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Research article Special Issues

Exploring variable-sensitive q-difference equations for q-SINE Euler polynomials and q-COSINE-Euler polynomials

  • Received: 13 November 2023 Revised: 19 April 2024 Accepted: 29 April 2024 Published: 14 May 2024
  • MSC : 11B68, 11B83, 26C10, 34A30, 65D20, 65L99

  • In this study, we introduced several types of higher-order difference equations involving q-SINE Euler (QSE) and q-COSINE Euler (QCE) polynomials. Depending on the parameters selected, these higher-order difference equations exhibited properties of trigonometric functions or related Euler numbers. Approximate root construction focused on the QSE polynomial, which was the solution of the q-difference equations obtained earlier. We also showed the structure of the approximate roots of higher-order polynomials among the QSE polynomials, understood them, and considered the associated conjectures.

    Citation: Jung Yoog Kang, Cheon Seoung Ryoo. Exploring variable-sensitive q-difference equations for q-SINE Euler polynomials and q-COSINE-Euler polynomials[J]. AIMS Mathematics, 2024, 9(6): 16753-16772. doi: 10.3934/math.2024812

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  • In this study, we introduced several types of higher-order difference equations involving q-SINE Euler (QSE) and q-COSINE Euler (QCE) polynomials. Depending on the parameters selected, these higher-order difference equations exhibited properties of trigonometric functions or related Euler numbers. Approximate root construction focused on the QSE polynomial, which was the solution of the q-difference equations obtained earlier. We also showed the structure of the approximate roots of higher-order polynomials among the QSE polynomials, understood them, and considered the associated conjectures.



    COVID-19 is one of the three Coronaviruses that has caused epidemic outbreaks over the last two decades. It can spread through close contact, coughing, sneezing, or talking. For COVID-19, there are two types of infected individuals: one is symptomatic infected individuals, defined as those who show symptoms (such as fever, cough, sore throat, etc.) after obtaining infected. The other is asymptomatic infected individuals, defined as those who do not show symptoms after obtaining infected [1,2]. Asymptomatic cases are not confirmed cases which are divided into two different states. The first one is the asymptomatic individuals who show no symptoms for the whole time of the infection. The second one is the asymptomatic individuals who exhibit symptoms after a period of the asymptomatic infection. In the generally tested subgroup, the proportion of asymptomatic individuals who are found to be positive for COVID-19 is alarmingly high. Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) shows that asymptomatic and mildly symptomatic infections may be the key to disease transmission [3,4]. Most asymptomatic infected individuals do not seek medical help because they have no obvious clinical signs which leads to the rapid spread of the disease. Therefore, the prevention and control of this specific type of patient on a global scale is a huge challenge and requires more attention from the world.

    The global health crisis of the Coronavirus Covid-19 has brought to light the role of mathematical modeling in political and health decision-making [5,6,7,8]. We will study the SIS model inspired by the current Coronavirus outbreak. Mathematical models of infectious diseases have been used for over a decade to study and understand the mechanism of disease spread, predict the future of epidemics, and determine the best treatment strategy to protect human health. The famous SIR and SEIR mathematical epidemic models are the most used models providing good descriptions of infectious diseases especially when taking account of either symptomatic and asymptomatic infectious or time delay [9,10,11,12].

    In this work, we have designed a compartmental ODE model to investigate the COVID-19 spreading dynamics incorporating non-linear saturated incidence rates for both asymptomatic and symptomatic infection. The incidence rate is an essential component for infectious disease transmission in a mathematical model for both symptomatic and asymptomatic infectious [13]. In the epidemiology perception, incidence rate measures the number of new cases of a disease within a specified time duration as a percentage of the number of persons at threat for the disease [14]. We have also included two control functions designing quarantine strategies which is used to reduce the contact among infected individuals. Here we have studied the qualitative nature of the model through investigation of local and global nature of equilibrium points. Finally we have characterised optimal control to reduce the outbreak size and the control implementation costs. Results of the optimal control model show that optimal control provides significant reduction in COVID-19 spread.

    The paper is organised as follows: In Section 2, we have formulated the proposed model and have studied the boundedness and positivity of solutions. Steady state analysis with constant controls have been done in Section 3 with respect to the basic reproduction number (R). In Section 4, we characterised the optimal control model and we give an efficiency analysis. Finally, some numerical simulations and discussions were given in Section 5. The conclusions of our study has been reported in Section 6.

    In this paper, we have studied a compartmental ODE model to investigate the COVID-19 spreading dynamics using non-linear saturated incidences. For this purpose we have formulated an ODE model by splitting the total host population Λ into three disjoint epidemiological classes specifically susceptible individuals S(t), asymptomatic infectious individuals Ia(t) and symptomatic infectious individuals Is(t) i.e. Λ=S(t)+Ia(t)+Is(t).

