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The currently circulating Corona virus disease 2019 (COVID-19) has reached Sudan by the mid of March 2020 or earlier. From the onset of the virus arrival, the government of Sudan has allocated isolation centres across the country and gradually placed a series of measurements to prevent the disease from widespread community transition. In this study, we present a mathematical model for the transmission of COVID-19 in Sudan, using a modified version of SEI framework. We placed emphasis on quantifying the impact of the control measurements and the rate of case detecting in slowing down the spread of the disease. The model estimated that the current case detection ratio is only 22.7%, the impact of the non-pharmaceutical interventions is about 23.1% and, alarmingly, the death rate of the undetected cases is higher than that of the detected ones. The models also showed the undetected infectious individuals spread the disease more than the detected ones, i.e, the major source of the disease transmission, an indication for the importance of social distancing. If the country continued with the present level of measurements, the model predicted that the peak of the cumulative
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A Mathematical Model for the Transmission of
Corona Virus Disease (COVID-19) in Sudan
Sara M-A. S. Elsheikh1, Mohamed K. Abbas2, Mohamed A. Bakheet3, and A. M. Degoot1,4
1Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Khartoum.
2Institute of Endemic Diseases, University of Khartoum.
3Faculty of Mathematical Sciences and Statistics, Al-Neelain University.
4African Institute for Mathematical Sciences (AIMS) Rwanda.
May 31, 2020
The currently circulating Corona virus disease 2019 (COVID-19) has reached Sudan by the
mid of March 2020 or earlier. From the onset of the virus arrival, the government of Sudan has
allocated isolation centres across the country and gradually placed a series of measurements
to prevent the disease from widespread community transition. In this study, we present a
mathematical model for the transmission of COVID-19 in Sudan, using a modified version of
SEI framework. We placed emphasis on quantifying the impact of the control measurements
and the rate of case detecting in slowing down the spread of the disease. The model estimated
that the current case detection ratio is only 22.7%, the impact of the non-pharmaceutical in-
terventions is about 23.1% and, alarmingly, the death rate of the undetected cases is higher
than that of the detected ones. The models also showed the undetected infectious individ-
uals spread the disease more than the detected ones, i.e, the major source of the disease
transmission, an indication for the importance of social distancing. If the country continued
with the present level of measurements, the model predicted that the peak of the cumulative
Corresponding author. E-mail:
detected cases would be around mid July with more than 150 thousands cases. Such figure
far exceeds the limited capacity of country’s health system. Therefore, a diverse and compre-
hensive strategies are required to improve the efficiency of the current measurements. Both
the control measurements and efforts of case detecting are important to flatten the curve, and
the best way to control the spread of the epidemic is simultaneous increase of both elements.
1 Introduction
The currently circulating Corona virus disease 2019 (COVID-19) has reached Sudan on mid
of March 2020 or earlier, and by the end of May 2020 there are 5026 confirmed cases and
286 deaths. It is likely that there are far more numbers of undetected cases due to, in
part, limited number of tests, public hesitancy to report and, even worse, denial attitudes.
From the onset of the virus arrival, the government of Sudan has allocated isolation centres
across the country and gradually placed a series of measurements to prevent the disease from
widespread community transition. Started with closure of schools and universities in 13th of
March, to partial lockdown in 23th of March, and finally complete lockdown and intensified
media campaign about social distancing in 13th of April. The outbreak is continuously
growing at a steady step, posing a huge burden on public health and socioeconomic crisis
that affect the daily lives of millions of people. The re-opening date is set to be in early June,
but with the daily new confirmed cases in rise it is likely to be extended further.
Mathematical models proved to play a significant role in the study of infectious diseases
[1, 2]. They can provide a deeper insight into the dynamics of the spread of diseases and also
suggest effective controlling strategies to help local public health authorities in the process
of control and decisions making [3]. In this study, we present a conceivable mathematical
model for the transmission of COVID-19 in Sudan. We placed emphasis on quantifying the
impact of control measurements and the rate of case detecting in slowing down the spread
of the disease. The control measurements include government actions, public reactions and
other measures, whereby collectively we term them as “non pharmaceutical interventions” or
simply “interventions”.
