key: cord-1030984-okfmtmq9
authors: Garba, Salisu M.; Lubuma, Jean M-S.; Tsanou, Berge
title: Modeling the transmission dynamics of the COVID-19 Pandemic in South Africa
date: 2020-08-04
journal: Math Biosci
DOI: 10.1016/j.mbs.2020.108441
sha: c9d6539c963b68f8f605afaf5a9cff7df1251f7d
doc_id: 1030984
cord_uid: okfmtmq9

Since its emergence late in 2019, the COVID-19 pandemic continues to exude major public health and socio-economic burden globally. South Africa is currently the epicenter for the pandemic in Africa. This study is based on the use of a compartmental model to analyse the transmission dynamics of the disease in South Africa. A notable feature of the model is the incorporation of the role of environmental contamination by COVID-infected individuals. The model, which is fitted and parametrized using cumulative mortality data from South Africa, is used to assess the impact of various control and mitigation strategies. Rigorous analysis of the model reveals that its associated continuum of disease-free equilibria is globally-asymptotically stable whenever the control reproduction number is less than unity. The epidemiological implication of this result is that the disease will eventually die out, particularly if control measures are implemented early and for a sustainable period of time. For instance, numerical simulations suggest that if the lockdown measures in South Africa were implemented a week later than the 26 March, 2020 date it was implemented, this will result in the extension of the predicted peak time of the pandemic, and causing about 10% more cumulative deaths. In addition to illustrating the effectiveness of self-isolation in reducing the number of cases, our study emphasizes the importance of surveillance testing and contact tracing of the contacts and confirmed cases in curtailing the pandemic in South Africa.

• Introduction of a starting time parameter for the lockdown measure, showing that its early implementation results in considerable decrease in the number of COVID-19 cases, and in not delaying the peak time.

• Incorporation of the role of environmental contamination by COVID-19 infected individuals.

• Computation of the attack rate of COVID-19, and the number of susceptible individuals who escaped infection at the end of the pandemic.

1 [58, 67] . In the same vein, another deterministic model was designed to analyze the 16 MERS-CoV outbreak in the Republic of Korea [31, 63] . The impact of the timing of 17 control measures associated with a reduction of the transmission rate and diagnostic de- 18 lays on the outbreak size and duration was also assessed [67] . Simulation of the model 19 reveals that the lack of personal hygiene and targeted control measures were the reasons 20 of the outbreak spread quickly. However, it was reported that strengthening personal 21 hygiene ability of susceptible and quickly isolating or monitoring close contacts are ef-22 fective measures to control the disease [63] . Furthermore, partial correlation analysis 23 shows that the infectivity and proportion of the asymptomatic infected cases have much 24 influence on the disease spread [1] . 25 26 Since its emergence in December 2019, numerous models have been designed and used 27 to determine effective ways to combat the pandemic [5, 18, 19, 27, 28, 37, 42, 66 ]. In The distinguishing aspect of the current work is the emphasis on the COVID-19 dy-population for compliance, and to implement existing non-pharmaceutical control mea- 23 sures, while strengthening the recovery strategy of the economy and providing support 24 to those who suffer the most during the lockdown. The remaining part of the paper is organized as follows. The model is formulated 3 in Section 2. Section 3 is devoted to "materials and results", including estimation of 4 the parameters and some analytical results (such as the asymptotic stability of the The total human population at time t, denoted by N (t), is divided into six mutuallyexclusive compartments, namely susceptible S(t), exposed E(t), asymptomatic infectious A(t), symptomatic infectious I(t), isolated/hospitalized J(t), and recovered individuals R(t). Thus,

This sub-division of the total population is consistent with South Africa's guidelines for 11 isolation from COVID-19 exposure and infection, which states that: 'Though isolation 12 is reserved for persons who are already sick and/or have tested positive for COVID-19 13 infections, it may in the context of the COVID-19 pandemic, include [53]:

