Modeling the spread of Covid-19 pandemic: case of Morocco

Coronavirus: definition

Coronaviruses are a family of many diverse and numerous viruses that can infect both humans and animals; it can produce a number of diseases (WHO, 2020)^[1]. The name coronavirus, which means “crown virus” is related to the fact that all viruses of this family have a crown-like shape when observed under an electron microscope.

Although the first disease caused by this type of virus was discovered in 1930 in the United States of America, nevertheless, it did not infect humans until the mid-1960s. Since that time, the virus has continuously developed and causes each time a further disease due to genetic changes or the high mutation rate of the virus. Such mutations have led to the development of more serious infectious diseases to the human being such as Severe Acute Respiratory Syndrome (SARS), Middle East respiratory syndrome (MERS), and Covid-19.

Covid-19: from epidemic to pandemic

Initially detected in Wuhan (China) in December 2019, the outbreak caused by Covid-19 spread rapidly around the world and affected several countries such as South Korea, Iran, Italy, Spain, and France. Indeed, the World Health Organization (WHO) estimated that Covid-19 could be qualified as a pandemic as early as March 11th 2020, since the number of Covid-19 cases detected outside China increased 13-fold to more than 118,000 cases with 4,291 deaths in the first two months alone. At that time, the number of affected countries reached 114 countries.

In April 2020, this highly contagious virus spread to the five continents of the Earth. It contaminated more than 1.3 million people in more than 180 countries, causing a heavy death toll of more than 75.000 deaths, and about 6% as mortality rate (See Figure 1).

Figure 1:

Worldwide number of Covid-19 contaminated cases.^[2]

Infection process

The new variant of the coronavirus has a high speed of transmission from one person to another. Covid-19 is transmitted primarily through close contact with an infected person (e. g., after shaking hands or through the air when coughing or sneezing). But what makes this virus dangerous is the fact that it remains active for several hours (or even days) on a different surface. Touching a surface contaminated with the virus and then touching the mouth, nose, or eyes with hands will transmit the virus. Climatic conditions can also have a significant impact on the wide spread of the disease. The Covid-19 assumed to develop more in the cold than in dry areas. In France, the High Council of Public Health (HCSP) notes in an opinion issued on March 05th 2020^[3] “that coronaviruses survive up to 3 h on dry inert surfaces and up to six days in wet environments”.

The next figure schematizes the different phases of the disease, its process of contagion, and virus transmission between individuals: (See Figure 2).

Figure 2:

Contagion process representation.^[7]

During a pandemic such as Covid-19, determining the incubation period (latent period) is critical to understanding its transmission dynamics even before the onset of symptoms. According to the WHO, the incubation period is “the time between infection and the appearance of symptoms of the disease.” ^[4] For the case of Covid-19, WHO estimates that this period usually lasts about 5 days, but can vary between 1 and 14 days.

However, the contagious period of Covid-19 (even before the onset of symptoms) is not easily determined. According to Alexandre Bleibtreu, an infectiologist at the Pitié-Salpêtrière in Paris, “We also take by analogy with other viral diseases where it is generally considered that in the 24 h preceding the onset of symptoms, the viral load is sufficient to be contagious.” ^[5]

Also, Covid-19’s recovery time is not really fixed. According to Dr. Gérald Kierzek, “after a fortnight when you have mild symptoms, the cure is spontaneous except when there is a serious form and you end up in resuscitation where you need oxygen, where you have pneumonia and then you need medication. Resuscitation patients stay in the hospital for a long time, which can be more than 20 days.” ^[6]

Data set

On March 02nd 2020, Morocco officially detected the first case infected by the new coronavirus. Since that time, the Covid-19 pandemic has continued to develop, mainly in large cities such as Meknes, Casablanca, Fez, and Marrakech. On March 16th, the government decided to close down all the schools, restaurants, cafés, etc. On March 20th, it declared a health emergency until further notice.

The number of infected cases^[8] increased exponentially during the month of March, rapidly rising to more than 555 infected cases by the end of March 2020. But this number almost doubled in the first six days of April 2020. Parallel to this development, the death toll is relatively high. The number of people who died from this virus exceeded 80 people on April 06th 2020, a mortality rate of 7.1% (See Figure 3).