Our model adopts the generic framework of SEI models and incorporates the effect of
interventions through a multi-valued parameter, a step-wise constant varying during different
phases of the interventions, designed to capture their impact in the model. We proof that
our model is mathematically consistent and present various simulation results using best
estimated parameters value. The model can be easily updated later when things have altered,
for example easing some restrictions, to accommodate such new changes. We hope that our
simulation results may guide the local authorities to make timely right decisions.
2 Description of the Model
We consider a modified version of Susceptible-Exposed-Infectious (SEI) model [4, 5] for
Corona virus transmission in a population of size Nindividuals. In this model, we divide the
total population into four sub-classes, susceptible “S”, exposed “E”, undetected infectious
Iu” and detected infectious “Id”, i.e, reported to the authorities. The latter class includes
those who went under self quarantine or hospitalized, and N(t) = S(t) + E(t) + Iu(t) + Id(t).
We assume that a susceptible individual acquires infection of corona virus following ef-
fective contact with an infected individual at a rate λand moves to the exposed class E.
An exposed individual becomes infectious at a rate σand joins the undetected class Iu. The
authorities identify an individual as detected infection at a rate αfrom the undetected class.
Both the detected and undetected infectious individuals either recover from the disease at
rates γIdand γIu, respectively; or die due to the infection at rates δId,and δIu, respectively.
The parameter αcan be viewed as the rate at which an individual leaves the undetected
infectious class. Its values range from zero to one and various factors govern the value of α
such as the level of testing capacity, diagnosis rate, and public reporting.
A diagram summarising the main structure of the model is presented in Figure S1 in the
Supplementary Materials. Given the above assumptions and notations, the model is written
as follows:
S(t) = λ(t)S(t),
E(t) = λ(t)S(t)σE(t),
Iu(t) = σE(t)(α+γIu+δIu)Iu(t),
Id(t) = αIu(t)(γId+δId)Id(t),
subject to the initial conditions:
S(0) = S00, E(0) = E00, Iu(0) = Iu00, Id(0) = Id00.
Where λ(t) denotes the force of infection and it is given by
λ(t) = (1 θ)β1Iu(t) + β2Id(t)
where β1and β2are the effective contact rates of an individual in the undetected and
detected infectious classes, respectively.
The values of the hyper-parameter θrange from 0 to 1, zero corresponds to the absence
of any intervention and one corresponds to a fully controlled scenario, which is an idealized
situation and nearly impossible to attain. Intermediate values of θbetween 0 and 1 quantify
the impact of the interventions. It reduces the number of susceptible individuals by factor
of (1 θ) and we used to account for the impact of the government actions, public reactions
and other non-pharmaceutical interventions. The higher the value of θthe smaller the size
of all classes, and the smaller the value of θthe larger the size of all classes.
Similarly, for α, the ratio of detecting the infectious individuals, the higher its value the
larger the size of detected infectious class, leading to a more controlled situation, and the
smaller the value of αthe larger the size of undetected infectious and thus resulting a less
controlled situation.
3 Analysis of the model
A complete mathematical analysis of Corona virus model in (1) from various perspectives is
given in Section S2 in the Supplementary Materials. In particular, given non-negative initial
conditions, the model is well-posed, has a non-negative solution for all t > 0, and its solution
is positively-invariant in the region
Ω = (S, E, Iu, Id)R4
(see the proof of Proposition S2.1). The model attains a stable disease free equilibrium
(DFE) at the point E0= (N, 0,0,0), which satisfies all required conditions listed in [6]. The
expression of the basic reproduction number R0, the expected number of secondary infections
caused by a single infection [7, 8, 9], derived from the model using the next generation matrix
approach [6] was follows
R0= (1 θ)β1
The first and second terms in R0represent the number of new infections produced by a
typical individual during the time she/he spends in the undetected and detected infectious
class, respectively.
The DFE point E0is is locally-asymptotically stable if R0<1, and unstable if R0>1.
