Modeling the dynamic of COVID-19 with different types of transmissions.

In this paper, we propose a new epidemiological mathematical model for the spread of the COVID-19 disease with a special focus on the transmissibility of individuals with severe symptoms, mild symptoms, and asymptomatic symptoms. We compute the basic reproduction number and we study the local stability of the disease-free equilibrium in terms of the basic reproduction number. Numerical simulations were employed to illustrate our results. Furthermore, we study the present model in case we took into consideration the vaccination of a portion of susceptible individuals to predict the impact of the vaccination program.

Introduction

Coronaviruses are a large family of viruses that have a crown. Viruses that cause common colds and life-threatening illnesses, such as Middle East Respiratory Syndrome (MERS-CoV) and Severe Acute Respiratory Syndrome (SARS-CoV-1) [11]. A new member of the coronavirus family is COVID-19 (Coronavirus disease 2019 or novel coronavirus), which is a highly contagious viral disease caused by the virus known to the virologist’s community as Severe Acute Syndrome Coronavirus-2 (SARS-CoV-2) [26]. The virus was first identified in December 2019 in Wuhan, China, and quickly spread to some other parts of China and even other countries [9].

Symptoms of COVID-19 include breathing difficulty, fatigue, fever, dry cough, tiredness, conjunctivitis, chest pain, loss of speech, diarrhea, and sore throat, and also in severe cases pneumonia, multi-organ failure, acute respiratory distress syndrome, septic shock, arrhythmias, myocarditis, blood clots, heart failure, seizure, encephalitis, stroke, and Guillain Barré syndrome, bilateral lung penetration [8], [31]. Also, some patients may suffer from diarrhea, loss of appetite, taste, or smell, without any signs of breathing disorder [19], [28]. Other symptoms, such as gastroenteritis and neurological diseases of varying severity, have also been reported [10]. However, some cases are asymptomatic.

On January 30, 2020, the World Health Organization (WHO) declared the outbreak as a public health emergency of international concern. Furthermore, the WHO announced the COVID-19 outbreak as a pandemic on March 11, 2020 [12], [48].

According to data reported by WHO, by the end of March 2020, there were more than 450,000 reported laboratory-confirmed human infection, including 20,000 cases that led to the death. Globally, as of March 07, 2021, there have been 116,135,492 confirmed cases of COVID-19, including more than 2,581,976 cases resulting in death, and 249,160,837 vaccine doses have been administrated across 217 countries and territories, including Morocco [52]. In Morocco, the first COVID-19 case was reported on March 16, 2020. As of March 01, 2021, there were a total of 483,766 confirmed positive cases, 8637 deaths, and 469,345 recoveries in the country [51].

Mathematical analysis and modeling is an essential part of infectious disease epidemiology. Application of mathematical models in epidemiology allows to analyze the dynamic of diseases and support control strategies [3], [20], [33]. The global problem of the pandemic COVID-19 has attracted the interest of mathematical modelers and epidemiologists to investigate many of the details of the disease relating to transmission, evolution, prevention, and control of the pandemic. In this way, several studies have been published to comprehend the dynamics of COVID-19 and to find the likely outcome of an outbreak that is beneficial for public health initiatives and awareness programs.

Precisely, several epidemiological modeling studies have been done to explore the transmission dynamics of COVID-19 in the human population: Wu et al. [49] employed a simple susceptible-exposed-infectious-recovered (SEIR) based model to forecast the potential of the COVID-19 to spread in China and beyond. They estimated the reproduction number () and epidemic doubling time, indicating the exponential growing nature of the COVID-19 outbreak. Tang et al. [44] developed and analyzed a deterministic model which incorporates quarantine and hospitalization to estimate the transmission risk of the COVID-19 and its implication for public health interventions. Their model was extended in Ngonghala et al. [37] by dividing infectious compartment into two essential compartments of hospitalized and isolated individuals, and those in intensive care units to assess the impact of the national provider identifier on curtailing the spread of the COVID-19 pandemic. Musa et al. [35] proposed a deterministic model to show the importance of timely quarantine and hospitalization to reduce the epidemic. In [34] the authors proposed to study the transmission dynamics of COVID-19 in Nigeria. Their model incorporates different hospitalization measures for mild and severe cases to assess the effect of awareness programs on the dynamics of COVID-19 infection.

