# Modelling and Simulating the Novel Coronavirus with Implications of Asymptomatic Carriers

Authors: Ghassane Benrhmach; Khalil Namir; Jamal Bouyaghroumni
Journal: International Journal of Differential Equations
 (2020)

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## Abstract

The World Health Organization declared that the total number of confirmed cases tested positive for SARS‐CoV‐2, affecting 210 countries, exceeded 3 million on 29 April 2020, with more than 207,973 deaths. In order to end the global COVID‐19 pandemic, public authorities have put in place multiple strategies like testing, contact tracing, and social distancing. Predictive mathematical models for epidemics are fundamental to understand the development of the epidemic and to plan effective control strategies. Some hosts may carry SARS‐CoV‐2 and transmit it to others, yet display no symptoms themselves. We propose applying a model (SELIAHRD) taking in consideration the number of asymptomatic infected people. The SELIAHRD model consists of eight stages: Susceptible, Exposed, Latent, Symptomatic Infected, Asymptomatic Infected, Hospitalized, Recovered, and Dead. The asymptomatic carriers contribute to the spread of disease, but go largely undetected and can therefore undermine efforts to control transmission. The simulation of possible scenarios of the implementation of social distancing shows that if we rigorously follow the social distancing rule then the healthcare system will not be overloaded.


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## Body

## 1. Introduction

By the end of 2019, the city of Wuhan, China, and the rest of the world have seen the apparition of a deadly virus, the most recently discovered coronavirus, named “SARS-CoV-2” [1]. Few weeks later, COVID-19 spread widely in China and other countries, making the World Health Organization (WHO) to declare a global pandemic in March 2020 [2]. There have been 3,090,445 confirmed cases and 217,769 confirmed deaths in 212 countries around the globe until 30 April 2020 according to the World Health Organization.In order to stop the rapid spread of the coronavirus, many countries have introduced widespread physical distancing measures, population-level movement restrictions, and health education knowledge [3, 4]. However, one of the biggest problems is the existence of asymptomatic infection (having very mild symptoms) among the infected people, and unaware of their contagious ability increases the transmission rate [5]. Recently, and to keep up with the biologists, many mathematical models have been developed to enable a more effective response to this hazard. In [6], Peng et al. have numerically studied a SEIR model that includes intrinsic impact of hidden exposed infected population to describe the epidemic of COVID-19. They have estimated the time of the outbreak of this epidemic in several cities in China. However, they did not incorporate the transmission paths and speed of such an epidemic. In [7], M. Peirlinck et al. have introduce the effect of asymptomatic infection on the outbreak of Covid-19 by proposing a SEIIR model and focusing in decomposing the infectious population into a symptomatic and an asymptomatic group and assuming a similar transmission rate β for both groups. They found that the large asymptomatic population is associated with a high risk of Covid-19 spread. On the other hand, the epidemic progresses across countries, and hospital staff are faced with a massive influx of patients and a large number of deaths, causing a huge problem in resource management. Motivated by these reasons, we propose a SELIAHRD model including eight states (Susceptible, Exposed, Latent, Symptomatic, Asymptomatic, Hospitalized, Recovered, and Death) taking into consideration the total population as susceptible, age, and population structure is not considered [7, 8]. This model was developed based on studies of COVID-19 and the H1N1 influenza epidemic model proposed by Lopez et al. [9] however, in this model, they suppose the existence of vaccinated people and used Generalized regression neural network for simulation without taking in consideration the social distancing scenarios.This article begins with a section that describes the proposed epidemic model and gives an idea about each compartment of the model. The second section presents a discussion of the well-posedness and equilibria of the proposed model. The stability of the disease-free equilibrium, the use of next-generation operator to get the basic reproductive numberR0, and the proof of the global stability of the disease-free equilibrium are presented in the third section. The results of the numerical simulation and their discussion are given in the fourth section.