The model has a unique and locally-asymptotically stable endemic equilibrium whenever
R0>1 (see Lemma S2.1 and S2.2).
4 Data
Accurate statistics for the total population of Sudan are not available, however the World
Bank and other sources [10, 11, 12] estimated the population to be around 43 million people.
We used this total number to estimate the size of susceptible class at the beginning of the
epidemic, S0.
COVID-19 rarely affect children and young people, therefore we excluded 40% of pop-
ulation and set S0= 0.6N. We obtained data of daily confirmed cases for the COVID-19
epidemic in Sudan from [13] and through official reports [14]. There is a rapid increase in the
daily new confirmed cases (see panel aof Figure 1). The death rate is above 5%, which is
relatively high, while the recovery rate is about 28% (see panel bof Figure 1). Information
for the onset of symptoms or the duration that recovered individuals spent in isolation centres
were not available.
5 Parameter Estimation
The transmission dynamics of COVID-19 varies among different communities; because the
contact rate depends on various factors that are unique for each particular community, such
as social behaviours and cultural norms. In addition, there are wide range variations of
parameters value for COVID-19 models developed thus far [15, 16, 17]. Therefore, we decided
(a) (b)
Figure 1: The cumulative number of (a) confirmed and (b) deceased and recovered COVID-
19 cases in Sudan. The red dashed vertical line in panel (a) represents the beginning of the
complete lockdown.
to estimate all parameters of the model from the cumulative confirmed cases, instead of relying
on known values from literature.
In the estimation process we used the least square method [18] to minimise the errors
between the observed (confirmed) cases and those predicted by the model. Table 1 presents
the estimated value for each parameter, and Figure 2 shows the model fit to the cumulative
confirmed data.
6 Results
Figure 2 illustrates that the model fits relatively well to the cumulative cases of COVID-19
in Sudan. We will use those optimally estimated parameter values to calculate the value of
R0, and make projections with the present measurements and other scenarios.
The value of R0obtained from the estimated parameters using Equation (2) was 2.50,
which is within the interval of internationally estimated range of R0values for the COVID-19
pandemic [17, 19, 20, 21]. R0is highly sensitive to the value of β1(see Section S3 in the
Supplementary Materials) indicating the importance of social distancing, and to lesser degree
to the values of β2and θ.
Table 1: Parameters in the model and their estimated values from the cumulative confirmed
Parameter Description Estimated value
β1Effective contact rate for the undetected infectious class 1.86
β2Effective contact rate for the detected infectious class 1.66
σ σ1is the average of initial latent period (in days) 0.27
αCase detecting rate 0.227
θimpact of non-pharmaceutical interventions 0.231
γIuRecovery rate of undetected infectious individuals 0.48
δIuDisease-induced death rate of undetected infectious individuals 0.061
γIdRecovery rate of detected infectious individuals 0.53
δIdDisease-induced death rate of detected infectious individuals 0.053
Figure 2: Model fit for the reported cumulative COVID-19 cases. The red points are the
cumulative confirmed number of COVID-19 cases and continuous blue curve represents the
model fit.
Also from the estimated parameter values in Table 1 we can deduce the followings. The
undetected infectious individuals spread the disease more than the detected ones; the mean
incubation period of the disease in the country is approximately about 4 days; case detection
ratio is 22.7% and impact of the non-pharmaceutical intervention is only 23.1%. The death
rate of the undetected cases is higher than that of the detected ones (see δIuand δIdin Table
1), but the detected cases recover faster than the undetected ones (see the inverse of γIuand
γIdin Table 1).
7 Simulating Scenarios
Having estimated values of the parameters, the model allows to consider many scenarios by
increasing and/or decreasing the value of control measurements (θ) and/or case detecting rate
(α), starting from the current situation. Continuing with the present level of measurements,
θ= 23.1% and α= 22.7%, the model predicted that the peak of the cumulative cases would
be around mid July and then the cases start to decrease. At the peak time there will be
around 390 thousands undetected cases and about 150 thousands detected cases (see panel
(a) in Figure 3). This number far exceeds the limited capacity of the fragile health system
of the country, and therefore a diverse and comprehensive strategies are required to improve
the efficiency of the current measurements.
