key: cord-0793701-61grfn8e
authors: ARIES, N.; OUNIS, H.
title: Mathematical Modeling of COVID-19 pandemic in the African continent
date: 2020-10-13
journal: nan
DOI: 10.1101/2020.10.10.20210427
sha: 207a4c3fe97c77defef435750702dde31ffd7b3d
doc_id: 793701
cord_uid: 61grfn8e

The present work aims to give a contribution to the understanding of the highly infectious pandemic caused by the COVID-19 in the African continent. The study focuses on the modelling and the forecasting of COVID-19 spread in the most affected African continent, namely: Morocco, Algeria, Tunisia, Egypt and South Africa and for the sake of comparison two of the most affected European country are also considered, namely: France and Italy. To this end, an epidemiological SEIQRDP model is presented, which is an adaptation of the classic SIR model widely used in mathematical epidemiology. In order to better coincide with the preventive measures taken by the governments to deal with the spread of COVID-19, this model considers the quarantine. For the identification of the models parameters, official data of the pandemic up to August 1st, 2020 are considered. The results show that the number of infections due to the use of quarantine is expected to be very low provided the isolation is effective. However, it is increasing in some countries with the early lifting of containment. Finally, the information provided by the SEIQRDP model could help to establish a realistic assessment of the short-term pandemic situation. Moreover, this will help maintain the most appropriate and necessary public health measures after the lockdown lifting.

The year 2020 is and will be forever marked by the COVID-19 pandemic caused by the highly 29 infectious disease SARS-CoV-2. According to the World Health Organization, since its first 30 appearance in late 2019 in the Wuhan region of China, the Covid-19 has affected more than 6 31 million people and has caused the death of almost 400 thousand people worldwide, during the 32 first half of 2020. However, the spread of the disease and the damage caused by it did not 33 happen in the same way and at the same time for all continents. Indeed, taking the case of the 34 African continent, the COVID-19 pandemic reached a significant point by exceeding 180,000 35 cases in early June with more than 5,000 deaths. Although the pandemic has reached a 36 worrying threshold, the spread of the disease in Africa has not followed the exponential path as 37 in the rest of the world (e.g. Europe and United States). According to WHO's analysis, the 38 relatively low mortality rate compared to other continents is likely due to the demographic 39 nature of the continent. Indeed, demographically, Africa is considered as the youngest 40 continent, with more than 60% of the population under the age of 25. However, the disease still 41 represents a danger for the African population, particularly, the South and African North. In 42 fact, according to the WHO, the above-mentioned regions account for more than 56% of the 43 cases recorded in Africa. 44 To date, in the absence of an effective vaccine or treatment against COVID-19, African 45 governments have taken several measures to limit the spread of the disease as much as possible 46 (e.g. quarantine, social distancing, lockdown, curfew, masks, etc.). Furthermore, during this [1-5], Europe [6] [7] [8] [9] [10] [11] and the United States [7, [11] [12] [13] , which is understandable because these 54 are the regions which were the first and most affected by . Nevertheless, studies on 55 the African continent are practically negligible compared to those dedicated to the above-56 mentioned regions [14] [15] [16] . 57 The present paper aims to present a contribution on the modelling and the forecasting of 58 COVID-19 spread in the most affected African continent, namely: Morocco, Algeria, Tunisia, 59 Egypt and South Africa. In addition, for the sake of comparison two European countries are 60 considered, namely: France and Italy. Thereby, an epidemic SEIQRDP model is presented, 61 which is an adaptation of the classic SIR model widely used in mathematical epidemiology. In 62 the next section, we present the data and the model. Section 3 presents the results of the Algeria, Morocco, Egypt, South Africa, Tunisia, France, and Italy. Figure 1 shows the 79 cumulative reported cases of COVID-19 in the considered countries. The data refer to daily 80 cumulative cases from January 22, 2020 until August 1st, 2020. It is important to note that 81 these data are influenced by the capacity and strategy of countries in case detection. Some 82 countries perform more tests than others. However, the provided data still provide indicator for 83 tracking the trajectories of the epidemic. It can be well seen from figure 1 The epidemic spread to Africa a few weeks after Europe, allowing its leaders to adopt 94 preventive measures well in advance. South Africa, Tunisia, Morocco and Algeria imposed the 95 lockdown and curfews before the epidemic had had time to spread. In addition, the low 96 population density has considerably limited the transmission of the virus. Indeed, the 97 inhabitants are generally concentrated in the capitals. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in Using a pandemic modeling makes it possible to understand, describe and forecast its behavior 104 and its spread. Indeed, the mathematical models assess hypotheses, indicate trends, and help 105 develop public health responses by estimating risks, in real time, during an epidemic.