    To formulate the model we have considered the following factors:

    ● Susceptible individual once infected has a probability 0<p<1 to be symptomatic and then a probability 0<1p<1 to be asymptomatic.

    ● Susceptible individual is infected by asymptomatic infected individuals at a nonlinear increasing rate fa(Ia)S and by symptomatic infected individuals at a nonlinear increasing rate fs(Is)S. The importance of these increasing incidence rates is that the rate of effective contacts between infected individuals and susceptible individuals increases with the increase of number of infective individuals.

    ● asymptomatic and symptomatic individuals are recovered at constant rates σa and σs, respectively.

    ● Recovered individuals can catch the diseases and then they are added to susceptible compartment.

    ● asymptomatic individuals become symptomatic individuals at a constant rate γ.

    ● Susceptible individuals are recruited at constant rate mΛ, natural mortality rate is m.

    According to the epidemiological assumptions, we proposed the following mathematical model

    {˙S=mΛS(fa(Ia)+fs(Is))+σaIa+σsIsmS,˙Ia=(1p)S(fa(Ia)+fs(Is))γIamIaσaIa,˙Is=pS(fa(Ia)+fs(Is))+γIamIsσsIs, (2.1)

    with positive initial condition (S0,I0a,I0s)R3+.

    Assumption 1. fa and fs are bounded, non-negative C1(R+), concave and increasing functions with fa(0)=fs(0)=0.

    Lemma 1. The general non-linear incidence rates fa and fs satisfy fa(I)Ifa(I)fa(0)I and fs(I)Ifs(I)fs(0)I,I>0.

    Proof. Let I,I1R+, and the function g1(I)=fa(I)Ifa(I). Since fa(I)0 (fa is increasing function) and fa(I)0 (fa is concave) then g1(I)=Ifa(I)0 and g1(I)g1(0)=0. Therefore, fa(I)Ifa(I). Similarly, let g2(I)=fa(I)Ifa(0) then g2(I)=fa(I)fa(0)0 once fa is a concave function. Thus g2(I)g2(0)=0 and fa(I)Ifa(0). Similarly for the function fs. It is necessary that the state variables S(t),Ia(t) and Is(t) remain non-negative for all t0.

    Proposition 1. Ω0={(S,Ia,Is)R3+/S+Ia+Is=Λ} is a positively invariant compact set for model (2.1).

    Proof. Assume that the initial condition (S0,I0a,I0s)R3+. If S=0 then ˙S=mΛ+σaIa+σsIs>0 therefore S(t)>0 for all t>0. Similarly, assume that Ia=0 then ˙Ia=(1p)Sfs(Is)0 therefore Ia(t)0 for all t>0. By the same way, assume that Is=0 therefore ˙Is=pSfa(Ia)+γIa0 then Is(t)0 for all t>0. Therefore the model (3.2) admits a non-negative solution.

    By summing the equations of (2.1), we get, for T=S+Ia+IsΛ, the following equation :

    ˙T=˙S+˙Ia+˙Is=mΛmSmIamIs=mT.

    Hence

    T(t)=T(0)emt. (3.1)

    Hence, Ω0 is invariant for the model (2.1) due to all variables are non-negative. Using the conservation principles, we can compute S as function of Ia and Is :

    S=ΛIaIs.

    Now, we can reduce the analysis of the original system (2.1) to the analysis of the following two-dimensional limiting system on the invariant set Ω :

    {˙Ia=(1p)(ΛIaIs)(fa(Ia)+fs(Is))γIamIaσaIa,˙Is=p(ΛIaIs)(fa(Ia)+fs(Is))+γIamIsσsIs, (3.2)

    with positive initial condition (I0a,I0s)R2+.

    Ω={(Ia,Is)R2+/Ia+IsΛ} is a positively invariant compact set for model (3.2). It is obvious that E0=(0,0) is the only disease-free equilibrium of system (3.2).

    The global behavior of our system inevitably depends on the basic reproduction number (R), that is, the average number of secondary cases produced by an infectious individual who is introduced into an established population only of susceptible.

    To drive the basic reproduction number (R) for complex compartmental models, we use the next-generation operator approach proposed by Diekmann et al. [15,16]. Following the approach of van den Driessche and Watmough, we can rewrite (3.2)

    (˙Ia˙Is)=((1p)(ΛIaIs)(fa(Ia)+fs(Is))p(ΛIaIs)(fa(Ia)+fs(Is)))((γ+m+σa)IaγIa+(m+σs)Is)=FV (3.3)

    where F denotes the rate of appearance of new infections, and V denotes the rate of transfer of individuals into or out of each population set. Furthermore,

    F=DF(E0)=((1p)Λfa(0)(1p)Λfs(0)pΛfa(0)pΛfs(0)),V=DV(E0)=((γ+m+σa)0γ(m+σs))
    V1=(1(γ+m+σa)0γ1(m+σs)),FV1=Λ((1p)fa(0)(γ+m+σa)+γ(1p)fs(0)(1p)fs(0)(m+σs)pfa(0)(γ+m+σa)+γpfs(0)pfs(0)(m+σs)).

