Mathematical Models in the Pandemic

Written by: He Jizhao (21-U5)

Designed by: Cheng Zhi Shan (22-U1)

In the time of the COVID-19 pandemic, mathematical modelling can be used as a tool in many areas to deal with the pandemic. In this article, I will talk about the use of statistics and probability involved in models with regards to  pandemic prediction and prevention. 

The presence of statistics is now evident in our daily life, which has been totally changed by the relentless pandemic. The number  of confirmed cases and deaths in the daily news has influenced people’s moods and social norms. Figure 1 is one of many examples causing much anxiety and fear worldwide  by presenting  statistics such as the total number of confirmed cases and the number of global deaths.

Figure 1: COVID-19 Dashboard by the Centre for Systems Science and Engineering at John Hopkins University

Different countries have implemented various measures to collect data of confirmed cases, such as  places y visited, so as to reduce the transmission of the COVID-19 virus. Technology has been  widely used in gathering and recording data during the pandemic period. For instance, Israel’s security agency Shin Bet is using citizens’ cell phone location data to monitor the movements of those infected. The Singaporean government has also made it compulsory for citizens to use a trace-together app or trace-together token for contact tracing during the pandemic to record places that a person has been to. If, unfortunately, one is diagnosed with the virus, the Ministry of Health (MOH) can immediately alert those in  close contactto do a swab test and take relevant measures to prevent the spread of the virus in the community.

To combat COVID-19 effectively, many scientists have been studying the data of confirmed cases as early as the beginning of the pandemic. Through thorough analysis of data collected in terms of age group, population density and human-controlled factors like social distancing, many conclusions, like older adults having increased risks of contracting more lethal symptoms  from COVID-19, have arrived opportunely to guide policy-making. Statistical modelling also plays a significant role in determining the finer details of  policies such as  the distance between people stipulated in social distancing and the maximum number per group during an outing. Thanks to statistical modelling, these measures are effective in curbing the spread of the coronavirus.

However, it is undeniable that statistical modelling is not without its limitations. To make sense of our complex real world, several assumptions have to be made before setting up a model. Inevitably, this will lead to inaccurate predictions. Moreover, reasons like poor data collection, high sensitivity of estimates, and lack of incorporation of epidemiological features also affect the accuracy of the results.

Despite these limitations, through data analysis, scientists can still  make predictions about the trend of  virus  development in the near future. In Figure 2, using China as an example, we can see that the gradient of the line representing confirmed and suspected cases is decreasing to zero, which means that the current situation in China is getting more and more stable. The gradient of the line, representing the currently active cases, tends to decrease, which also means that the situation is getting stable since there are fewer active cases confirmed.

Figure 2: Model shows the trend of novel coronavirus cases in China. Data from Feb 09,10 and 11 were removed from the model because the cases were underreported and then included all at once on Feb 12. 

OK but…how does it work?

To analyse the spread of the virus, the SIR(Susceptible, Infectious, Recovered) model has been  introduced. Based on the SIR model, scientists have introduced an important index to estimate the number of second-generation affected cases caused by a first-generation confirmed case. This index is called the Basic Reproduction Number (R). If the value of R is greater than one, it shows that the probability of extinction is low and the virus will spread globally. However, if R is smaller than one, the probability of extinction is high and the pandemic will gradually die down and disappear over time. For a modelling process using the SIR model, the calculation of the index R will be based on the formula R=1+r/b, where r is the rate of the increase of cases and b is the rate of the virus becoming infectious when in human bodies. 

According to Figure 3 below, taking Austria and Brazil as two examples to illustrate the use of index R in the analysis of the COVID-19 situation, we can see that the reproductive number line (the pink line) is under the green line of R=1 for Austria after 30th March while the reproductive line for brazil is above R=1. This means that the situation in Austria is getting stable while the pandemic still seems to be severe in Brazil. We can also see that the gradient of the reproductive line suddenly increases when it is close to R=1 at the end of March in Brazil, this provides  information that suggests countries should  be more alert in dealing with COVID-19 as the situation may get worse unexpectedly. 

Figure 3: Curves of daily confirmed cases and instantaneous reproductive number of some countries with the most cumulative cases

With the help of R, scientists can also estimate the effectiveness of vaccination by calculating the new value of R after people are vaccinated. This is decided by comparing the vaccination coverage rate to the marginal value of 1-1/R, when the value of the vaccination coverage rate is greater than this marginal value, the Efficient Reproductivitive Index will be less than one. This indicates  that the probability of extinction is high and hence, we  will be  more likely  to resolve  the spread of the Covid-19 . 

In conclusion, the application of mathematical modelling in pandemic prediction and prevention has made the strategies for dealing with the virus more time-efficient and effective. As scientists know more about the coronavirus, the model built will become more accurate in predicting future trends to better inform policy-makers of relevant measures required to curb the spread of  transmission. It is also critical to note that the parameters used in the model have to be changed as the COVID-19 situation evolves. Only by keeping pace with the dynamic situation can mathematical models remain relevant and accurate allowing us to  take effective measures to emerge victorious  in the long war against COVID-19!


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