(a) (b)
Figure 3: Simulation results of the model in case of (a) continuing with the current level of
measurements, the peak will be in mid July and about 150 thousands people will be detected;
and in case of (b) increasing θby 10% while maintaining the current case detecting rate, the
peak will be in early August and about a 73 thousands people will be detected.
Improving control measurements by increasing the current level of interventions (θ) by
factor of 10% while maintaining the present case detecting rate decreases the number of
detected and undetected cases by factor of 52% and 51%, respectively, by the peak period,
which is predicted to be around early August (see panel (b) in Figure 3).
(a) (b)
Figure 4: Simulation results of the model in case of (a) increasing αby 10% while maintaining
the current level of interventions, the peak will be around mid July and about 180 thousands
people will be detected; and in case of (b) increasing both θand αby factor 10%, the peak
will be in mid August and about 85 thousands people will be detected.
But, increasing the case detecting efforts by factor of 10% while keeping the current
level of restrictions, yields an increase in the number of detected cases by factor of 20% and
decreases the number of undetected infections by 17% at the peak time (see panel (a) in
Figure 4). Increasing both the case detecting rate and non pharmaceutical interventions by
factor of 10% will drastically decrease both the number of undetected individuals (about
61%) and the number of detected cases (about 44%), but will postpone the peak time, which
is projected to be after mid August (see panel (b) in Figure 4).
Considering the opposite direction, relaxing the current control measurements by factor of
10% and keeping the current level of case detecting rate would result 55% and 56% increase
in the total number of the detected and undetected cases, respectively (see panel (a) in
Figure 5). However, panel (b) in Figure 5 shows that decreasing the current level of control
measurements by factor of 10% and increasing the rate of case detecting by factor of 30%
would maintain the total number of undetected infections in vicinity of 400 thousands cases
and more than a twice the number of detected cases.
Table 2 summarizes the results of the above discussed scenarios. It presents the actual
(a) (b)
Figure 5: Simulation results of the model in case of (a) decreasing θby 10% while maintaining
the same case detecting; and in case of (b) decreasing θby 10% and increasing case detecting
efforts by factor of 30%.
numbers of detected and undetected infected, the corresponding R0value, and the peak time
for each scenario.
%θIncrement %αIncrement Values R0# Undetected # Detected Peak period
0% 0% 2.5 390360 150845 mid July
0% 10% 2.46 325549 181180 mid July
10% 0% 2.17 189226 73309 early August
10% 10% 2.14 152348 84992 mid August
10% 0% 3.82 607779 233962 early July
10% 30% 2.72 391870 350282 mid July
Table 2: Simulation results of the model under different scenarios.
8 Discussion
Mathematical models are integral part in the process of controlling the spread of infectious
diseases. They can provide a global picture of a disease dynamics in a community under differ-
ent scenarios and propose the best feasible strategies to contain the epidemic. In this study,
following the generic framework of SEI models, we have presented a simple deterministic
model for transmission of COVID-19 disease in Sudan, placing emphasis on how the control
measurements and effort of case detecting affect the dynamics of the disease. We also have
proved that the model is mathematically concise and well-posed (see the Supplementary
We utilized the cumulative number of confirmed cases and quantified the level of control
measurements and case detecting rate. The model estimated the current level of control
measurements to be around 23.1%, effort of case detecting is about 22.7% and R0= 2.5.
If the country continues with current level of interventions, the model predicted that at the
peak time there would be around a 150 thousands cumulative cases will be detected. This
number is beyond the capacity of the country’s health system, and that the current effort is
not enough to flatten the curve, more robust strategies are needed.
The model shows that both the control measurements and efforts of case detecting are of
equal importance to flatten the curve. The best way to control the spread of the epidemic
is simultaneous increase of both elements. The model also showed that the control mea-
surements delays the peak time but lowers the overall number of infections. The opposite is
typically true -relaxing the current level of control measurements accelerates the peak period
but increases the number of infections.
