key: cord-0821624-f9jd6ek8
authors: Balike, Z. D.
title: Mathematical model for the mitigation of the economic effects of Covid-19 in the Democratic Republic of the Congo
date: 2020-12-16
journal: nan
DOI: 10.1101/2020.12.14.20248182
sha: b511eb2f4866d185c7d50feaec40f9afecb30fff
doc_id: 821624
cord_uid: f9jd6ek8

A mathematical model of the spread of the Covid-19 in the Democratic Republic of the Congo taking into account the vulnerability of the economy is proposed. The reproduction number of the Covid-19 is calculated and numerical simulations are performed using Python software. Clear advice for the policymakers is deduced from the forecasting of the model.

In December 2019 a novel Coronavirus appeared in Wuhan, China [9] . The international 2 committee for taxonomy of virus has attributed the name SRAS-Cov-2 to that 3 disease [10] , [4] . On January 30, 2020; the World Health Organization (WHO) declared 4 it to be an epidemic of international concern [19]. Four months later, the virus was 5 spread out worldwide, only less than 10 countries were not yet touched by the disease. 6 This is why WHO declared it to be a pandemic since March 11, 2020 [19] . 7 Countries affected by this outbreak have envisaged several measures to mitigate its 8 negatives effects on their health care systems. Those measures include national 9 lockdown and cancellation of travels to and from outside their borders [14] . Many 10 low-income countries, including African countries, have also adopted the above measures 11 without taking into account of the vulnerability of their economies which rely mainly on 12 informal system [23] . 13 On 10 March 2020, the first infected individual has been detected in Democratic 14 Republic of Congo (DRC, [20] ). On March 18, 2020 ; the DRC President has announced 15 draconian measures to mitigate the circulation of the pandemic. These measures include 16 cancellation of flights out and from Kinshasa (the capital city) to other provinces, and infected people are quarantined. The meaning of 'suspicious' will be precised in the 27 following section.

Since the apparition of this virus, several models have been 29 proposed [1] , [2] , [3] , [5] , [6] , [7] , [8] and [13] . At my knowledge, there is no model which 30 has been performed to help understanding the spreading of this pandemic in DRC and 31 taking into account the economic impact of the virus on the populations.

To fill in this gap and help addressing this challenge, an SEIR model (see [18] , [15] 33 and [17] for details of this model) with additional compartments is proposed in this 34 paper. In this model, full lockdown or national quarantine is not envisaged as it may 35 jeopardize the fragile economic system of the country. The following compartments have 36 been added to the traditional SEIR model : Quarantined and Hospitalized. infected and not immunized against the disease. They are recruited at rate θ and 41 transferred into the Exposed group at the rate β.

In DRC, only one laboratory can declare positive individuals amid a Covid-19 43 testing. It is the National Institute of Biomedical Research (NIBR) located in 44 Kinshasa, the capital city.

In many cases, after testing, individuals have to wait for many days before they 46 can get the result as the NIBR is far away from many provinces (for example, 47 Kinshasa is 2000 km from Bukavu) and the transportation system is bankrupted. 48 To prevent potential infected individuals to spread the disease, authorities should 49 quarantine all suspicious individuals from susceptible population at rate . who are able to spread the disease. In general such individuals are at the onset of 59 the illness. Thus, they will be transferred into the hospital (H) at rate ν and are 60 removed at rate η. This removal is due either to death or recovery without being 61 transferred at the hospital. This will particularly happen for young people who 62 are resilient to the disease and some unbelievers who never accept that the virus is 63 deadly and are currently propagating into the population. 

In this section, an analysis of the model (system 1) will be performed.

It is clear that this system under study satisfies the Lipschitz (see [12] ) and theorem 78 1 (see [22] , page 343) conditions. This guarantees the existence and the uniqueness of 79 the solution. The model is performed for understanding the spread of the SRAS-Cov-2 into a human 82 population. Thus, the parameters are expected to be positive.