In [38], the authors presented a mathematical model qualitatively using stability theory of differential equations and the basic reproduction number that represents an epidemic indicator obtained from the largest eigenvalue of the so-called next-generation matrix to find the likely outcome of an outbreak that is beneficial for public health initiatives. Other mathematical models with ordinary time derivatives can be found in literature [4], [14], [29], [30], [31], [38].

In recent years, fractional-order dynamical systems have been appeared in several areas of science and engineering as a field of mathematical analysis [22]. It deals with the investigation and applications of integrals and derivatives of arbitrary order instead of classical integer-order and integration, see [5], [23], [25], [32], [39], [45], [46].

Nowadays, many researchers have focused their interest on investigating the dynamics of fractional-order in connection with the pandemic COVID-19, since it can more accurately explain natural phenomena than the differential equations of the integer-order, especially those associated with hereditary properties and history-based phenomena [40]: In [1], Akgul, Ahmed et al. analyzed a model of differential equation related to COVID-19. They used fractal-fractional derivatives, and they analyzed the equilibria of the model and its stability in detail and solved the model numerically. In [17], the authors considered the model (SIRU): susceptible asymptomatic infectious reported symptomatic and unreported symptomatic infectious they used the operator called Caputo fractional operator to the reported and unreported cases by analyzing a time-fractional model and finding its solution, see also [53]. The authors of [18] studied the model (SEIARM): susceptible exposed infected asymptotically-infected recovered and reservoir defined by a system of six equations, they generalized this model to incorporate memory consequences and hereditary properties, using the fractional derivative in the sense of Caputo. Singh et al. [43] proposed a mathematical model (ABCDE): susceptible exposed infected asymptomatic and recovered They replaced time-derivative in model with fractional-order time-derivative. They studied the COVID-19 infection by fractional-order model, following the Grünwald-–Leitnikov fractional derivative.

In literature, several different fractional-operators related to COVID-19 are being proposed to study the dynamics of fractional-order epidemic models, see [2], [6], [7], [16], [21], [36], [41], [42].

Motivated by the afore-mentioned works, we intend first to develop a new deterministic model, which extends the models developed in Musa et al. [34] and Ngonghala et al. [37]. We do this by taking into account several essential properties of the pandemic COVID-19, such as the existence of individuals tested positive for COVID-19 with severe, mild, or asymptomatic symptoms, and dividing infectious compartment into two essential compartments of hospitalized individuals and those in intensive care units. We study the stability properties of the solutions of a proposed nonlinear mathematical model with nine compartments, namely, susceptible-exposed-infectious with severe symptoms-infectious with mild symptoms-asymptomatic infectious-hospitalized-intensive care unit-dead infectious corpses-recovered to investigate the current outbreak of coronavirus disease (COVID-19) in Morocco and beyond. We hope that this study will provide better hospitalist bed management and a clear guidance for public health measures to combat the spread of COVID-19. The manuscript is organized as follows: In Section 2, we introduce a new model for COVID-19. In Section 3, we investigate a qualitative analysis of the model and compute the basic reproduction number of the COVID-19 system model; we study the local stability of the disease-free equilibrium in terms of basic reproduction number. In Section 4, we add vaccination into consideration in our model, and we study its stability under vaccination. Section 5, concerns data fitting, we illustrate our model by numerical simulation, and we compared it with actual data of Morocco. Lastly, we give a conclusion and future work in Section 6.

The model proposed

In this section, we present a new model which is a generalization of the models studied in Musa et al. [34] and Ngonghala et al. [37]. The model take into account the existence of individuals with severe, mild, or asymptomatic symptoms, we propose a new epidemiological compartment model that takes into consideration the difference between individuals with severe symptoms, mild symptoms, and without symptoms. The model under consideration subdivides the population of humans at time into five compartments. That is, susceptible class exposed class severe symptoms infectious individuals mild severe symptoms infectious individuals infectious but asymptomatic individuals hospitalized intensive cure unit class recovery with immunity class and dead class . Before presenting the model, we put some assumption as given by the following assertions:

The disease is transmitted by contact between an infected individual and a susceptible individual.

All susceptible individuals are equally susceptible, and all infected individuals are equally infectious.

The population size is constant and equal to . This means that no births or migration occurs, and all deaths are taken into account.

Initially, everyone in the population is susceptible to the contagious disease, except for a small number of individuals who are already infected.

