Using maths to combat COVID-19

‘The pandemic has now reached a level in which no human can make an optimal decision without the aid of a computer. Therefore, we need to start working on quantitative models to identify optimal decisions, instead of pointing fingers for not making proper decisions, when decision making is literally beyond the capacity of a human.’ – Senior Lecturer, Department of Mathematics, University of Colombo,    

Dr. Anuradha Mahasinghe

by Sajitha Prematunge

What has math got to do with a pandemic? At the outset, it might seem the two are completely unrelated. One has only to observe that the number of infected in certain districts is higher than that of others and making informed decisions based on those numbers could mean the difference between stifling a cluster and a full-blown third wave. Senior Lecturer, Department of Mathematics, University of Colombo, Dr. Anuradha Mahasinghe knows only too well how important the numbers are in combating COVID-19. This is why, in June, he and his colleagues proposed an optimization model, aimed at minimizing the damage to the economy, while confining the COVID-19 incidence to a level endurable by the available healthcare capacity in the country, while their compartment model projected COVID-19 transmission. Their study investigated the effectiveness of the control process with the aid of epidemiological models.

Epidemiological models

Mahasinghe explained that an epidemiological model is a model that simulates and describes an epidemic. “Modelling is essential if you want to describe a phenomenon. From the spin of an electron to the rotation of the heavenly bodies, these phenomena are understood with the aid of models. It is the models that help us describe the changes in economies and fall of financial markets,” said Mahasinghe. And pandemics are no exception. He explained that an epidemiological model, based on a reasonable theory and supported by evidence, is a collection of entities and their operations, that when put together simulate and describe an epidemic, which provides important insights into the transmission of the disease.

But how does maths factor comes in when dealing with a pandemic such as COVID-19? Mahasinghe pointed out that math is inevitable whenever dealing with numbers or quantities. “Aren’t we really sensitive to numbers in this COVID era more than ever? Every person is anxious to know the numbers of reported cases and deaths.” One might say the numbers are governing us because decisions are also made based on these numbers. However, these numbers are only the smoke, warns Mahasinghe. “One should be able to make a better decision if he sees the fire. Therefore, to make the best decisions during the pandemic you have to look into a mathematical model that can best describe the phenomenon.”

Numbers may govern us but what governs the numbers? According to Mahasinghe, this can only be uncovered by a model that captures the quantitative aspects of the pandemic. “It’s what we call a mathematical model which provides us with an explanation to the occurrence of these numbers.” Such a model can also forecast how these numbers are going to change in the future.

But how credible are these models? Hopefully, they are nothing like the local weather forecast. “Such models are based upon very fundamental and well-accepted laws in nature such as energy conservation, which cannot be falsified,” reiterated Mahasinghe. “Such models are supported and validated by empirical evidence. People use such models very often to make decisions in industry to make profit. So, why not look into the numbers and the math behind them to make optimal decisions accordingly, in a pandemic scenario?”

Where we went wrong

When asked where Sri Lanka went wrong in attempting to contain the pandemic, Mahasinghe said, “I guess we didn’t see the fire, we only saw the smoke. More precisely, we didn’t pay enough attention to the transmission dynamics or to optimal decision making. We were able to make some good decisions in a qualitative sense, but I seriously doubt we had the insight to make quantitatively sound decisions.” He pointed out that even when the decisions were made, the outcome could not be predicted due to a lack of a mechanism to forecast.

When asked whether the authorities were too quick to lift the lockdown, Mahasinghe answered in the negative. “I don’t think it was too early. Lifting strict lockdowns was essential at that moment. We were struggling to achieve two conflicting goals; containing the disease and sustaining the economy. Stepping down from strict curfew to partial lockdowns is indeed a good decision in such a context.” But was it methodical? Was there a mechanism to decide on the nature of the partial lockdowns? Did we know how to optimally restrict mobility in order to achieve those conflicting goals? Did we know to what extent the lockdown of a district should be optimally eased? Did we estimate the potential increase in positive cases from a district when its lockdown would be relaxed? Did we know the magnitude of the economic loss caused by shutting down a region? These are questions Marasinghe believes that authorities should have paid attention to, when easing the lockdown.