Figure 3:

Evolution of the number of people infected by Covid-19 in Morocco until April 08th 2020.^[9]

The analysis of the temporal evolution of the Covid-19 virus as well as its transmission speed in the Kingdom will enable us to anticipate its future development and to forecast the expected end date of this pandemic.

Modeling

Covid-19 pandemic modeling has three main objectives:

• To understand better the virus chains transmission.

• To make reliable predictions about the imminent development of the disease.

• To measure the impact of the measures taken by the public authorities to control the rapid spread of the virus.

• To predict the period of time needed to stop the pandemic.

In epidemiology, the calculation of contagion probabilities is based mainly on compartmental models. These models divide the population into epidemiological classes. Moreover, they have proved their effectiveness in the study of several epidemics (SARS, MERS, etc.). Thus, this form of modeling proves to be very well adapted to our problem, which makes them a useful decision-making tool for public authorities.

Two types of compartmental models are more widely used in epidemiology. These are notably:

1. Deterministic models which are based on differential equation(s) for each compartment.

2. Stochastic Models that aim to estimate parameters on the basis of their stochastic processes.

In the case of Covid-19, Liu et al. (2020) showed that mathematical methods provided more reliable results and more accurate estimates than those of stochastic and statistical models^[10]. Although, stochastic modeling can be useful in numerous other study cases like in Aoun and El Afia (2014a, 2014b, 2018) and El Afia and Aoun (2017).

Epidemiological modeling

The virus spreads are a dynamic phenomenon depending on the number of contacts between an infected individual and a healthy one. The prevalence of the disease depends mainly on this. Similarly, an infected person with the virus becomes also contagious.

Such a phenomenon can be modeled by differential equations, the resolution of which will help us to understand its behavior and determine the dynamics of this system. For this purpose, some key parameters estimation for the epidemic has paramount importance to make good predictions, and this based on different propagation scenarios.

However, we will use the SIR model developed by Kermack and McKendrick (1927). There are also some model alternatives in literature like SIRS and the SIS models (Korobeinikov and Wake 2002). The chosen SIR model divides the population into three compartments, denoted S, I, and R, where S is the set of individuals who are Healthy or Suspected of being infected, the symbol I indicates those who are Infected, and R refers to those who are Recovered. Compartment R, therefore, includes both recovered individuals (considered immune), and deceased individuals (See Figure 4).

Figure 4:

Representation of the SIR model.^[12]

The number of people in each of these three compartments varies over time. As a result, the model will consist of these three variables as a function of time.

where N is the total population. This model assumes a short time interval so that births and deaths (other than deaths due to this disease) can be neglected.^[11]

The work of Kermack and McKendrick (1927) showed that the differential equations describing this model could be derived for the first time by:

Thus, the evolution of the number of the population in each compartment of the SIR model will be determined according to the following parameters β and γ .

β describes the likelihood that the disease is transmitted by exposure, i. e., an infected individual can infect β   N other individuals (obviously after contact). A negative sign in the first equation means that the number of people who are healthy or likely to be infected (in compartment S) is reduced after contact with an infected person. This same number of individuals will be added to the number of infected persons so that the second equation can describe the increase in the number of infected persons minus the number of recoveries. Thus, the parameter γ expresses the average cure rate.

Determination of reproduction number

The basic reproductive number, denoted by R 0 , has become a key concept in epidemiology. It is defined as the average number of cases likely to be infected due to an infected person. Thus, it is of crucial importance in predicting the development of the epidemic, as it tells us the expected number of new infections. Flahault et al (1988) obtained a formula for R 0 based on three main parameters:

where “β” represents transmission probability, “c” is the number of contacts and “d” indicates the period of contagiousness.

However, three main statements can be made depending on the value of this parameter:

• If R 0 > 1 : each infected individual can infect more than one other individual. Thus, there can be an epidemic spreading through the population;

• If R 0 < 1 : an infected individual cannot infect more than one other individual. Therefore, there is no risk of an epidemic;

• If R 0 = 1 : an infected individual can infect, on average, only one other individual. So, the disease persists without an epidemic (See Figure 5).

Figure 5:

Reproduction rate of a virus^[14].

SIR model parameters can be sensitive to numerous factors, as change in population behavior or even environmental factors. Authors Liu and Stechlinski (2012) used even a time time-varying parameters for the SIR model. By using the White et al. (2009) model, the reproduction number R 0 can also be calculated using the maximum likelihood estimator: ^[13]

where:

• N: the total population;

• Γ ( x ) : the gamma function;

• p: provides estimators for reproduction number and serial interval.