The recovered individuals do not return to susceptible class; that is, recovered individuals have immunity against the disease; they cannot become infected again and cannot infect susceptible either.

Infected individuals in the hospital or intensive care unit (ICU) are isolated, then they do not contribute to the transmission of the infection.

The model takes the following form summarizing the main structure of our model.

where is the human-to-human transmission coefficient per unit time per person; represents the rate at which an individual leaves the exposed clan by becoming infectious. The period is called an incubation period. The parameter is the probability that an individual leave exposed compartment and become symptomatic infectious with severe symptoms ; is the probability at which exposed individuals become infectious with mild symptoms ; while is the probability at which exposed individuals went to asymptomatic clan ; will be the rate at which an individual leave the compartment ; is the probability at which a person in went to the compartment of hospitalized individuals; is the recovery rate of people with mild symptoms and asymptomatic people without being hospitalized; is the death rate of hospitalized patients without intensive care; is the death rate of hospitalized patients with intensive care .

Under the diagram in Fig. 2, the evolution of the compartments mentioned above is modeled by the following system of ordinary differential equations.with initial conditions: . Where and denote the number of susceptible individuals, exposed individuals but not yet infectious, infectious individuals with severe symptoms, infectious individuals with mild symptoms, asymptomatic individuals, hospitalized individuals, individuals in intensive cure unit, recovered by immunity individuals and dead individuals, at time respectively.

Fig. 2

Flowchart of model (4.1).

The reproduction number

In epidemiology, the basic reproduction number denoted of infection can be thought of as the expected number of cases directly generated by one case in a population where all individuals are susceptible to infection. The method to compute the basic reproduction number using the next-generation matrix is given by Diekmann et al. (see [13]) and elaborated by van den Driessche and Watmough see [15]. In our system there exists a disease-free equilibrium denoted which is given by

In order to calculate the basic reproduction number based on this steady state, we consider the following subsystem:

Let the matrix associated to the rate of the appearance of new infections and The matrix associate with the net rate out of the corresponding compartments. Then the generation matrices and which are the Jacobian matrices of and at respectively, are given byandThe basic reproduction number is obtained as the spectral radius of precisely,

The disease free equilibrium of system (2.1) , is locally asymptotically stable if and unstable if .

From the system (2.1) we have that :Furthermore, since the total population size is constant, one hasTherefore, the local stability of model (2.1) can be studied through the remaining coupled system of state variables, namely, the variables and in system (2.1). The system associated to these variables at the disease-free equilibrium is the following:The Jacobian matrix associated to these variables of system (3.3) at is given by :The Jacobian matrix of system (3.3) at is The matrix has always two negative eigenvalues the other eigenvalues of are determined by the equationwhere

Next, by using the Routh-Hurwitz criterion, all the roots of are negative or have negative real part if, the following conditions are satisfied:

and

.

We have the following two cases:

If then the conditions (1)–(4) are satisfied, hence real parts of all the eigenvalues of the matrix are strictly negative, thus the disease free equilibrium of the system (2.1) is locally asymptotically stable. Hence the disease will decay.

If the condition (4) is not satisfied, then at least one eigenvalue has a positive real part, thus the disease-free equilibrium of the system is unstable saddle point. In this case, the disease can resist.

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Epidemic model with vaccination

Vaccination has been established as a powerful tool in managing and controlling infectious diseases by providing protection to susceptible individuals [47]. This section aims to study the dynamics of the model with vaccination. We assume that a certain proportion of individuals in the susceptible class are vaccinated. In this case, vaccinated individuals are moved to a new compartment . Let be the proportion of the population vaccinated per unit of time. Since the vaccine does not provide immunity to all vaccine recipients, vaccinated individuals may become infected but at a lower rate than unvaccinated. In this case, let such that be the vaccine efficacy.

The diagram 1 becomes:

Fig. 1

Flowchart of model (2.1).

This diagram can be translated to the following system of ordinary differential equations.with initial conditions: . Where is number of vaccinated individuals at time and is the rate at which vaccine wanes. Then, the system (4.1) has one equilibrium point . The infection components in this model are and . Then the infection matrix and the transition matrix are given byDifferentiating and with respect to and and evaluating at the disease-free equilibrium respectively, we getandThus, the reproduction number for the vaccinated model is:This means that the disease free equilibrium will be asymptotically stable if and unstable if . The critical percentage of the population necessary to achieve herd immunity is: represents the proportion for which the basic reproductive number under vaccination is equal to 1.

