1 Introduction and motivation
Deterministic models are important decision tools that can be useful to forecasting different scenarios. The first motivation of studying such models is based on the use of the theory of ordinary/partial differential equations and a low computational complexity which can permit a better calibration of the model characteristics. Furthermore, deterministic approached are the only suitable methods that can be used when modeling a new problem with few data. For more details, we refer the readers to[2, 30, 11, 49] and references therein. The use of the mathematical models as a predictive tool in the simulation of complex problems arising in a broad range of practical applications in biology, environmental fluid mechanic, chemistry and applied mathematics (for example: mathematical model in population biology and epidemiology, mixed Stokes-Darcy model, Navier-Stokes equations, nonlinear time-dependent reaction-diffusion problem, heat conduction equation and unsteady convection-diffusion-reaction equations) represents a good candidate for developing efficient numerical schemes in the approximate solutions of such problems [40, 34, 9, 10, 39, 38, 36, 14, 15, 32, 41, 28, 21, 43, 45, 53]. For parabolic partial differential equations (PDEs) which present strong steep gradients (for instance: shallow water flow and advection-diffusion equations), numerical algorithms are needed with good resolution of steep gradients [12, 8, 20, 22, 31, 44, 27, 29].
Early in an epidemic, the quality of the data on infections, deaths, tests and other factors often are limited by undetection or inconsistent
detention of cases, reporting delays, and poor documentation, all of which affect the quality of any model output. Simpler models may provide
less valid predictions since they cannot capture complex and unobserved human mixing patterns and other time-varying parameters of infectious
disease spread. Also, complex models may be no more reliable than simpler ones if they miss key aspects of the biological entities (either ions,
molecules, proteins or ceils) . At a time when numbers of cases and deaths from coronavirus (Covid-19) pandemic continue to
increase with alarming speed, accurate forecasts from mathematical models are increasingly important for physicians, politicians, epidemiologists,
the public and most importantly, for authorities responsible of organizing care for the populations they serve. Given the unpredictable behavior
of severe acute respiratory syndrome Covid-19, it is worth mentioning that efficient numerical approached are the best tools that can be used to
predict the spread of the disease with reasonable accuracy. These predictions have crucial consequences regarding how quickly and strongly the
government of a country mores to curb a pandemic. However, assuming the worst-case scenario at state and national levels will lead to
inefficiencies (such as: the competition for beds and supplies) and may compromise effective delivery and quality of care, whereas supposing
the best-case scenario can conduct to disastrous underpreparation.
Covid-19 is a rapidly spreading infectious disease caused by the novel coronavirus SARS-Cov-2, a betacoronavirus which has provided a global epidemic. Up today, no drug to treat the Covid-19 disease is officially available (approved by the World Health Organization (WHO)) and a vaccine will not be available for several months at the earliest. The only approaches widely used to slow the spread of the pandemic are those of classical epidemic control such as: physical distancing, contact tracing, hygiene measure, quarantine and case of isolation. However, the primary and most effective use of the epidemiologic models is to estimate the relative effect of various interventions in reducing disease burden rather than to produce precise quantitative predictions about extent on duration of disease burdens. Nevertheless, consumers of such models including the media, the publics and politicians sometimes focus on the quantitative predictions of infectious and mortality estimates. Such measures of potential disease burden are also necessary for planners who consider future outcomes in light of health care capacity. The big challenge consists to assess such estimates.
In this paper, we develop an efficient numerical scheme for solving a mathematical model well adapted to Covid-19 pandemic subjected to special
characteristics (effect of undetected infected cases, effect of different sanitary and infectiousness conditions of hospitalized people and
estimation of the needs of beds in hospitals) and considering different scenarios . Specifically, the proposed technique should
provide the numbers of detected infected and undetected infected cases, numbers of deaths and needs of beds in hospitals in countries (for
example, in Cameroon) where Covid-19 is a very serious health problem. It is Worth noticing that the model of Covid-19 considered in this work
has been obtained under the asumption of ”only within-country disease spread” for territories with relevant number of people infected by
SARS-Cov-2, where local transmission is the major cause of the disease spread (for instance: case of Cameroon). Furthermore, the parameters
of the model used in this note are taken from the literature [24, 25, 19]. Our study also relates the disease fatality rate with
the percentage of detected cases over the real total infected cases which allows to analyze the importance of this percentage on the impact
of Covid-19. In addition, to demonstrate the efficiency and validity of the new approach when applied to the mathematical problem of coronavirus
epidemic, we consider the case of Cameroon, the country of the central Africa where one can observe the highest number of people infected
by the new virus SARS-Cov-2. We compare the results produced by the numerical method to the data obtained from this country and those provided by
the World Health Organization in its reports . Finally, it important to mention that the considered area (Cameroon) in the
numerical experiments can be replaced by any territory worldwide.