Nevertheless, modeling is a simplified representation of reality. Its precision is limited here by 107 the ignorance of certain factors and mechanism of propagation of In this work, the SEIQRDP epidemiological model with seven different components, proposed 109 by [2] is used. The model is an adaptation of the classic SEIR model [18, 19] , widely used to 110 study the COVID-19 pandemic in many countries with variations in components and 111 parameters for adaptations to regions and study period [16, [19] [20] [21] [22] [23] ]. The present model 112 considers the effect of quarantine adopted by many countries as an effective means of 113 preventing the spread. The stochastic SEIQRDP model is specified by the following system of 114 differential equations: 

The protection rate  , represents the population that takes into consideration security measures 128 and the actions of health authorities, it was introduced assuming that the sensitive population 129 decreases steadily. Moreover, beside the parameter  , all the remaining parameters depend on 130 All rights reserved. No reuse allowed without permission.

perpetuity.

preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in

The copyright holder for this this version posted October 13, 2020. . https://doi.org/10.1101/2020.10.10.20210427 doi: medRxiv preprint 5 the evolution of the epidemic, the health care and the screening capacities and are calculated 131 based on the daily numbers of confirmed, recovery and death cases. The infection rate  , 132 represent the average number of contacts per-capita per time, multiplied by the probability of 133 successfully getting infected when coming into contact with an infected individual. While, the 134 latent rate 1   , represent the average time for a latent individual to become infectious, which is 135 estimated within several days [4, 24] . Furthermore, the quarantined rate 1   represent the 136 average time for an infectious individual to enter in quarantine, which is considered to be 137 between 2 and 14 days [2] . Finally, to consider the measures taken by the different countries in 138 this study, we assume that the recovery rate t  and the death rate t  are time dependent 139 function, as confirmed by [2, 21] .

In case of a new disease as the COVID-19, the medical staff must learn new therapeutic 141 procedures and treat patients with new symptoms every day. Hence, the time of recovery, 142 cannot be a constant, because the recovery time at the start of the disease is longer, this means a 143 slower recovery rate which gradually increasing with time, considering the measures taken by 144 the governments. Consequently, we will assume that the recovery rate is modeled by following 145 function,

Unlike the recovery rate, the mortality rate quickly decreases over time. This is due to the 149 medical assistance, the adaptation of the pathogen and the development of new treatments.

Hence, we will assume that the mortality rate is modeled by an exponential decay function 151 given as follows, If the 0 R is above 1, each infection breeds more, and the outbreak will continue to grow. When 165 it falls below 1, the outbreak will continue but at a lower death rate.

166 All rights reserved. No reuse allowed without permission.

perpetuity.

preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in

The copyright holder for this this version posted October 13, 2020. . https://doi.org/10.1101/2020.10.10.20210427 doi: medRxiv preprint 6 In this study, the R 0 is calculated by the next generation matrix method [22] , based on a 167 endemic equilibrium, given by :

, gives the best parameters for data fitting. It should be noted that the use of the cumulative data 179 should be avoided, since the cumulative incidence data are highly correlated, and the least 180 squares approaches relies on the fitting of independent data. Therefore, the use of daily new 181 data should be preferred, since these data are independent. Once all the parameters are known,

we can solve the system of differential equations (1.1).

The estimated parameters for the considered countries are given in perpetuity.

preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in

The copyright holder for this this version posted October 13, 2020. . https://doi.org/10.1101/2020.10.10.20210427 doi: medRxiv preprint 7 is, the less an infected individual will be able to infect healthy individuals. However, Tunisia 197 shows a high infectious potential of the disease, compared to other African countries, which 198 confirms the result of the protection rate. The obtained latent rates (Table 1) countries, which coincides with the results of the quarantine time obtained (Table 1) . This is perpetuity.

preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in

The copyright holder for this this version posted October 13, 2020. As can be seen, R t becomes less than 1 between early April and early June, about four to seven 234 weeks after the lockdown in the Maghreb countries (figure 2a-c). In fact, this shows the effects 235 of the lockdown, the quarantine and the strict curfews adopted to reduce the effect of virus 236 transmission, while for Egypt and South Africa, the R t has not crossed the line R t = 1 237 (mitigation phase), till end of June for Egypt and mid-July for South Africa. However, the 238 lowest infection kinetics is recorded in Tunisia and has generated a first R 0 = 4.690, much 239 lower than those of other African countries and that of Italy especially. For Egypt, the number 240 R t decreased more slowly due to relative delayed lockdown compared to the other countries 241 and an early lifting of the lockdown. However, the number of effective reproduction number 242 depends on the parameter δ, which governs the rate of transfer from the infectious class to the 243 quarantine class. Therefore, the use of quarantine to control the disease not only decreases the 244 size of the endemic infectious class, but also reduces the reproduction number R t below 1, so 245 that the disease vanishes. 246 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 13, 2020. . https://doi.org/10.1101/2020.10.10.20210427 doi: medRxiv preprint 9

The estimation results (Table 1 and Figure 2 ) indicate that the measures taken to prevent, and 247 control the epidemic taken by the governments have strengthened over time. Indeed, the 248 admission and follow-up of suspect cases, quarantine and treatment of confirmed cases 249 significantly affect the parameters values. As a result, the number of active (quarantined), 250 suspect, cured cases and the deaths predicted by the SEIQRDP model tend to decrease over 251 time. On the other hand, the accuracy of long-term forecasts will also decrease. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 13, 2020. . perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 13, 2020. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 13, 2020. .

Based on this model, the epidemic in Italy is in its final phase and should end by September. to decrease after reaching its maximum. We note that there was a slight decrease in daily cases 316 in April which seems to be explained by a low number of tests and the random rules of the 317 lockdown. With approximately 6.500 tests performed, only cases with advanced symptoms 318 appear to be counted. After this plateau of active cases in April, there was a worrying return 319 until the end of May. However, according to Algerian COVID-19 forecasts by the SEIQRDP 320 model (Figure 4b) , the situation will begin to stabilize. the number of recovery cases will 321 increase exponentially, and the number of death cases will also stabilize despite the fact that it 322 has not grown significantly compared to the European countries. The gradually flattening in 323 deaths is may be due to the use of the chloroquine treatment protocol [29] . there is a decrease in active cases after a peak in mid-April, followed by a second significant 337 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 13, 2020 . . https://doi.org/10.1101 13 peak predicted approximately in mid-October, due to the lockdown lifting. The curve of active 338 cases (Figures 3c and 4c ) describes well this reassuring trend, with a peak in mid-April and a (Figure 3a-e) . Regarding the number of deaths, we note that it is stable so far, 379 this stability is explained using the treatment protocol with chloroquine and the CST. In 380 addition, the experience of the Ebola epidemic in South Africa which taught caregivers and 381 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 13, 2020. . https://doi.org/10.1101/2020.10.10.20210427 doi: medRxiv preprint 14 populations the best practices in place, such as, isolation of the sick, precautions during care 382 and basic hygiene such as hand washing. For the forecast of the COVID-19 pandemic in South

Africa by the SEIQRDP model (Figure 3e) , the situation will stabilize in terms of infected and 384 recovered cases, while the dead experienced a certain increase in mid-July, due to the lifting of 385 the lockdown of certain economic [35] . On the other hand, according to the statistical, fitted and predicted curves, we noticed that the 402 epidemic peaks are between mid-April and mid-August, after which the epidemic declines.

This situation can be explained by the policy adopted by African countries, aimed at effectively and interurban. Nevertheless, the three Maghreb countries will have to strengthen the control of 420 health measures, to avoid the epidemic resumption. The crucial element will remain the 421 generalization of tests which will remain one of the main ingredients in the fight against 422 epidemics in the country. All these precautions will effectively reduce the number of basic 423 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 13, 2020. . https://doi.org/10.1101/2020.10.10.20210427 doi: medRxiv preprint 15 reproductions at each interaction, which is considered a crucial parameter during a pandemic 424 used to estimate the risk of an outbreak of COVID-19 and assess the effectiveness of the 425 measures implemented. 426 Finally, the epidemiological situation is reassuring in the three Maghreb countries, while for 427 South Africa, the situation seems controllable, mainly due to the CST operation. However, in 428 Egypt the number of infected cases continues to increase, the situation comes mainly from 429 ignorance of physical distancing measures and the violation of curfews after the last relaxation 430 of the restrictions in mid-March. In fact, without control of the population and lack of effective 431 treatment and vaccine, there will be no way to prevent the death of many people. Furthermore, 432 hospitals will be flooded and will never be able to manage the pandemic. 

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