The parameter estimation is important for modeling and prediction in biological systems. In the case of our model, it is necessary to do experimental design and to estimate the model parameters using historical data; in order to fit model simulations to the observed dataset. The Reproduction numbers can also be estimated by several methods such as fish regression, Markov chains, mathematical models (Lotfi and Karim 2016a, 2016b), statistical exponential growth models (Lotfi and Karim 2016a, 2016b, 2017), etc. The maximum likelihood method seems to be adapted to our case, as it is a powerful statistical tool for coupling mathematical models to data sets. Moreover, this method quite often yields good estimates based on several studies of epidemiological models (Zhu and L. Ying (2016)).

The reproduction rate of Covid-19 in Morocco

Aiming to identify how coronavirus 2019 is spreading among the Moroccan population, how quickly and to which extent. We have estimated the reproduction rate of this virus since its first observation on the national territory. The idea is to find out how many people can be infected because of an infected person in Morocco. This will enable us to predict the development of the pandemic on a national scale and also evaluate the effectiveness of the containment and border closure measures taken by the public authorities in order to encircle it and limit its spread.

During the first 18 days of the pandemic, i. e., between March 02nd and March 20th 2020 (date of the beginning of containment), the estimated R 0 reproduction number in Morocco is 1.72 on average (95% CI 1.34–2.15). It means that during this period, 10 people infected with the virus can infect almost 17 people, who in turn, can infect nearly 290 people and so on. This evolution shows that the epidemic is spreading quite rapidly in Morocco. The graph below describes the evolution of the probability of transmission of the virus (reproduction number R 0 ) in Morocco just after containment (See Figure 6).

Figure 6:

Evolution of the number of reproductions.

Thus, this graph reflects the speed at which the epidemic is spreading over time. However, if the curve of R 0 is below the green line (where R 0 = 1 ), it can be concluded that the epidemic is in recession. Otherwise, the disease is still expanding.

After the decision to close schools, restaurants, etc., the probability of transmission of the virus has significantly decreased with a delayed effect of five days. The drop in this probability accelerated after the date of containment following the various health measures taken by the public authorities, even recording a reproduction number below one on March 24th 2020 (where R 0 =0.98). Consequently, this value means that Morocco had a strong chance of controlling the spread of this virus from the first month of its appearance. Unfortunately, that day coincided with the organization of a few demonstrations in several cities, which (most likely) caused the probability of transmission of the virus to rise for a second time. Despite this situation, the probability of transmission of the disease decreased to an R 0 of 1.36 (95% CI 1.25–1.48) after two weeks post-confinement, even less than the lower bound of the Covid-19 reproduction number posted by the WHO. Therefore, this numerical evaluation demonstrates the effectiveness of the various measures taken by the public authorities to limit the spread of this virus throughout the Kingdom.

However, this value remains relatively low in comparison with other countries such as Canada with an R 0 of 6.47. The table below gives some estimates of the R 0 reproduction number of Covid-19 in selected countries (See Table 1):

In fact, the spreading rate of Covid-19 in Morocco slowed down significantly after containment. However, the epidemic still persists. Hence the importance of estimating the evolution of infections in Morocco using the Covid-19 SIR model.

Forecast assessment

Using the system of differential equations provided by the SIR model, we will try to predict the future development of the pandemic caused by Covid-19 in Morocco. This modeling is primarily based on the historical evolution of the number of people in each compartment (S, I or R). Based on COVID-19 evolution data between March 02nd and April 08th 2020, the scenario of the spread of the coronavirus 2019 in Morocco is schematized by the following chart realized by the python language (See Figure 7):

Figure 7:

Forecast of Covid-19 transmission in Morocco.

This graph gives a reassuring result of the evolution of Covid-19 in Morocco. Firstly, it is estimated that this epidemic will reach its peak in mid-April 2020, i. e. after 45 days from the appearance of the first infected case. On that date, it is expected that the number of infected people will reach nearly 2,700. After this period, the number of newly infected people will decrease gradually. On the other hand, there will be a noticeable increase in the number of people who have recovered. It is then estimated that 95% of the virus should disappear on May 25th 2020, and should completely disappear on June 12, 2020. But it is important to highlight that these predictions are conditioned by full respect of the health instructions required by the public authorities.