Let Ω be the set of all feasible solution to the system 1. Each solution to a single 84 equation lie in R + . 85 We hence have

where N is the total size of the population under study. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted December 16, 2020. ; https://doi.org/10.1101/2020.12.14.20248182 doi: medRxiv preprint

Substituting each derivative by the corresponding value from system 1 and 90 simplifying, I get :

This equation can be rewritten in the following way:

Equation 4 can be solved using an integrating factor.

Multiplying both sides of that equation by e λdt , we get:

We finally get

where C is an arbitrary constant. For t = 0, we have C = N 0 − θ λ which we substitute 96 in equation 5 to find

Thus, we have N ≥ θ λ as t → +∞, provided equation 2.

This establishes that Ω is positively invariant and attracts all solutions in R 6 + .

In addition, it can be proven that the solution of system 1 has a positive solution 100 provided the initial data set (S 0 , E 0 , I 0 , H 0 , Q 0 , R 0 ) ≥ 0 ∈ Ω.

For example, we have

The right side of this inequality is always positive, that is the function S is also positive 102 and consequently its initial value.

A similar computation can show that

where K i , (i ∈ {1, · · · , 5}) are arbitrary constants.

December 14, 2020 4/11

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted December 16, 2020. ; https://doi.org/10.1101/2020.12.14.20248182 doi: medRxiv preprint 1.2 The disease free equilibrium (DFE) 105 From the equation 1, we set f i = 0, i = 1, · · · , 6 to get the new algebraic system below. 106

From equations (a) to (f), one successively get

If E = 0 (that is β = 0) and = 0, then I = H = Q = R = 0 and accordingly, the DFE 108 is P 0 The infection components in this model are E, I, H and Q. In accordance with the notations in [16] , [13] and [2] , the new infection matrix F and the transition matrix V are given by

The basic reproduction number of model 1 is then defined as the spectral radius of 117 the next generation matrix F V −1 , i.e

If the inequality R 0 ≤ 1 is satisfied, then no epidemic outbreak is possible, otherwise an 119 epidemic occurs. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this this version posted December 16, 2020. ; https://doi.org/10.1101/2020.12.14.20248182 doi: medRxiv preprint After substituting I * from equation 7 and simplifying, we get:

This finally yields

Provided 2 and its result, I have

Hence, I get

.

the new reproduction number.

This is exactly the spectral radius of the new next generation matrix obtained after 127 ignoring the fifth equation of the system 1 in the infection compartments. 

This jacobian evaluated at the DFE yields

has all its eigenvalues negative whether is null or not.

Thereby, the DFE is locally asymptotically stable.

December 14, 2020 6/11

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Due to lack of available data in DRC, some parameters have been reasonably 141 assumed (without either overlooking or exaggerating them). Provided some parameters 142 after fitting the model with data, some other parameters have been calculated and 143 others were borrowed to [2] . To determine the value of the calculated parameters, I 144 proceed in the following way: At the beginning of the epidemic, E = I and second 145 equation of system 1 becomes

whose solution is

where k is an arbitrary constant. Using the value got from data fitting, I get

Since the value of λ is known from the data on life expectancy in DRC 148 (λ = 1 12×Life expectqncy , see [15] and [24] for details) and provided there some others 149 parameters borrowed from [2] , I get the values in Table 1 . Using the values of parameters like provided in Table 1 , a simulation of the model 151 has been performed in Python. The model is designed such that Susceptible stratum 152 collapses as early as possible. This makes the Exposed and infectious groups to explode 153 quickly.

The impact of can be evaluated on Fig 3 and Fig 4. When the number of individuals 155 in group Q is high, the number of positive cases drops but the number of individuals 156 hospitalized increases. This is a proof that organized quarantine allows a rapid 157 detection of positive cases and enables the timely management of patients who need to 158 be hospitalized. The fact that the infectious curve reaches its peak without ever 159 affecting the maximum of the population under study is proof that many individuals are 160 freely circulating as long as they have not been deemed suspicious for quarantine. This 161 allows the economy to run normally in the midst of a virus crisis. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted December 16, 2020. ; https://doi.org/10.1101/2020.12.14.20248182 doi: medRxiv preprint 

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