## 2. Proposed Epidemic Model

As per the recent report from the WHO, a significant number of COVID-19 patients have very mild symptoms. This means the existence of asymptomatic infections, which increase the transmission rate dramatically in some countries.To take into consideration the effect of undetectable infected persons, we propose the SELIAHRD model including eight compartments shown in Figure1.Figure 1
SELAIHRD model.The susceptible state is a person without the disease at timet. The exposed state refers to those near the infected person and has higher risk to get infected. The exposed persons who got infection but not yet infectious are called latent people. From the latent compartment, we can classify the people as symptomatic and asymptomatic people; the symptomatic people will be admitted to the hospital. Asymptomatic and hospitalized people have the probability to go either to the recovered compartment or to the death compartment.The model equations are listed below (system 1):(1)dStdt=−βStLt+At+ItN,dEtdt=βStLt+At+ItN−μEt,dLtdt=μEt−μ1α1+μ2α2Lt,dItdt=μ1α1Lt−μ3σ1+1−σ1It,dAtdt=μ2α2Lt−μ4σ2+1−σ2At,dHtdt=μ3σ1It−μ5γ1+1−γ1Ht,dRtdt=μ4σ2At+μ5γ1Ht,dDtdt=1−σ1It+1−σ2At+1−γ1Ht−δDt.Table1 shows the definition of symbols in the proposed model.Table 1
Definition of symbols.
SymbolDefinitionSNumber of susceptible personENumber of exposed personLNumber of latent personINumber of symptomatic personANumber of asymptomatic personHNumber of hospitalized personRNumber of recovered personDNumber of death personNTotal number of personβTransmission rateμExposed rateα1Symptomatic infection rateα2Asymptomatic infection rateσ1Symptomatic hospitalized rateσ2Asymptomatic recovery rateμ1The velocity of latent person becomes symptomatic personμ2The velocity of latent person becomes asymptomatic personμ3The velocity of symptomatic person becomes hospitalized personμ4The velocity of asymptomatic person becomes recovered personμ5The velocity of hospitalized person becomes recovered personγ1Recovery rateδNormal death rate


## 3. Well-Posedness and Equilibria

The existence, the positivity, and the boundedness of solutions of the proposed system (1) need to be proved to ensure that the model has a mathematical and biological meaning.System (1) can be rewritten as follows:(2)dZ=PZt,where(3)Zt=StEtLtItAtHtRtDt,and P is a C1 function mapping ℛ8 into itself, defined by(4)PZ=p1S,L,E,I,A,H,R,Dp2S,L,E,I,A,H,R,Dp3S,L,E,I,A,H,R,Dp4S,L,E,I,A,H,R,Dp5S,L,E,I,A,H,R,Dp6S,L,E,I,A,H,R,Dp7S,L,E,I,A,H,R,Dp8S,L,E,I,A,H,R,D=−βSL+A+INβSL+A+IN−μEμE−μ1α1+μ2α2Lμ1α1L−μ3σ1+1−σ1Iμ2α2L−μ4σ2+1−σ2Aμ3σ1I−μ5γ1+1−γ1Hμ4σ2A+μ5γ1H1−σ1I+1−σ2A+1−γ1H−δD.By the fundamental theory of functional differential equations, system (1) has a unique solution St,Et,Lt,It,At,Ht,Rt,Dt with respect to the initial data Z0 such that(5)St≥0,Et≥0,Lt≥0,It≥0,At≥0,Ht≥0,Rt≥0,Dt≥0.If we putS+E+L+I+A+R+H+D≤N, then the following theorems hold.Theorem 1.
The following biological feasible region of system (1) B=S,E,L,I,A,H,R,D∈ℛ+8;S+E+L+I+A+H+R+D≤N is positively invariant and attracting.Theorem 2.
Lett0>0 and the initial conditions satisfy St0>0,Et0>0,Lt0>0,It0>0,At0>0,Ht0>0,Rt0>0,Dt0>0, then the solution St,Et,Lt,It,At,Ht,Rt,Dt of system (1) are positive for all t≥0.Proof.
From the first equation of system (1), we have(6)St=S0e−β/N∫0tLu+Iu+Audu.
Hence,St is nonnegative for all t≥0. From the other equations of system (1), we set(7)Et=E0e−μt+∫0teu−tμβNSuLu+Iu+Audu,Et=E0+βN∫0teμuSuLu+Iu+Audue−μt,Lt=L0+μ∫0teuμ1α1+μ2α2Eudue−μ1α1+μ2α2t,It=I0+μ1α1∫0teuμ3σ1+1−σ1Ludue−μ3σ1+1−σ1t,At=A0+μ2α2∫0teuμ4σ2+1−σ2Ludue−μ4σ2+1−σ2t,Ht=H0+μ3σ1∫0teuμ5γ1+1−γ1Iudue−μ5γ1+1−γ1t,Dt=D0+∫0teuδ1−σ1Iu+1−σ2Au+1−γ1Hudue−δt.
Therefore,Et,Lt,It,At,Ht, andDt are all nonnegative for all t≥0.
From the seventh equation of system (1), we can easily deduce the positivity of Rt for all t≥0.
Hence, the positivity of solution has been proved.


