A model of the spread of Covid-19

The reason for addressing this issue is twofold. On the one hand, Covid-19 is the most important challenge for humanity at the moment, but on the other hand the process of expansion of the virus is an example of how nature establishes models based on information processing.

The analysis of the dynamics of the virus expansion and its consequences will be based on a model implemented in Python, which for those who are interested can be downloaded, being able to make the changes that are considered appropriate to analyze different scenarios.

The model

The model is based on a structure of 14 states and 20 parameters, which determine the probabilities and the temporal dynamics of transitions between states. It is important to note that in the model the only vectors for virus spread are the “symptomatic” and “asymptomatic” states. The model also establishes parameters for the mobility of individuals and the rate of infection.

Some simplifications have been made to the model. Thus, it assumes that the geographical distribution of the population is homogeneous, which has contributed to a significant reduction in computational effort. In principle, this may seem to be a major limitation, but we will see that it is not an obstacle to drawing overall conclusions. The following figure represents in a simplified way the state diagram of the model. The conditions that establish the transitions can be consulted in the model.

The parameters have been adjusted according to experience gained from the progression of the virus, so information is limited and should be subject to further review. In any case, it seems clear that the virus has a high efficiency in infiltrating the cells to perform the copying process, so the viral load required for the infection seems to be small. This presupposes a high rate of infection, so it is also assumed that a significant part of the population will be infected.  

Scenarios for the spread of the virus can be listed in the following sections:

  • Early action measures to confine the spread of the virus  
  • Uncontrolled spread of the virus.
  • Exceptional measures to limit the propagation of virus.

The first scenario is not going to be analyzed as this is not the case in the current situation. This scenario can be analyzed by modifying the parameters of the model.

Therefore, the scenarios of interest are those of uncontrolled propagation and exceptional measures, as these represent the current state of the pandemic.

The natural evolution

The model dynamics for the case of uncontrolled propagation are shown in the figure below. It can be seen that the most important vectors in the propagation of the virus are asymptomatic individuals, for three fundamental reasons. The first is the broad impact of the virus on the population. The second is determined by the fact that it only produces a symptomatic picture in a limited fraction of the population. The third is directly related to the practical limitations in diagnosing asymptomatic individuals, as a consequence of the novelty and rapid spread of Covid-19.  

For this reason, it seems clear that the extraordinary measures to contain the virus must be aimed at drastically limiting contact between humans. This is what has surely advised the possible suspension of academic activities, which includes the child and youth population, not because they are a risk group but because they are the most active population in the spread of the virus.

The other characteristic of the spreading dynamics is the abrupt temporary growth of those affected by the virus, until it reaches the whole population, initiating a rapid recovery, but condemning the groups at risk to be admitted to the Intensive Care Unit (ICU) and probably to death.

This will pose an acute problem in health systems, and an increase in collateral cases can be expected, which could easily surpass the direct cases produced by Covid-19. This makes it advisable to take extraordinary measures, but at the same time, the effectiveness of these measures is in doubt, since their rapid expansion may reduce the effectiveness of these measures, leading to late decision-making.  

Present situation

This scenario is depicted in the following figures where quarantine is decreed for a large part of the population, restricting the movement of the propagation vectors. To confirm the above, two scenarios have been modeled. The first, in which the decision of extraordinary measures has been taken before the curve of diagnosed symptoms begins to grow, which in the figure occurs around day 40 from patient zero. The second in whom the decision has been taken a few days later, when the curve of diagnosed symptoms is clearly increasing, around day 65 from patient zero.

These two scenarios clearly indicate that it is more than possible that measures have been taken late and that the pandemic is following its natural course, due to the delay between the infected and symptomatic patient curves. Consequently, it seems that the containment measures will not be as effective as expected, and considering that economic factors will possibly have very profound consequences in the long and medium term for the well-being of society, alternative solutions should be considered.

It is interesting to note how the declaration of special measures modifies the temporal behavior of the pandemic. But once these have not been taken at an early stage of the virus’ emergence, the consequences are profound.

What can be expected

Obviously, the most appropriate solution would be to find remedies to cure the disease, which is being actively worked on, but which has a developmental period that may exceed those established by the dynamics of the pandemic.

However, since the groups at risk, the impact and the magnitude of these are known, a possible alternative solution would be:

  • Quarantine these groups, keeping them totally isolated from the virus and implementing care services to make this isolation effective until the pandemic subsides, or effective treatment is found.
  • Implement hospitals dedicated exclusively to the treatment of Covid-19.
  • For the rest of the population not included in the risk groups, continue with normal activity, allowing the pandemic to spread (something that already seems to be an inevitable possibility).  However, strict prophylactic and safety measures must be taken. 

This strategy has undeniable advantages. Firstly, it would reduce the pressure on the health system, preventing the collapse of normal system activity and leading to a faster recovery.  Secondly, it would reduce the problems of treasury and cash management of states, which can lead to an unprecedented crisis, the consequences of which will certainly be more serious than the pandemic itself.  

Finally, an important aspect of the model remains to be analyzed, such as its limitation for modeling a non-homogeneous distribution of the population. This section is easy to solve if we consider that it works correctly for cities. Thus, in order to model the case of a wider geographical extension, one only has to model the particular cases of each city or community with a time lag as the extension of the pandemic itself is showing.

One aspect, namely the duration of the extraordinary measures, remains to be determined. If it is considered that the viral load to infect an individual is small, it is possible that the remnants at the end of the quarantine period may reactivate the disease, in those individuals who have not yet been exposed to the virus or who have not been immunized. This is especially important considering that cured people may continue to be infected for another 15 days.

1 thought on “A model of the spread of Covid-19

  1. Pingback: Covid-19: Interpretation of data | Information and Reality

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s