The lockdown cannot be in place for so long time as it has unbearable economic burden on
the daily-based workers and informal traders. If the country had to be open, increasing the
case detecting rate by 30%, as the model predicted, may compensate the relaxation of control
measurements. This could be achieved various ways including expanding testing capacity,
active and in time contact tracing, and intense media campaign.
The model presented here endows all limitations of deterministic dynamical models. The
precise impact of control measurements is hard to quantify in exact numbers [17], and that the
transmission dynamics of a highly contagious disease like COVID-19 can not fully captured
by a mathematical model. Nonetheless results of the model are plausible to some degree
-actual figures might be different but in general will follow similar trajectories. We hope that
authorities may consider our findings in making practical decisions.
This project is funded by the Ministry of Higher Education And Scientific Research of Sudan,
through a grant to the Institute of Endemic Diseases at University of Khartoum, and by
the African Institute for mathematical Sciences (AIMS)-Rwanda. We are grateful to Prof
Muntaser Ibrahim, Dr Mohamed Abdalla, and Prof Wilfred Ndifon for their advice and
encouragements. We also would like to acknowledge the invaluable conversations we had
with Dr Aymen Hussein.
Author Contributions
All authors contributed equally.
Conflicts of Interest
Authors declare no conflict of interest.
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Supplementary resource (1)

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SciPy is an open-source scientific computing library for the Python programming language. Since its initial release in 2001, SciPy has become a de facto standard for leveraging scientific algorithms in Python, with over 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories and millions of downloads per year. In this work, we provide an overview of the capabilities and development practices of SciPy 1.0 and highlight some recent technical developments. This Perspective describes the development and capabilities of SciPy 1.0, an open source scientific computing library for the Python programming language.
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Background: The initial cases of novel coronavirus (2019-nCoV)-infected pneumonia (NCIP) occurred in Wuhan, Hubei Province, China, in December 2019 and January 2020. We analyzed data on the first 425 confirmed cases in Wuhan to determine the epidemiologic characteristics of NCIP. Methods: We collected information on demographic characteristics, exposure history, and illness timelines of laboratory-confirmed cases of NCIP that had been reported by January 22, 2020. We described characteristics of the cases and estimated the key epidemiologic time-delay distributions. In the early period of exponential growth, we estimated the epidemic doubling time and the basic reproductive number. Results: Among the first 425 patients with confirmed NCIP, the median age was 59 years and 56% were male. The majority of cases (55%) with onset before January 1, 2020, were linked to the Huanan Seafood Wholesale Market, as compared with 8.6% of the subsequent cases. The mean incubation period was 5.2 days (95% confidence interval [CI], 4.1 to 7.0), with the 95th percentile of the distribution at 12.5 days. In its early stages, the epidemic doubled in size every 7.4 days. With a mean serial interval of 7.5 days (95% CI, 5.3 to 19), the basic reproductive number was estimated to be 2.2 (95% CI, 1.4 to 3.9). Conclusions: On the basis of this information, there is evidence that human-to-human transmission has occurred among close contacts since the middle of December 2019. Considerable efforts to reduce transmission will be required to control outbreaks if similar dynamics apply elsewhere. Measures to prevent or reduce transmission should be implemented in populations at risk. (Funded by the Ministry of Science and Technology of China and others.).
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The basic reproduction number (R0), also called the basic reproduction ratio or rate or the basic reproductive rate, is an epidemiologic metric used to describe the contagiousness or transmissibility of infectious agents. R0 is affected by numerous biological, sociobehavioral, and environmental factors that govern pathogen transmission and, therefore, is usually estimated with various types of complex mathematical models, which make R0 easily misrepresented, misinterpreted, and misapplied. R0 is not a biological constant for a pathogen, a rate over time, or a measure of disease severity, and R0 cannot be modified through vaccination campaigns. R0 is rarely measured directly, and modeled R0 values are dependent on model structures and assumptions. Some R0 values reported in the scientific literature are likely obsolete. R0 must be estimated, reported, and applied with great caution because this basic metric is far from simple.