This paper is organized as follows: Section 2 considers some preliminaries together with the mathematical formulation of Covid-19 spreading. In section 3, we provide a full description of the two-level second-order explicit scheme for solving the problem indicated in 2. Section 4 analyzes the stability and the convergence rate of the new procedure while a large set of numerical experiments are presented and critically discussed in Section 5. We draw in section 6 the general conclusion and we provided our future investigations.
2 Preliminaries and mathematical model of SARS-Cov-2 infectiousness
We use a mathematical formalism  that describes how infectiousness varies as a function of time since infections for a representative
cohort of infected persons. We assume that transmission of SARS-Cov-2 is contagious from person to person and not point source. Furthermore, it is
also assumed that, at the initial phase of Covid-19 disease, the proportion of the population with immunity to SARS is negligible
[3, 16, 4, 48]. At the beginning of a contagious epidemic, a small number of infected people start passing the disease to a large
population. Individuals can go through nine states. They start out susceptible (: the person is not infected by the disease pathogen),
exposed (: the person is in the incubation period after being infected by the disease pathogen, but has no clinical signs), infected
(: the person has finished the incubation period, may infected other people and start developing the clinical signs. Here, people can
be taken in charge by sanitary authorities of this country (hospitalized persons) or not detected by the authorities and continue as infectious),
infectious by undetected (: the person can still infect other individuals, have clinical signs, but is not detected and reported by the
authorities (these people will not die)), hospitalized or in quarantine at home (: the person is in hospital or in quarantine at home,
can still infect other people, but will recover), hospitalized but will die (: the person is hospitalized and can infect other people,
but will die), recovered after being previously infectious but undetected (: the person was not previously detected as infectious,
survived the disease, is no longer infectious and has developed a natural immunity to the disease), recovered after being previously detected
as infectious (: the person survived the disease, is no longer infectious and has developed a natural immunity to the virus, but she/he
remains in hospital for a convalescence period days), dead by SARS-Cov-2 ().
The proposed model is based on thirteen parameters.
denotes basis reproductive number, that is, the expected number of new infectious cases per infectious cases,
is the number of persons in a considered country before the starting of the pandemic,
designates the natality rate () in the considered country (the number of births per day and per capita),
represents the mortality rate () in the considered country (the number of deaths per day and per capita),
denotes the case fatality rate in the considered territory at time (the proportion of deaths compared to the total number of infectious people (detected or undetected). Here, and are the minimum and maximun case fatality rates in the country, respectively),
means the fraction of infected people that are detected and reported by the authorities in the country at time For the convenience of writting, we assume that all the deaths due to Covid-19 are detected and reported, so ,
for are the disease contact rates () of a person in the corresponding compartment , in the country (without taking into account the control measures),
represents the disease contact rates () of a person in compartment , in the country (without taking into account the control measures), where the fraction of infected individuals that are detected is ,
designates the transition rate () from compartment to compartment . It’s the same in all the countries,
is the transition rate () from compartment to compartments , or at time . It can change from a country to another,
, and denote the transition rate () from compartments , or to compartments , and , respectively, in the considered country at time ,
for are functions representing the efficiency of the control measures applied to the corresponding compartment in the considered country at time
is the person infected that arrives in the territory from other countries per day. is the person infected that leaves the territory from other countries per day. Both can be modeled following the between-country spread part of the Be-CoDis model, see .
isolation: infected people are isolated from contact with other persons. Only sanitary professionals are in contact with them. Isolated patients receive an adequate medical treatment that reduces the Covid-19 fatality rate,
quarantine: movement of people in the area of origin of an infected person is restricted and controlled (for instance: quick sanitary check-points at the airports) to avoid that possible infected people spread the disease,
tracing: the aim of tracing is to identify potential infectious contacts which may have infected an individual or spread SARS-Cov-2 to other people. Increase the number of tests in order to increase the percentage of detected infected persons,
increase of sanitary resources: number of operational beds and sanitary personal available to detect and treat affected people is increased, producing a decreasing in the infectious period for the compartment
Furthermore, the mathematical model of coronavirus epidemic considers the following assumptions:
the population at risk is large enough and time period of concern is short enough that over the time period of interest, very close to of the population is susceptible,
the pandemic is at the early stage and has not reached the point where the susceptible population decreases so much due to death or post-infection immunity that the average number of secondary cases falls,
unprotected contact results in infection,
the epidemic in the population of interest begins with a single host (note that the equations used in computing cases and deaths are easily modified if this is not the case),
infectivity occurs during the incubation period only,
the models are deterministic, that is, the thirteen parameters of Covid-19 spread cited above are constant values.
Under these assumptions, the mathematical formulation of Covid-19 disease is given by the following system of nonlinear ordinary differential equations:
with the initial conditions
where all the unknowns dependent on the time Setting and where
the system of nonlinear equations - is equivalent to
In the modeling point of view, the term corresponds to the apparent fatality rate of the disease (obtained by considering only the detected cases) in the considered area at time whereas is the real fatality rate of coronavirus disease.