    Since det(FV1)=0, then one of the eigenvalues is zero. Therefore, we can deduce the second eigenvalue since their sum is equal to the trace of the matrix FV1. Then, the basic reproduction number which is the spectral radius of FV1 (the maximum of the absolute values of its eigenvalues) is given by

    R=(1p)Λfa(0)(γ+m+σa)+γ(1p)Λfs(0)+pΛfs(0)(m+σs)=(1p)Λfa(0)(γ+m+σa)+(p+γ(1p)(m+σs))Λ(m+σs)fs(0)R1+R2. (3.4)

    Proposition 2. The model (3.2) admits a unique disease-free equilibrium E0=(0,0) and a unique endemic equilibrium E=(Ia,Is) such that Ia,Is>0.

    Proof. Equilibria of (3.2) satisfy

    {0=(1p)(ΛIaIs)(fa(Ia)+fs(Is))γIamIaσaIa,0=p(ΛIaIs)(fa(Ia)+fs(Is))+γIamIsσsIs, (3.5)

    which reduces to

    (ΛIaIs)(fa(Ia)+fs(Is))=(γ+m+σa)Ia(1p)=γIa+(m+σs)Isp, (3.6)

    Thus

    {Is=γ+p(m+σa)(1p)(m+σs)Ia(ΛIaIs)(fa(Ia)+fs(Is))=(γ+m+σa)Ia(1p), (3.7)

    We conclude that from (3.6)

    (ΛIaγ+p(m+σa)(1p)(m+σs)Ia)(fa(Ia)+fs(γ+p(m+σa)(1p)(m+σs)Ia))=(γ+m+σa)Ia(1p).

    We can write this equation on the form

    Iag(Ia)=0

    where the function g given by

    g(Ia)=(ΛIaγ+p(m+σa)(1p)(m+σs)Ia)(fa(Ia)Ia+fs(γ+p(m+σa)(1p)(m+σs)Ia)Ia)(γ+m+σa)(1p).

    ● If Ia=0, then Is=0. This equilibrium named the disease-free equilibrium will be noted here by E0=(0,0).

    ● If Ia0, then g(Ia)=0. Let us calculate the derivative of the function g given by

    g(Ia)=(1+γ+p(m+σa)(1p)(m+σs)Ia)(fa(Ia)Ia+fs(γ+p(m+σa)(1p)(m+σs)Ia)Ia)+(ΛIaγ+p(m+σa)(1p)(m+σs)Ia)(fa(Ia)Iafa(Ia)I2a+fs(γ+p(m+σa)(1p)(m+σs)Ia)fs(γ+p(m+σa)(1p)(m+σs)Ia)IaI2a).

    The functions fa and fs satisfy fa(I)Ifa(I) and fs(I)Ifs(I),I0 and all variables are non-negative, hence g is an increasing function and satisfies g(Ia)<0.

    An easy calculation gives

    limIa0g(Ia)=limIa0(ΛIaγ+p(m+σa)(1p)(m+σs)Ia)(fa(Ia)Ia+fs(γ+p(m+σa)(1p)(m+σs)Ia)Ia)(γ+m+σa)(1p)=Λ(fa(0)+γ+p(m+σa)(1p)(m+σs)fs(0))(γ+m+σa)(1p)=(γ+m+σa)(1p)(Λ(1p)(γ+m+σa)fa(0)+Λ(γ+p(m+σa))(γ+m+σa)(m+σs)fs(0)1)=(γ+m+σa)(1p)(R1),

    and

    g(Λ)=γ+p(m+σa)(1p)(m+σs)Λ(fa(Λ)Λ+fs(γ+p(m+σa)(1p)(m+σs)Λ)Λ)(γ+m+σa)(1p)<0.

    Since R>1, g(Λ)<0 and limIa0g(Ia)<0, then g(Ia)=0 has a unique positive solution Ia in (0,Λ) and therefore the equilibrium state E=(Ia,Is) is unique with Is=γ+p(m+σa)(1p)(m+σs)Ia.

    In this subsection, the local stability behaviours of equilibria are discussed.

    Theorem 1. E0 is LAS when R<1 and it is unstable when R>1.

    Proof. The Jacobian matrix given at the equilibrium point E0 is

    J0=((1p)Λfa(0)(γ+m+σa)(1p)Λfs(0)pΛfa(0)+γpΛfs(0)(m+σs)).

    The trace is given by

    Trace (J0)=(1p)Λfa(0)+pΛfs(0)(γ+m+σa)(m+σs).