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We derive and analyze the dynamic of a stochastic SEI epidemic model for disease spread. Fluctuations in the transmission rate of the disease bring about stochasticity in model. We discuss the asymptotic stability of the infection-free equilibrium by first deriving the closed form deterministic ( R0 ) and stochastic ( R0 ) basic reproductive number. Contrary to some author’s remark that different diffusion rates have no effect on the stability of the disease-free equilibrium, we showed that even if no epidemic invasion occurs with respect to the deterministic version of the SEI model (i.e., R0<1 ), epidemic can still grow initially (if R0>1 ) because of the presence of noise in the stochastic version of the model. That is, diffusion rates can have effect on the stability by causing a transient epidemic advance. A threshold criterion for epidemic invasion was derived in the presence of external noise.
Background: Since Dec 31, 2019, the Chinese city of Wuhan has reported an outbreak of atypical pneumonia caused by the 2019 novel coronavirus (2019-nCoV). Cases have been exported to other Chinese cities, as well as internationally, threatening to trigger a global outbreak. Here, we provide an estimate of the size of the epidemic in Wuhan on the basis of the number of cases exported from Wuhan to cities outside mainland China and forecast the extent of the domestic and global public health risks of epidemics, accounting for social and non-pharmaceutical prevention interventions. Methods: We used data from Dec 31, 2019, to Jan 28, 2020, on the number of cases exported from Wuhan internationally (known days of symptom onset from Dec 25, 2019, to Jan 19, 2020) to infer the number of infections in Wuhan from Dec 1, 2019, to Jan 25, 2020. Cases exported domestically were then estimated. We forecasted the national and global spread of 2019-nCoV, accounting for the effect of the metropolitan-wide quarantine of Wuhan and surrounding cities, which began Jan 23-24, 2020. We used data on monthly flight bookings from the Official Aviation Guide and data on human mobility across more than 300 prefecture-level cities in mainland China from the Tencent database. Data on confirmed cases were obtained from the reports published by the Chinese Center for Disease Control and Prevention. Serial interval estimates were based on previous studies of severe acute respiratory syndrome coronavirus (SARS-CoV). A susceptible-exposed-infectious-recovered metapopulation model was used to simulate the epidemics across all major cities in China. The basic reproductive number was estimated using Markov Chain Monte Carlo methods and presented using the resulting posterior mean and 95% credibile interval (CrI). Findings: In our baseline scenario, we estimated that the basic reproductive number for 2019-nCoV was 2·68 (95% CrI 2·47-2·86) and that 75 815 individuals (95% CrI 37 304-130 330) have been infected in Wuhan as of Jan 25, 2020. The epidemic doubling time was 6·4 days (95% CrI 5·8-7·1). We estimated that in the baseline scenario, Chongqing, Beijing, Shanghai, Guangzhou, and Shenzhen had imported 461 (95% CrI 227-805), 113 (57-193), 98 (49-168), 111 (56-191), and 80 (40-139) infections from Wuhan, respectively. If the transmissibility of 2019-nCoV were similar everywhere domestically and over time, we inferred that epidemics are already growing exponentially in multiple major cities of China with a lag time behind the Wuhan outbreak of about 1-2 weeks. Interpretation: Given that 2019-nCoV is no longer contained within Wuhan, other major Chinese cities are probably sustaining localised outbreaks. Large cities overseas with close transport links to China could also become outbreak epicentres, unless substantial public health interventions at both the population and personal levels are implemented immediately. Independent self-sustaining outbreaks in major cities globally could become inevitable because of substantial exportation of presymptomatic cases and in the absence of large-scale public health interventions. Preparedness plans and mitigation interventions should be readied for quick deployment globally. Funding: Health and Medical Research Fund (Hong Kong, China).
A precise definition of the basic reproduction number, R o , is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if R o < 1, then the disease free equilibrium is locally asymptotically stable; whereas if R o > 1, then it is unstable. Thus, R o is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super-and sub-threshold endemic equilibria for R o near one. This criterion, together with the definition of R o , is illustrated by treatment, multigroup, staged progression, multistrain and vector-host models and can be applied to more complex models. The results are significant for disease control.