Since the mathematical model of Covid-19 provided by the system of equations - is too complex and because both natality and mortality (not from SARS-Cov-2) do not seem to be useful factors for this pandemic (at least for relatively short periods of time), we assume in the rest of this paper that
It is worth mentioning that the aim of this paper is to compute the following Covid-19 characteristics:
the model cumulative of coronavirus cases at day given by
the model cumulative number of deaths (due to Covid-19) at day , which is given by ,
and which are the basic reproductive number and effective reproductive number of Covid-19
the number of people in hospital is estimated by the following equation
where represents the fraction, at time of people in compartment that are hospitalized and days is the period of convalescence (i.e., the time a person is still hospitalized after recovering from Covid-19). This function can help to estimate and plan the number of clinical beds needed to treat all the SARS-Cov-2 cases at time ,
the maximum number of hospitalized persons at the same time in the territory during the time interval which is defined as
can help to estimate and plan the number of clinical beds needed to treat all the Coronavirus cases over the interval
the number of people infected during the time interval by contact with people in compartments , and , respectively. They are defined as
(17) (18) (19)
We recall that the basis reproduction number is defined as the number of cases an infected individual generates on average over the
course of its infectious period, in an otherwise uninfected population and without special control measures. It depends on the considered
population, but does change during the spread of the disease, while the effective reproduction number is defined as the number of
cases one infected person generates on average over the course of its infectious period. A part of the population can be already infected
and/or special control measures that have been implemented. It depends on the spread of the disease. In addition, and
the evolution of the epidemic slow down when
Now, applying the next generation method  to the nonlinear system - to get
where for the sake of simplicity of notations, all previous coefficients correspond to their particular values at times and
In the literature [26, 52], it is established that the observed patterns of Covid-19 are not completely consistent with the hypothesis
that high absolute humidity may limit the survival and transmission of the virus, whereas the lower is the temperature, the greater is the
survival period of the SARS-Cov-2 outside the host. Since there is no scientific evidence of the effect of the humidity and the temperature
on the SARS-Cov-2, these factors are not included in our model.
Focusing on the application on the control strategies, the efficiency of these measures indicated in  satisfies equations
for , where measures the intensity of the control measures (greater value implies lower value of disease contact
rates), (in ) simulates the efficiency of the control strategies (greater value implies lower value of disease
contact rates) and , , denotes the first day of application of each control strategy.
is the first day of application of a control measure that was being used before if any. In this
work, represents the number of changes of control strategy. In general, the values of are typically taken
in the literature (using dates when the countries implement special control measures). It is important to remind that some of the values
of can be also sometimes known. The rest of the parameters needed to be calibrated.
In the following, we assume that the case fatality rate , depends on the considered country, time , and it can be affected by the application of the control measures (such as, earlier detection, better sanitary condition, etc…). Thus, it satisfies equation
where is the case fatality rate when no control measures are applied (i.e., ) and is
the case fatality rate when implemented control measures are fully applied ().
the transition rate from to depends on the disease and, therefore is considered constant, that is
the value of can be increased due to the application of control measures (that is, people with symptoms are detected earlier). As a consequence, the values of , and can be decreased (i.e., persons with symptoms stay under observation during more time),
for the sake of readability, the infectious period for undetected individuals is the same than that of hospitalized people that survive the disease. So . Furthermore, we suppose that the additional time a person is in the compartments and is constant, so it comes from  that
(24) (25) (26)
where represents the decease of the of the duration of the function due to the application of the control measures at time is the maximum number of days that can be deceased due to the control measures.
Finally, the disease contact rate is defined by
where and are suitable lower and upper bounds, respectively. For the convenience of writing, we assume that In addition, the people in compartments , , and are less infectious than people in compartment (due to their lower virus load or isolation measures). This fact results in
3 Construction of the two-level explicit numerical scheme
In this section, de develop the robust two-level explicit scheme for solving the mathematical problem - modeling the spread
of Covid-19 with undetected cases.
Let be the step size, is a positive integer. Set
for and be a regular partition of
. Let be the grid functions space defined on
Define the following norms
where designates the norm defined on the field of complex numbers . Furthermore, denote
where the function is given by
be a polynomial of degree interpolating the function at the node points . According to
equations -, it’s important to remind that is not necessarily the interpolation polynomial of
degree of the function at the node points .
Now, integrating both sides of equation at the node points and , this yields
which is equivalent to
For is a linear polynomial approximating the function at the points and . Using equations and , it is easy to observe that
where the error term is given by
where (respectively, each component of ) is an unknown function which is between the maximum and the minimum of the numbers , and (respectively: , and ). But equation can be rewritten as
Substituting approximation into the ith equation of the system , we obtain
which is equivalent to the following system
The integration of both sides of equation provides
where the last two equalities come from the identities , and To get the desired first-level of the new algorithm, we should approximate the sum by the term , in which the coefficients , , and , are real numbers and are chosen so that the Taylor expansion
The application of the Taylor series expansion for and about and , respectively, with step size using forward difference representations gives