• Isolation at a personal home known as self-isolation. This is the preferred option, 15 subject to the person meeting the self-isolation criteria. 16 • Isolation in a health facility or at a designated isolation facility. People who cannot 17 self-isolate at home should be considered for admission to such a facility.' 18 As far as the compartments of human populations are concerned, we use an extension 19 of the standard SEIR model [26] , modified by the incorporation of the A and J classes 20 to account for asymptomatic transmission and isolation/hospitalization. However, re-21 search works [10, 17, 52, 63] , specifically the recent study in [10] The population of susceptible individuals is decreased following infection with COVID-7 19, which can be acquired at the rate (force of infection)

in which the first term expresses direct transmission, while the second term accounts for 9 the indirect (environmental contamination-based) transmission. Here, the parameter 10 β(t) is the time-dependent effective contact rate (contact, per person per unit time,

capable of leading to COVID-19 infection). Due to the introduction of social-distancing 12 policy (e.g., lockdown or stay-at-home), it is reasonable to assume that the contact rate 13 will be a decreasing function of time. Unlike the exponential decay function used in 14 [18], we consider the following slow-decaying continuous function β(t):

where τ 0 is the time for the onset of the community lockdown. The parameter ω > 0 is 16 a measure of the compliance of the population with the interventions, mostly the social- 17 distancing, and also the wearing in public of face-masks that has been introduced as of 18 1 May 2020 in South Africa. The larger ω is, the faster the contact rate β(t) decays to 19 β 1 , which represents the desired contact rate for COVID-19 to be controlled. individuals to the asymptomatic-infectious (at the rate rσ) and symptomatic-infectious 1 (at the rate (1 − r)σ) classes, where 0 ≤ r ≤ 1 is the proportion of exposed individuals The population of asymptomatic infectious individuals is increased by the progression 5 of infected individuals from the exposed class (at the rate rσ). It decreases by isolation 6 at the rate γ 1 ), and by recovery (at the rate τ 1 ). This gives

The population of symptomatic infectious individuals (with clinical symptoms of COVID-8 19) in I class increases, following the development of clinical symptoms by individuals 9 in the exposed class, at the rate (1 − r)σ. This population decreases by isolation (at the 10 rate γ 2 ), recovery (at the rate τ 2 ), and COVID-19 induced mortality (at the rate δ 1 ).

Hence,

The population of individuals that are isolated or hospitalized (J(t)) is generated by the 13 isolation of infectious individuals in the asymptomatic class (at the rate γ 1 ) and those with clinical symptoms of COVID-19 (at the rate γ 2 ). It is decreased by recovery (at 1 the rate τ 3 ) and disease-induced death (at the rate δ 2 ). Hence,

The recovered population is generated by the recovery of individuals in A, I and J 3 classes at the rates τ 1 , τ 2 and τ 3 , respectively. This gives:

Infectious individuals in the A, I and J classes contaminate the environment with 5 COVID-19 at the rates ξ 1 , ξ 2 and ξ 3 , respectively. The virus is cleared from the con-6 taminated environment at the rate υ. Hence,

In summary, the COVID-19 transmission model is given by the following system of 8 nonlinear differential equations (the flow diagram of the model is depicted in Figure 2 , 9 and the associated parameters and state variables are described in Table 2 ):

Like in the case of many other models for COVID-19 [18, 42] , the model (2) assumes 11 homogeneous mixing, and recovery induced permanent natural immunity against future 12 infections.

The system is solved subject to the following (generalized) nonnegative initial conditions 16 17 Adding the first six equations of the model (2) gives: Effective contact rate before the community lockdown. β 1

Targeted effective contact rate for disease control. ω

Measure of social-distancing effectiveness.

Isolation rates of asymptomatic (symptomatic) infectious individuals.

Modification parameters for reduction in infectiousness of asymptomatic (isolated) individuals in comparison to symptomatic infectious individuals. η 3

The per capita rate of people who interact with the environment daily. τ 0

Starting day of community lockdown.

Recovery rates of asymptomatic (symptomatic) (isolated) infectious individuals. rσ ((1 − r)σ) Progression rate of exposed individuals to asymptomatic (symptomatic) infectious classes.