“Lockdowns could have been relaxed with the aid of proper optimization models capable of providing answers to these questions.” He repeated that such models are used very often in industrial decision making and provide promising solutions. “You can’t bring the COVID incidence down to zero even with such models, but at least you know what’s going on and the effectiveness of a decision so the health sector can take relevant measures.

According to Mahasinghe, authorities have overlooked the significance of data. “Even now I don’t think enough attention is paid to data.” According to him, some important data were not gathered. For example, he pointed out that, despite Western Province residents being advised against crossing borders, some invariably did, as there were no strict rules against it. “There is no point in regretting the fact, but we could have counted the number of vehicles that crossed the borders and used it to estimate the impact on transmission.”

He explained that the entire country can be regarded as an epidemiological network, where the nodes are the cities and the interconnections are the roads. “There are elegant models in network theory to gain many insights into transmission through such a network.” He also noted another pertinent issue, that even if data were gathered, they were not used. “Much effort was made to gather and organize COVID related data such as incidence per region etc, and that is really commendable. However, have we used them; what were the insights we gained into transmission from them except for some trivial speculations?” questions Mahasinghe. He reiterated that such insights can only be gained through an extensive study that involves the collaboration between mathematicians, computer scientists, epidemiologists and economists. “The mathematician’s part alone includes exhaustive algorithmic development and computational modelling challenges,” explained Mahasinghe.

Criteria

When asked what factors were taken into consideration in their optimization model, Mahasinghe reminded that a delicate balance must be struck between two conflicting goals. “We need to find the optimal compromise between containing the disease and sustaining the economy. As a developing country, we can’t afford beyond a certain level of the control process, so budgetary constraints must be considered.” It is obvious that COVID-19 is transmitted through human mobility. He pointed out that, consequently, inter-regional travel plays a significant role. “On the other hand, transmission dynamics can be modelled to a certain extent by well-known compartment models. However, human mobility affects the compartments and the relevant model has to be moderated accordingly to reflect that reality.” The optimization model considers all factors, such as medical capacity to deal with the pandemic, economic concerns, transmission dynamics, regional contribution to the economy, and generates a lockdown relaxation strategy that keeps the level of incidence below a desired threshold, while minimizing damage to the economy.

However, Mahasinghe pointed out that this was a prototype and it can be made closer to reality by incorporating more constraints. “For instance, I haven’t considered the fact that most agricultural activities are done in the North Central province. But, if required, that too can be incorporated without difficulty.” According to him epidemiologists and economists can introduce more constraints to the optimization model, and the applied mathematician’s job is to overcome the computational challenges posed by incorporating them.

Relevant?

There is no point in closing the stable doors after the horse has bolted. Months after the lifting of the lockdown are such models even relevant? “The compartment model that captures the transmission of COVID-19 is still applicable, irrespective of any lockdowns, unless it is quite certain that there is absolutely no community transmission. I think we were in such a stage only at the very beginning of the first wave,” said Mahasinghe. According to him, the network-based model that captures human mobility is also applicable irrespective of lockdowns or any other preventive measures. In contrast to these, the optimization model is applicable in its existing form only when lockdowns are in force. “Having said that, this model may still be useful with some changes in the present context where small regions are isolated. For instance, a slightly changed variant of that model can determine which areas should undergo isolation. Moreover, it is possible to modify the optimization model further to be used in the process of making decisions on identifying the persons to be quarantined.”

Human mobility is a critical factor in the spread of a pandemic as well as any models targeted at managing such, how could a mathematical model factor this in? “Not only COVID-19 but even dengue is transmitted mainly due to human mobility. A mosquito doesn’t travel very far during its lifetime. Humans are more responsible for carrying diseases.” Mahasinghe pointed out that COVID-19 is not very different. “If you know the way humans move from place to place, and also know the level of incidence in each place, it is not that difficult to model how the disease is transmitted through humans.” He observed that most preventive measures are also focused on restricting human mobility, which he deemed commendable. “A mathematical model can prescribe the optimal way to restrict mobility.”