## 4. Stability of Disease-Free Equilibrium

This section is devoted to studying the stability of the diseases-free equilibriumU0 of system (1).The equilibrium of the model is obtained by settingdSt/dt=dEt/dt=dLt/dt=dIt/dt=dAt/dt=dHt/dt=dRt/dt=dDt/dt=0, and the given system has a disease-free equilibrium given by U0=N,0,0,0,0,0,0,0. In the proposed model, we have multiple types of infected individuals, and then to investigate the stability of the disease-free equilibrium, we will define the basic reproductive number (R0) using the next-generation operator [10], who define R0 as the spectral radius of the next-generation operator.Theorem 3.
IfR0<1, then DFE (disease-free equilibrium) U0 is locally asymptotically stable (LAS). If R0>1, then the DFE is unstable.LetY be the vector of infected classes, such as infectious, exposed, carrier, etc., and X be the vector of uninfected classes, such as susceptible, recovered, etc.(8)dXdt=WX,Y,dYdt=fX,Y=FX,Y−VX,Y.LetU0=X∗,0∈ℛ+8 denote the disease-free equilibrium, that is, fX∗,0=WX∗,0=0, where FX,Y is the vector of new infection rates (flows from X to Y) and VX,Y is the vector of all others rates (not a new infection). These rates include flows from Y to X (for instance, recovery rates), flows within Y, and flows leaving the system (for instance, death rates). For each compartment, inflow in V is negative and outflow in V is positive.The formula of the basic reproductive number (R0) is defined by the maximum eigenvalue of ℱV−1, where ℱ=∂F/∂YX∗,0 and V=∂V/∂YX∗,0, and ℱV−1 is called the next-generation matrix.(9)ℱ=0βββ00000000000000000000000000000000,V−1=1μ000001J1J0000μ1α1Jdμ1α1Jd1d000μ2α2Jeμ2α2Je01e00μ1α1μ3σ1Jdmμ1α1μ3σ1Jdmμ3σ1dm01m0CCT1−σ2eδ1−σ2mδ1δ,where(10)J=μ1α1+μ2α2,d=μ3σ1+1−σ1,e=μ4σ2+1−σ2,m=μ5γ1+1−γ1,C=μ1α11−σ1Jdδ+μ2α21−σ2Jeδ+1−γ1μ1α1μ3σ1Jdδ,T=1−γ1μ3σ1mdδ+1−σ1dδ.Multiplyingℱ and V−1 together and calculating the spectral radius, we have(11)R0=βJ+βμ1α1Jd+βμ2α2Je,R0=βμ1α1+μ2α2+βμ1α1μ1α1+μ2α2μ3σ1+1−σ1+βμ2α2μ1α1+μ2α2μ4σ2+1−σ2.


