and by adjusting their values, we can also make

close to the real values.

In this section, we verify the effectiveness of our Markov method by simulating for several countries. Assuming that the parameters remain stable during the short-term observation period, we use the historical data in the observation period to predict the trend of active cases in the future. After that, the impact of three key parameters (i.e., the initially observed number k , the expansion batch size d and the infection rate  ) on the transmission of COVID-19 is discussed. Note that the three parameters can be obviously influenced by the non-drug intervention strategies.

Our observation and analysis for the three parameters in the transmission process of COVID-19 is studied by means of our Markov processes of COVID-19; while some necessary and useful data of COVID-19 is taken up to the end of December, Fig. 3 .

We further simulated the growth processes of India in Asia (Nov. 1 to 20, 2020, r =0.566, respectively. These parameters' decrease clearly reflects the effect of the non-drug intervention strategies on the control of COVID-19 transmission. From Fig. 9 , it can be seen that the influence of the initial value of active cases, k, on the newly infected cases is smaller, compared to the explosive transmission process within the next periods; and the influences of the infection rate,  , and the infection batch size, d, are significant.

Therefore, it is better to reduce  and d, rather than k in order to control COVID-19 transmission effectively. To reduce the infection rate  , the infection batch size d, and the initial value of active cases k, it is observed from Figures 9 and 10 that we need to further sufficiently adopt some non-drug intervention strategies, such as travel restrictions, community isolation, reducing parties, wearing masks, etc.

In this research, we develop a Markov system to model the COVID-19 transmission process. To the best of our knowledge, the transmission of COVID-19 has been mainly studied by SIR models and their extensions in the existing research.

Our research represents the first time that the theory of Markov processes is used to study such a problem. That is, we propose a new direction in the study of COVID-19

transmission. We believe that the methodology and results given transmission. In addition, in the history of applications of Markov processes, our research is the first one to combine multiple Markov processes to describe and analyze a practical stochastic system.

In our study, we consider three values of the infection batch size, d: transmission.

The transmission intensity of COVID-19 in a country or city may change over time, because of the adoption of the non-drug intervention strategies. This can be seen from the change points of these parameters in the transmission process of COVID-19.

These change points quantitatively evaluate the control effect of non-drug intervention strategies. The change points can be set in different groups, which enables our simulation to well reflect the short-term transmission processes of COVID-19 under non-drug intervention strategies.

Note that the simulation of the transmission process of COVID-19 by using our

Markov system can fit the actual situation of COVID-19 very well, and our method can also be well applied to the prediction of transmission processes of COVID-19.

Such a forecast can be developed in two types: (1) No intervention strategy. We focus on predicting the future transmission process using our simulation approach based on changing the value of the initial infection size k, while keeping the other three parameters (  ,  , and d ) unchanged. (2) The non-drug intervention strategies are adopted. When there are non-drug intervention strategies are considered to implement, we can roughly forecast the trend of the resulted future transmission process. This trend can be used to help decision making on COVID-19 intervention strategies within a country/city. Our Markov system and associated simulation technique can be used to support controlling the transmission processes.

Currently, some countries have adopted the COVID-19 vaccines. Therefore, our future research efforts will be devoted to adapting the proposed approach to not only evaluate the effect of the vaccines on the prevention of COVID-19, but also predict the new transmission trend during the future process of COVID-19 mutation. 

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