Disease-induced death rates for symptomatic (isolated) infectious individuals.

Contamination rates of environment by asymptomatic (symptomatic) (isolated) infectious individuals. 

To the question 'will COVID-19 ever disappear?' several sources suggest that the sce- Care Units (ICUs) when the worse case scenario comes. We will fit the model using 26 South African data and predict the evolution of the epidemic. To this end, we start the 27 next subsection by estimating the epidemiological parameters of the model (2) relevant 28 to COVID-19 data for South Africa obtained from [9] . lowed by a nationwide lockdown from 26 March 2020. We, therefore, take the time of By fitting the model (using mortality data), we take the effective contact rate β = β 0 1 before the lockdown to be 0.492 per day. The desired minimum contact rate β to which 2 the contact rate should decay is fitted to be β = β 1 = 0.166 per day (so as to achieve 3 the target of bringing the control reproduction number to a value below unity). Since 4 the incubation period for COVID-19 ranges from 5-6 days [42] , with about 70% of ex-5 posed individuals becoming infected, we assume the rate at which exposed individuals 6 become asymptomatically infectious to be rσ = 0.6 per day, so that the rate at which 7 exposed individuals becomes symptomatic is (1 − r)σ = 0.4 per day. It should be noted 8 that determining the portion associated with the spread of COVID-19 by asymptomatic 9 individuals is a challenge, as highlighted in [1, 24] . It is assumed that there is a short for COVID-19 to range from 6-14 days, so we set the rates at which asymptomatic, 15 symptomatic infectious and isolated individuals recover from COVID-19 (τ 1 , τ 2 and τ 3 ) 16 to be τ 1 = 1/6, τ 2 = 1/10 and τ 3 = 1/14 per day, respectively. 17 18 While some studies assumed the modification parameters (η 1 and η 2 ) for the relative Contaminated environment is reported to be a substantial route for the transmission of 27 SARS-CoV-2. This is the essence of the campaign to disinfect surfaces, buttons, hands, 28 knobs and other places touched often, apart from scientific reports such as [10, 17, 52] .

In fact, in South Africa, a number of hospitals have been closed and the scaling down The cumulative number of disease-induced deaths denoted by D = D(t) will be estimated from the following differential equation that results from recording death contributions in the model (2):

We now fit the model (2) using data obtained from [9] for South Africa for the period of 2 three months (21 March to 29 June 2020). Given the inability to realistically measure 3 the size of the asymptomatically-infectious pool, which makes most of COVID-19 case 4 data suspect, we chose to fit the model with the mortality data (which is more reliable). 5 The estimated, assumed and fitted parameters are tabulated in Table 3 (see Table 5 for 6 some sensitivity analysis). Figure 3 shows a reasonably good fit for total actual deaths implemented. With K 1 = γ 1 + τ 1 , K 2 = γ 2 + τ 2 + δ 1 and K 3 = τ 3 + δ 2 , the vector F, of 

The Jacobian matrices F of F and V of V are computed at the point E with respect to Simple computations show that R c can be rewritten as the sum of two main contribu-8 tions (viz. humans and environment) as follows:

where,

It should be mentioned that we do not make the substitution S(0)/N (0) = 1 in order 11 to link the control reproduction number R c to the effective reproduction number intro-12 duced in Remark 2 below. The effective reproduction numbers for some countries with COVID-19 cases are given 1 on Table 4 as of 13 March 2020. (2), the continuum of disease-free equilibria is both a stable set and 15 a global attractor. 16 Theorem 4 If R c < 1, then the continuum of disease-free equilibria of the model (2) 17 is globally asymptotically stable in the positively-invariant and absorbing compact set Ω 18 defined in (5).

