Its good to do better than guesswork, says S.Ananthanarayanan.

Lockdown has retarded the spread of COVID-19. When and how lockdown can be lifted is a question that administrations around the world are striving to answer.

In the current crisis, we do not have the benefits of smaller populations, lesser mobility and less globalized economics, which prevailed during past pandemics. However, we have more sophisticated capability in science and technology. A bit of math showed us where the spread of infection was headed - and got the world to act, with total lockdowns, more stringent than measures taken in the past. Math may again help optimize our crawl back to normalcy.

Thomas Rawson, Tom Brewer, Dessislava Veltcheva, Chris Huntingford, and Michael B. Bonsall, from the University of Oxford and the UK Centre for Ecology and Hydrology, Wallingford, near Oxford, describe in the journal, Frontiers in Public Health, mathematical modeling of how different approaches to lifting the lockdown may pan out. The conclusions are not different from what can be imagined, but the formal exercise puts down the numbers that help time the decisions and trim the sails as the situation develops. “Although the simulations may lack absolute precision, predictions will have some level of robustness,” the study says.

“As the number of new daily con?rmed cases begins to decrease, governments must decide how to release their populations from quarantine as ef?ciently as possible without overwhelming their health services,” the report says. Quarantine has high economic costs and cannot continue for long. But lifting restrictions too soon would lead to increasing numbers that would go out of control. The problem of timing is hence one of optimizing, which mathematical methods can solve.

The results show, as we can imagine, that release of the whole population would be disastrous. And gradual reintegration would be more reliable. Although the study deals with the case where the number of new infections has started falling, and daily new cases are below a threshold, the methods developed are useful to devise strategies even when cases are rising.

Optimisation problems are common in mathematics, economics, transport planning, etc., where the objective is to select the best from a range of solutions. Typical areas are in stocking raw materials – buying in bulk at one time would lock up capital. Repeated, smaller purchases have higher costs and may increase the risk of stocks running out. A Finance manager may need to keep cash assets in deposits and maximise interest. Short-term deposits would reduce the interest yield, while long-term, better paying deposits may lead to a liquidity crunch. The classic transportation problem is to plan the routes that city buses should follow, so that passengers have the least travel time and the least waiting time and cost, given the number of coaches, people and places to reach!

The Oxford group modelled a population facing an epidemic as consisting of four groups – the Susceptible, the Exposed, the Infected and the Recovered (or dead) – SEIR framework, to investigate the ef?cacy of two potential lockdown release strategies, focusing on the UK population as a test case. The gradual release strategy consists of allowing different fractions of those in lockdown to re-enter the working, non-quarantined population. Mathematical optimisation methods, combined with the SEIR model, lead to ways to maximise the number of those working while preventing the health service from being overwhelmed. The other strategy modelled was “on-off”, of releasing everyone, but re-establishing lockdown if infections become too high. The study concludes that the worst that can happen with a gradual release is more manageable than the worst possibility of an on-off strategy.

The modeling of how the numbers, S, E, I, R change with time was done separately for the population under quarantine and those who were not. In total lockdown, the second category consists of only front-line, essential workers, like medical staff, staff of utilities like power, water, etc.

The rate of change of the numbers was captured in eight relations - the number, S, of the susceptible, that is to say, those not yet exposed, would naturally reduce, depending both on the value of S and on the fraction of population that is infected. The number, E, of those exposed, would increase, depending on the same fraction as before, but tempered by its own increase. The same is true of I and R, depending on E and I, respectively, and tempered by their own increase. All these rates of change, separately for those in quarantine and the others, would then be affected by a factor that changes with time and reflects the policy of release of persons from quarantine.

The first factor of dependence to be considered, the transmission rate, of the conversion of the susceptible to the exposed, could be estimated from the data of Wuhan or Italy. As these rates, however, did not include the effect of asymptomatic persons, transmission rates that are twice as high have been considered. The relations take into account the natural death rate in the community and this is the rate that reduces the numbers of the recovered (R), even as the number increases as persons are cured. The lag, or the time that persons who are exposed (E), take to become infected (I) is considered, and then, how long the infected take to recover

Solving the relations, for different policies of release of people from quarantine to work gives us the progression patterns of numbers, of the rise in the numbers infected persons, who would need medical attention. The model can thus show us the highest work force that we can release, given the medical facilities available. In affluent societies, ample medical facilities may allow large numbers to be released. This would help the economy, no doubt, but may be at the cost of many lives. The numbers released would hence be lesser and the results would enable refining the factors considered and hence more sensitive control.

The model developed considers cases where the increase in infections has begun to slow, or after methods of containment have shown effects. This is true of many countries, Iran, South Korea, the UK, France, Italy. The model, however, can be used in India too, where the numbers are continuing to rise. We have three times the cases of China and are ahead of Italy. The daily increase, week to week, has risen every week, for the last ten weeks. Against the time for doubling, 35 days in UK and USA, 55 days in Italy and 56 days in Spain, it is a quick 15 days in India. There are reasons - differences in the method of testing, criteria of considering a person infected, etc., which make comparisons misleading, and that the testing in India is less, and limited to those at risk. And then, there is the large population – fertile ground for new infections. For all this, however, India’s rate of mortality is low, at 2.8%, compared to the global average of 6.8%.

Large population and rising numbers are reasons for great caution while relaxing the lockdown, which could lead to spiraling numbers. On the one hand, in conditions of crowding, as in slums, the lockdown may even prevent social distancing. On the other, it is mobility that gives wings to infection. The SEIR framework, adapted to Indian conditions, could chart an optimal course that is based on objective criteria.

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