### 4.1. Global Stability Condition for Disease-Free Equilibrium

In this section, we present two conditions, and if they are satisfied, they allow us to say that the disease-free equilibrium is globally asymptotically stable. First, system (1) must be written in the following form:(12)dXdt=WX,Y,dYdt=GX,Y,GX,0=0.(H1) for dX/dt=WX,0,X∗ is globally asymptotic stable (GAS), (H2) for GX,Y=QY−G^X,Y,G^X,Y≥0 for X,Y∈B, where Q=∂G/∂YU0 is an Metzler matrix (M-matrix, the off-diagonal elements of Q are nonnegative). If the system satisfies these two conditions, then the following lemma holds.Lemma 1.
The disease-free equilibriumU0=X∗,0 of system (2) is globally asymptotically stable (GAS) provided that R0<1 and assumptions (H1) and (H2) are satisfied.Now, we state the following theorem.Theorem 4.
The disease-free equilibrium of system (1) is globally asymptotically stable if R0<1.Proof.
LetX=S,R and Y=E,L,I,A,H,D. We will have(13)WX,Y=−βSL+I+ANμ4σ2A+μ5γ1H.
At the pointX,0,WX,0=0,X∗=N,0 is globally asymptotically stable for dX/dt=WX,0. On the other hand,(14)GX,Y=βSL+I+AN−μEμE−μ1α1+μ2α2Lμ1α1L−μ3σ1+1−σ1Iμ2α2L−μ4σ2+1−σ2Aμ3σ1I−μ5γ1+1−γ1H1−σ1I+1−σ2A+1−γ1H−δD.
In addition, this follows thatQ=∂G/∂YU0,(15)Q=−μβββ00μ−μ1α1+μ2α200000μ1α1−μ3σ1+1−σ10000μ2α20−μ4σ2+1−σ20000μ3σ10−μ5γ1+1−γ10001−σ11−σ21−γ1−δ.
The off-diagonal elements ofQ are nonnegative, i.e., an M-matrix and (16)G^X,Y=QY−GX,Y=βL+I+A−μEμE−μ1α1+μ2α2Lμ1α1L−μ3σ1+1−σ1Iμ2α2L−μ4σ2+1−σ2Aμ3σ1I−μ5γ1+1−γ1H1−σ1I+1−σ2A+1−γ1H−δD−βSL+I+AN−μEμE−μ1α1+μ2α2Lμ1α1L−μ3σ1+1−σ1Iμ2α2L−μ4σ2+1−σ2Aμ3σ1I−μ5γ1+1−γ1H1−σ1I+1−σ2A+1−γ1H−δD,G^X,Y=βL+I+A1−SN00000.
SinceS≤N,1−S/N≥0, and thus G^X,Y≥0. So, the conditions (H1) and (H2) of Lemma 1 are satisfied, and then this completes the global asymptotic stability of the disease-free equilibrium of the proposed system (1) for R0<1.


## 4.1. Global Stability Condition for Disease-Free Equilibrium

In this section, we present two conditions, and if they are satisfied, they allow us to say that the disease-free equilibrium is globally asymptotically stable. First, system (1) must be written in the following form:(12)dXdt=WX,Y,dYdt=GX,Y,GX,0=0.(H1) for dX/dt=WX,0,X∗ is globally asymptotic stable (GAS), (H2) for GX,Y=QY−G^X,Y,G^X,Y≥0 for X,Y∈B, where Q=∂G/∂YU0 is an Metzler matrix (M-matrix, the off-diagonal elements of Q are nonnegative). If the system satisfies these two conditions, then the following lemma holds.Lemma 1.
The disease-free equilibriumU0=X∗,0 of system (2) is globally asymptotically stable (GAS) provided that R0<1 and assumptions (H1) and (H2) are satisfied.Now, we state the following theorem.Theorem 4.
The disease-free equilibrium of system (1) is globally asymptotically stable if R0<1.Proof.
LetX=S,R and Y=E,L,I,A,H,D. We will have(13)WX,Y=−βSL+I+ANμ4σ2A+μ5γ1H.
At the pointX,0,WX,0=0,X∗=N,0 is globally asymptotically stable for dX/dt=WX,0. On the other hand,(14)GX,Y=βSL+I+AN−μEμE−μ1α1+μ2α2Lμ1α1L−μ3σ1+1−σ1Iμ2α2L−μ4σ2+1−σ2Aμ3σ1I−μ5γ1+1−γ1H1−σ1I+1−σ2A+1−γ1H−δD.
In addition, this follows thatQ=∂G/∂YU0,(15)Q=−μβββ00μ−μ1α1+μ2α200000μ1α1−μ3σ1+1−σ10000μ2α20−μ4σ2+1−σ20000μ3σ10−μ5γ1+1−γ10001−σ11−σ21−γ1−δ.
The off-diagonal elements ofQ are nonnegative, i.e., an M-matrix and (16)G^X,Y=QY−GX,Y=βL+I+A−μEμE−μ1α1+μ2α2Lμ1α1L−μ3σ1+1−σ1Iμ2α2L−μ4σ2+1−σ2Aμ3σ1I−μ5γ1+1−γ1H1−σ1I+1−σ2A+1−γ1H−δD−βSL+I+AN−μEμE−μ1α1+μ2α2Lμ1α1L−μ3σ1+1−σ1Iμ2α2L−μ4σ2+1−σ2Aμ3σ1I−μ5γ1+1−γ1H1−σ1I+1−σ2A+1−γ1H−δD,G^X,Y=βL+I+A1−SN00000.
SinceS≤N,1−S/N≥0, and thus G^X,Y≥0. So, the conditions (H1) and (H2) of Lemma 1 are satisfied, and then this completes the global asymptotic stability of the disease-free equilibrium of the proposed system (1) for R0<1.