Proof. It was proved that the system (2) is a dynamical system in the biologically feasible region Ω which contains the set [DFE]. Define on Ω the candidate Lyapunov function:

where the positive constants a 0 , a 1 , a 2 , a 3 will be determined shortly. Then the direc-20 tional derivativeL (where a dot represents derivative with respect to t) of L in the 21 

J o u r n a l P r e -p r o o f direction of the vector-function defined by the right-hand side of the system (2), (i.e. 1 the derivative along the trajectories), is given by

Since S/N ≤ 1 and S ≤ N 0 in Ω, some lengthy computations lead the following estimate 4 ofL:

The constants a 0 > 0, a 1 > 0, a 2 > 0, a 3 > 0 and a 4 > 0 are then chosen to be the unique solution of the following algebraic system:

This simplifies the above estimate ofL intȯ

where the control reproduction number is given in (7). 

where S ∞ is the unknown final size of the epidemic to be determined. 14 

where β = β 0 or β 1 , x 0 = x(0) and the inverse matrix V −1 was used in the com-15 putation of R c . 16 To simplify the final size relation (13), it is usual to set some of the initial conditions 17 to be equal to zero [2, 42] . 

For the analysis in the next section, Theorem 5 will be used as follows. The number 21 α := 1 − S ∞ S 0 , called the "attack rate or ratio" of the epidemic, is a measure of its sever-22 ity, apart from the number S ∞ of susceptible individuals who escaped the epidemic [4].

The larger the attack rate is, the more severe the epidemic is, in terms of the cumulative Table 3 . The simulations show a decrease in the numbers of exposed, asymptomatic, Table 3 with various values of ω.

lockdown or stay at home) by South Africa's government has a significant community-1 wide impact in mitigating the transmission of COVID-19. Table 3 with various values of the compliance parameter ω.

study strongly suggests that absolute caution should be exercised before terminating the 1 current strict social-distancing protocols or lowering the COVID-19 alerts, so as to avoid Table 3 . The effect of isolation of individuals infected with COVID-19 is monitored by simulating 2 the model (2) using parameter values given in Table 3 Table 3 .

as shown on Figure 9 (B). Furthermore, the computed attack rate (α) for β = β 0 is found Table 3 . These results confirm that the infection is more severe for high contact rate β. Notice 6 that, for the case when β = β 0 , the total number of COVID-19 cases at the end of the 7 pandemic is obtained to be S 0 − S ∞ = 50, 500, 000, which is more than three times the 8 number obtained for β = β 1 (15,000,000) (see Figure 9 ). Thus, our predictions in terms Table 3 . The parameters with significant negative and 1 positive PRCC values are seen from Table 5 . In particular, the environmental transmis-2 sion factor η 3 and the rate of virus cleaning from the environment (υ) are among the 3 most influential parameters. Table 5 : PRCC values of the model: -with the control reproduction number (R c ) as the response function for (A) β = β 0 and (C) β = β 1 ; -with the total number of infected humans (I(t)+J(t)) as the response function for (B) β = β 0 and (D) β = β 1 . Parameter values and ranges used are as given in Table 3 . Table 3 . Table 3 .

Since the beginning of March 2020, South Africa has been hit by the COVID-19 pan- The main findings of the study, which in a nutshell suggest that the COVID-19 pan-2 demic can be controlled in South Africa provided that all the envisaged measures are 3 implemented effectively, include the following: and its capital Cape Town (65% of the entire country) [14] . Western Cape has been 5 declared the epicenter for COVID-19 in South Africa. However, this province has been 6 exemplary in conducting the highest number of tests in the country (a total of 194939 7 tests i.e. 20%), and in identifying hotspot transmission areas or super-spread events.

8 When designing the model for Western Cape, we need to take into account the framework 

The big unknown: 21 the asymptomatic spread of COVID-19

A final size 24 relation for epidemic models

Essai d'une nouvelle analyse de la mortalité causée par la petite vérole

A Modified SIR Model for the COVID-19

Contagion in Italy

Dynamical models of tuberculosis and their appli-3 cations

Asymptotically autonomous epidemic models

Mathematical Population 6 Dynamics: Analysis of Heterogeneity

Differences 8 and similarities between Severe Acute Respiratory Syndrome (SARS)-Coronavirus 9 (CoV) and SARS-CoV-2

The presence of SARS-CoV-2 RNA in the feces of COVID?19 16 patients

A mathematical model for simulating the phase-18 based transmissibility of a novel coronavirus