Numerical simulations: the case study of Morocco

In Fig. 3 the daily confirmed, dead, and recovered cases of COVID-19 have been depicted from July 01, 2020, to March 01, 2021, which becomes seven months (or 210 days). We perform numerical simulations to compare the results of our model with the actual data in Fig. 3. The predicted evolution of the outbreak of COVID-19 without and with vaccination in Morocco can be seen in Fig. 4 and Fig. 5 , respectively. The parameters of the mathematical model were fitted with the data provided from [50] and collected in Table 1 . We enlarge the plots by taking the maximum number of people 600,000.

Fig. 3

Evolution of COVID-19 confirmed, recovered and dead cases in Morocco per day.

Fig. 4

Epidemic evolution predicted by the model.

Fig. 5

Epidemic evolution predicted by the model vaccinated.

Table 1

Values of the model parameters corresponding to the situation of Morocco.

Name	Description	Value	Units
S(0)	Initial susceptible population	37,000,000 [50]	dimensionless
E(0)	Initial Number of exposed population	185 (Estimated)	dimensionless
I_ss(0)	Initial Number of infected people with severe symptoms	100 (Estimated)	dimensionless
I_ms(0)	Initial Number of infected people with mild symptoms	100 (Estimated)	dimensionless
I_a(0)	Initial Number of infected people without symptoms	3 (Estimated)	dimensionless
H(0)	Initial Number of hospitalized people	370 (Estimated)	dimensionless
I_cu(0)	Initial Number of population in the ICU	37 (Estimated)	dimensionless
D(0)	Initial Number of dead people	0 (Estimated)	dimensionless
R(0)	Initial Number of recovered people	0 (Estimated)	dimensionless
β	Transmission coefficient from infected individuals	0.45 [54]	dimensionless
k	Rate at which exposed people become infectious	1/5.1 [27]	day^−1
p_1	Rate at which exposed people become infectious with sever symptoms	0.3 (Estimated)	day^−1
p_2	Rate at which exposed people become infectious with mild symptoms	0.4 (Estimated)	day^−1
h	Rate at which infectious people with sever symptoms leave the compartment	0.5 (Estimated)	day^−1
q_1	Rate at which infectious people with sever symptoms become hospitalized	0.8 (Estimated)	day^−1
δ_1	Death rate of hospitalized people	0.035 [50]	day^−1
γ_1	Death rate of ICU people	0.6 (Estimated)	day^−1
γ_3	Recovery rate of asymptomatic infectious people	0.4[24]	day^−1
(1−σ)	Efficacy of the vaccine	0.9 (Estimated)	dimensionless

From Figs. 4 and 5, we observe that all the trajectories adapt the same pattern and converge to the 0 point. In Fig. 4, we observe from the plot that exposed asymptomatic infected with severe symptoms infected with mild symptoms hospitalized and people in ICU, increased to the peak which is after 150 days from July 01, 2020. This is compatible with the data represented in Fig. 3. Thus, we show that our COVID-19 model describes well the real data of daily confirmed, recovered, and dead cases during these seven months (from July 01, 2020, to March 01, 2021). In Fig. 5, we observe that the plot is flatter than the one in Fig. 4; also the curve of asymptomatic people is almost identic to the x-axis, which shows the importance of vaccination program to reduce the epidemic.

Conclusion and future directions

Many models have been considered to study the new epidemic COVID-19. Here we have taken into consideration the different characteristic of COVID-19, and their relation with entering the hospital, the intensive care units or not, we propose a good model that describes the evolution of COVID-19 in Morocco, giving a good approximation of the reality of the Moroccan outbreak (see Fig. 4), and giving a simulation of this model under vaccination (see Fig. 5). In our future work, we intend to generalize the above model using fractional calculus, which can be more accurate to explain natural phenomena more than the classical differential. Furthermore, the model will be generalized to incorporate memory consequences and hereditary properties.

Funding

This work was carried out as part of the project “Gestion de la COVID-19: approche systéme dynamique” financed by the research program related to “COVID-19” 2020 and supported by MENFPESRS, Morocco grant, and CNRST, Morocco grant.

CRediT authorship contribution statement

Mohamed Amouch: Writing - original draft, Writing - review & editing. Noureddine Karim: Writing - original draft, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.