## 5. Numerical Results and Discussion

In this section, we present the simulation of our SELIAHRD model in two different scenarios. In order to adjust the social distance, we multiply the transmission rateβ by a social distancing factor in order to obtain the two scenarios.(a)
The mean incubation period was 5.2 days (95% confidence interval) [9]. Therefore, μ1=0.1923.(b)
There is a mean 5-day delay from symptom onset to detection/hospitalization of a case. The duration from illness onset to the first medical visit for the 45 patients with illness onset before January 1 was estimated to have a mean of 5.8 days [11]. So, we set μ3=0.1724.(c)
The recovery rateγ1=0.15 is determined by the average duration of recovery D of infection. After this period, they enter the recovery phase.(d)
Since there were no data on the proportion of asymptomatic infection of the virus, we simulatedμ1=μ2 and μ4=0.08.(e)
Symptomatic and asymptomatic infection ratesα1andα2 are defined as follows:α1=(symptomatic infected)/susceptible.α2=(asymptomatic infected)/susceptible.In this first simulation, we assume that there is no social distancing and a total population of 1.5 million. In this scenario, Figures2 and 3 show that the model predicts in the next 85 days the total symptomatic infected person will reach a peak of 31000 and the healthcare system will be overload (only 15,869 of them get treatment). As a result, the model predicts that the peak of deaths would be 15,665.Figure 2
Simulation for the number of Exposed, Symptomatic Infected, and Asymptomatic Infected.Figure 3
Simulation for the number of Dead, Hospitalized, and Symptomatic infected.In the second simulation, we assume that90% of population obey the rule of social distancing.As can be seen in Figures4 and 5, the peak of symptomatic infected cases drops significantly to 50 and asymptomatic infected cases to 6, making the healthcare system not overloaded at all and only around 5 deaths.Figure 4
Simulation for the number of Exposed, Symptomatic Infected, and Asymptomatic Infected with90% of population obeying the rule of social distancing.Figure 5
Simulation for the number of Dead, Hospitalized, and Symptomatic Infected with90% of population obeying the rule of social distancing.The simulation results show that after 150 days, with the presence of asymptomatic infection the basic reproduction numberR0 of coronavirus was 3.25 if there is no social distancing and the healthcare system will be overloaded. On the other hand, if the lockdown is fully operational (90% of population respect of social distancing) even with the presence of asymptomatic infection, the basic reproduction number becomes R0=0.68 after 150 days and the healthcare system will not be overloaded. The paper [12] proposed a SEIAR model to simulate the epidemic dynamics of “SARS-COV-2” and demonstrate the efficiency of quarantine and government measure but didn’t take in consideration the velocity of transition between different stages and the effect of lockdown on the healthcare system.


## 6. Conclusions

The world continues to pay a heavy price as the COVID-19 pandemic spreads. This paper proposes the mathematical model SELIAHRD (Susceptible, Exposed, Latent, Symptomatic, Asymptomatic, Hospitalized, Recovered, and Death), with two kinds of infected people, that is, symptomatic and asymptomatic. We assume that asymptomatic people will not be hospitalized and have the probability to go either recovered or dead. We calculate the basic reproduction numberR0 and dynamics of disease-free equilibrium for the “SARS-CoV-2” using the proposed model.The SELIAHRD model will provide in the presence of asymptomatic infected the number of hospitalized and of deaths for the public authorities to develop the prevention policies (cities lockdown, social distancing, testing, contact tracing, etc.). The simulation shows the importance of the social distancing in making the number of infected and deaths decrease significantly even with the presence of asymptomatic infection and not overloading the healthcare system.



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