SARS out-21 breaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isolation 22 as a control mechanism

Coronavirus South Africa (Wits in partnership with NRF, iThemba Labs and Data 26

COVID-19 South Africa Dashboard

On the definition and computation 28 of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous 29 populations

The construction of next-31

Novel Coronavirus (COVID-19) pandemic: built environment considerations 2 to reduce transmission

To mask or not to mask: Modeling the potential for face mask use by the 6 general public to curtail the COVID-19 pandemic

Impact 10 of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and 11 healthcare demand, London: Imperial College COVID-19 Response Team

Mod-15 elling strategies for controlling SARS outbreaks

Will COVID-19 ever go away? Time

Impacts of migration and immigration on disease transmission 19 dynamics in heterogeneous populations, Discrete and Continous Dynamical Sys-20 tems

Mathematical model

Early dynamics of transmission and control of COVID-19: a mathematical 2 modelling study

Herd immunity-estimating 4 level required to halt the COVID-19 epidemics in affected countries

Stability Analysis of Non-7 linear Systems

The stability of dynamical systems

A dynamic compartmental model for the mid-11 dle east respiratory syndrome outbreak in the republic of Korea: A retrospective 12 analysis on control interventions and superspreading events

Substantial undoc-15 umented infection facilitates the rapid dissemination of novel coronavirus

Transmission dynamics andcControl of severe acute respiratory syndrome

The reproductive number of 21 COVID-19 is higher compared to SARS coronavirus

The Epidemiology and Control of Malaria

Presence 26 of SARS-Coronavirus-2 RNA in sewage and correlation with

rrevalence in the early stage of the epidemic in The Netherlands

Transmission potential of the novel coronavirus

Mathematical assessment of the impact of non-pharmaceutical 11 interventions on curtailing the 2019 novel Coronavirus

Coronavirus pandemic: daily updated research and data

SARS-CoV-2 RNA in wastewater anticipated COVID-19 occurrence in a 17 low prevalence area

The Asian countries that beat COVID-19 have to do it again

The Prevention of Malaria

Mathematical analysis of a disease transmission model with [51] South African COVID-19 Modelling Consortium (Prepared by MASHA, HE2RO, 1 and SACEMA), Estimating cases for COVID-19 in South Africa. (Modelling 2 Consortium-20 and Update)

South Africa Health Department, COVID-19 environmental health guidelines

South Africa Health Department, Guidelines for quarantine and isolation in re-6 lations to COVID-19 exposure and infection

South Africas policy response to the COVID-19 pandemic

Estimation 11 of the transmission risk of the 2019-nCoV and its implication for public health 12 interventions

Coronaviruses and acute respiratory syndromes (COVID-19, MERS 15 and SARS), MSD Manuals on COVID-19

Proof of Concept -Sewage Surveillance for Covid-19 in South Africa

Modeling the transmission 20 dynamics of the Middle East Respiratory Syndrome Coronavirus (MERS-CoV) with 21 latent immigrants

Reproduction numbers and sub-threshold en-23 demic equilibria for compartments models of disease transmission

Simulating the SARS outbreak in Beijing with limited data

Global overview of an emerging novel coronavirus

2013. (World Health Assembly-20)

World Health Organization, China's latest SARS outbreak has been contained, but 30 biosafety concerns remain

World Health Organization, Laboratory testing strategy recommendations for COVID-19: interim guidance

Worldometer-20)

Nowcasting and forecasting the potential 4 domestic and international spread of the 2019-nCoV outbreak originating in Wuhan

China: a modelling study

Modeling the transmission 7 of Middle East respiratory syndrome corona virus in the Republic of Korea

A mathematical model for the novel coronavirus epi-10 demic in Wuhan, China

Staggered release policies for 13

COVID-19 Control: costs and benefits of sequentially relaxing restrictions by age

Description and 16 clinical treatment of an early outbreak of severe acute respiratory

The authors of the above-mentioned manuscript submitted to Mathematical Biosciences (Elsevier), hereby declare that there is no conflict of interest.