What are the implications of mobility? For example are people of certain districts more inclined to travel and therefore may contribute more to the spread of the disease and are such implications reflected in the numbers? “As long as the model is deterministic and you can overcome the computational challenges by necessary algorithm development, closed-form and conclusive solutions can be generated.” Mahasinghe implied that math helps to see the big picture. “Consequences of travel from the Western to other provinces is obvious. However, considering the transport network, Southern and North Western Provinces are also at high risk.” He observed that less attention has been paid to those regions. This begs the question, are the Southern and North Western provinces a time bomb waiting to go kaboom? He reiterated that special attention must be paid to regions that are relatively less danger, such as North Central, which contributes significantly to economic growth, as the Western province is not capable of contributing to the economy in its full capacity. “It is important to keep the incidence at a low level in such places.”

The study predicted that easing lockdown in the Western Province would have adverse repercussions. “As long as vehicles cross inter-provincial barriers, the disease is transmitted to those regions. But in what magnitude? We had access to certain transport data, so we knew to a certain extent how people would mobilise within the country. Also the epidemiological data were available. So we had enough inputs to be fed into our algorithm.” The results were appalling. In fact, this computer experimentation was done in the early days when Sri Lanka was hit by the first wave and there were no strict measures to curtail inter-provincial mobility. During the days in question, Mahasinghe ranked the provinces according to their vulnerability to COVID-19, using another model, by adopting some ideas from network theory. Recently, upon perusing a map that indicated the countrywide spread of the disease Mahasinghe came to realize that the ranking has been validated, eventually. “What I don’t understand is why we failed to foresee this.”

Mahasinghe and his team had access to certain transport data, such as the number of buses, trains and bus routes. However, his models were prototypes. To make the prediction more accurate they would need current transport data, such as the number of private vehicles crossing provincial borders. “There are a number of police barriers between borders, so a vehicle count would not be impossible. If health planners are willing to use that type of model, these could be extremely valuable datasets.”

Quantifying the qualitative

In their model they quantify the degree of social distancing. But can criteria so human in nature be quantified? Moreover, how can something as complex as a pandemic, with so many variables, human in nature, be simplified into ones and zeroes? Mahasinghe maintained that it is possible to estimate the degree of social distancing observed, if provided with sufficient data. “I understand that it sounds quite unrealistic. It is because we think of individuals.” Mahasinghe emphasised the importance of noting that they are not modelling an individual, but rather a population. “Though a population consists of individuals, the dynamics of the population is not merely the sum of the dynamics of an individual. When you single out a person, the behaviour of that person is surely very uncertain and unpredictable. Take two persons, they may have certain things in common, so it is not that unpredictable. If you take a thousand people, a lot of commonalities can be extracted and the situation becomes predictable now.” He explained that, therefore, it is possible to assign a value to the degree of distancing with the aid of necessary data.

“Interestingly, it is true that we mathematicians seek certainty in an uncertain world. However, an event that looks uncertain from one point of view looks certain from another.” The toss of a coin is a simple example. “If you toss a coin, the outcome of it being head or tail is widely believed to be uncertain. However, it is the lack of data that makes it uncertain. Suppose the initial speed, the weight, the angle of projection and such were provided, then the outcome may be predictable by basic equations of motion.” Mahasinghe emphasised that math does not guarantee elimination of uncertainty. “That is definitely not the direction the mathematical sciences are moving, specially with the recent developments in quantum physics and unconventional computing. However, where macroscopic events like pandemics or human behaviour are concerned, there are many certainties that we misinterpret under the cover of uncertainty due to our lack of knowledge, eventually missing an opportunity to gain crucial insights into the scenario.”

Mahasinghe pointed out that many decisions are binary in nature. Let alone policy decisions, many behavioural decisions are inherently binary. “For instance, you may decide whether to wear a mask or not. So the one-zero nature of the action is inherent and not artificially imposed by a mathematician.” He further explained that some non-binary decisions can still be quantified. “For instance, if you decide to wear the mask on three days and go unmasked on four days of the week, it can be quantified using numbers and interpreted using probability.” Mahasinghe elaborated that, with recent developments in non deterministic models, applied mathematicians do not hesitate to incorporate uncertainty. “Consequently, uncertainty is no longer immeasurable. It is possible to confine uncertainty of the solution within reasonable limits.

What next

With all the talk on vaccination, Mahasinghe emphasised the importance of developing two mathematical models prior to vaccination. The first is a compartment model that explains the post-vaccination dynamics of the disease. “This is pretty standard in mathematical epidemiology. The second, developing a model to capture the effects of the interactions between individuals and predict the outcomes, is subtler and challenging.” He explained that once a phase of vaccination is over, persons in society can be divided into two categories: vaccinated and unvaccinated. “Take a random encounter between two persons. What type of interaction would it be? Is it a vaccinated encountering another vaccinated, an unvaccinated encountering another unvaccinated or a vaccinated encountering an unvaccinated? Obviously, the consequences of these encounters are essentially different.”

The discipline of mathematics referred to as game theory is a promising tool in modelling this type of scenario and forecasting the outcomes. In addition, once vaccination commences, there will be the issue of free riding. Due to different reasons, some people in the high risk category will also choose to remain unvaccinated, eventually resulting in a significant number of potential free riders. Mahasinghe explained that this has already been addressed in the game theory in particular, under evolutionary games. “As a nation we can’t be content with an elementary formula for herd immunity. Instead, we need to develop and upgrade elegant vaccination strategies using compartment models and game theory.” Mahasinghe is of the view that, in this pre-vaccination phase, these two are the immediate concerns that need to be addressed by applied mathematicians.

Benefits

When asked what are the drawbacks of not using a mathematical model are and the benefits of using one, Mahasinghe pointed out that in a scenario of conflicting goals and monetary restrictions, it is impossible to make decisions without seeing where the optimal compromise is. “It is easy to put the blame on politicians and other policy makers for not making the right decisions, but how can a human make an optimal decision in this entangled web of parameters, conflicting goals and constraints? Plainly speaking, we need computers to generate the best decisions for us.” That’s indeed what the computers are intended to do primarily, according to Mahasinghe, although they are more frequently used to watch YouTube videos and log into Facebook!

But to perform the intended task using a computer, models and algorithms that can be read by the computer must be created. “That’s why you need to look into optimization, mathematical programming, computational modelling and game theory. This way, you may be able to keep the numbers within certain limits. Also, you can pre-assess a decision quantitatively. Our health workers and armed forces have already committed much and continue to do so and to receive the full benefit of their commitments, the willingness to switch from qualitative to quantitative methods, is essential.

When asked if such models are used successfully in other countries to counter the pandemic, Mahasinghe answered in the affirmative. Since the very beginning, an extensive mathematical modelling process has been done and that’s how the predictions were made. In fact, vaccination models had long been applied to control epidemics even in African countries. In Sri Lanka, there are many misconceptions about mathematical models.” Mahasinghe has observed certain non-mathematicians presenting elementary regressions, numerical approximations and statistical tests, erroneously referring to them as mathematical models.

“Perhaps that’s why some policy makers have lost faith in math. As mentioned earlier, a mathematical model is based on an unfalsifiable conservation law. It cannot be compared to a trivial curve fitting cakewalk. Our people get easily carried away by exotic words. People tend to admire words like machine learning, artificial intelligence and such, but how many are aware of the maths behind these words?” He observed that a closer examination of news reports on machine learning or AI being used in some country to counter the pandemic, would reveal that they are mathematical models and machine learning techniques are used due to the toughness of generating a closed-form solution. “Even to apply computational heuristics, the problem has to be formulated mathematically. Correct problem formulation is a major component of a so-called AI-powered decision.

Mahasinghe explained that the subject of operations research emerged in the new industrial era to enable industrial decision making using computers, as the number of industrial parameters exceeded human ability to process. “The pandemic has now reached this level so that no human can make an optimal decision without the aid of a computer. Therefore, we need to start working on quantitative models to identify optimal decisions, instead of pointing fingers for not making proper decisions, when decision making is literally beyond the capacity of a human.”