    Since R=pΛfs(0)(m+σs)+(1p)Λfa(0)(γ+m+σa)+(1p)γΛfs(0)(γ+m+σa)(m+σs) then when R<1, it is obvious that pΛfs(0)(m+σs)<1, (1p)Λfa(0)(γ+m+σa)<1 therefore Trace (J0)<0. The determinant is given by

    Det (J0)=((1p)Λfa(0)(γ+m+σa))(pΛfs(0)(m+σs))(1p)Λfs(0)(pΛfa(0)+γ)=p(γ+m+σa)Λfs(0)+(γ+m+σa)(m+σs)(1p)(m+σs)Λfa(0)(1p)γΛfs(0)=(γ+m+σa)(m+σs)(1pΛfs(0)(m+σs)(1p)Λfa(0)(γ+m+σa)(1p)γΛfs(0)(γ+m+σa)(m+σs))=(γ+m+σa)(m+σs)(1R)>0 if R>1.

    If R<1, then we have negative eigenvalues. Therefore, E0 is LAS. Whereas, if R>1, E0 is therefore unstable.

    Theorem 2. E is LAS when R>1.

    Proof. For the equilibrium point E, the Jacobian is given by J=(a11a12a21a22) where

    a11=(1p)(fa(Ia)+fs(Is))+(1p)(ΛIaIs)fa(Ia)(γ+m+σa)=(1p)(fa(Ia)+fs(Is))+(1p)(ΛIaIs)fa(Ia)(1p)(ΛIaIs)(fa(Ia)+fs(Is))Ia=(1p)(fa(Ia)+fs(Is))+(1p)(ΛIaIs)(fa(Ia)fa(Ia)Ia)(1p)(ΛIaIs)fs(Is)Ia(1p)(fa(Ia)+fs(Is))(1p)(ΛIaIs)fs(Is)Ia,a22=p(fa(Ia)+fs(Is))+p(ΛIaIs)fs(Is)(m+σs)=p(fa(Ia)+fs(Is))+p(ΛIaIs)fs(Is)p(ΛIaIs)(fa(Ia)+fs(Is))IsγIaIs=p(fa(Ia)+fs(Is))+p(ΛIaIs)(fs(Is)fs(Is)Is)p(ΛIaIs)fa(Ia)IsγIaIsp(fa(Ia)+fs(Is))p(ΛIaIs)fa(Ia)IsγIaIs,a12=(1p)(fa(Ia)+fs(Is))+(1p)(ΛIaIs)fs(Is),a21=p(fa(Ia)+fs(Is))+p(ΛIaIs)fa(Ia)+γ.

    The trace is given by

    Trace (J)(fa(Ia)+fs(Is))(1p)(ΛIaIs)fs(Is)Iap(ΛIaIs)fa(Ia)IsγIaIs<0

    and the determinant is given by

    Det (J)((1p)(fa(Ia)+fs(Is))(1p)(ΛIaIs)fs(Is)Ia)×(p(fa(Ia)+fs(Is))p(ΛIaIs)fa(Ia)IsγIaIs)((1p)(fa(Ia)+fs(Is))+(1p)(ΛIaIs)fs(Is))×(p(fa(Ia)+fs(Is))+p(ΛIaIs)fa(Ia)+γ)=p(1p)(ΛIaIs)(fa(Ia)+fs(Is))(fa(Ia)Is+fa(Is)Ia)+p(1p)(ΛIaIs)2(fs(Is)Iafa(Ia)Isfa(Ia)fs(Is))+γ(1p)(fa(Ia)+fs(Is))(IaIs+1)+γ(1p)(ΛIaIs)(fs(Is)Isfs(Is))+p(1p)(ΛIaIs)(fa(Ia)+fs(Is))(fa(Ia)+fs(Is))>0

    By Lemma 1, the functions fa and fs satisfy fa(I)Ifa(I) and fs(I)Ifs(I),I0 and since (ΛIaIs)>0, then, all terms of Det(J) are either positive or non-negative and therefore Det(J)>0 and the equilibrium point E (which exists only if R>1) is LAS.

    Here, we discuss the global behaviour of the equilibrium points. For proving the global stability of the positive equilibrium points E0 and E, we will exclude the existence of periodic solutions of system (3.2) by using generalized Bendixson-Dulac theorem. We first prove that the system (3.2) has no periodic orbit nor poly-cycle on Ω

    Theorem 3. System (3.2) has no periodic orbits nor poly-cycles on Ω.

    Proof. Consider a solution of system (3.2) on Ω. Let ξ1=ln(Ia) and ξ2=ln(Is) then the transformation of (3.2) gives the following new system:

    {˙ξ1=h1(ξ1,ξ2):=(1p)(Λeξ1eξ2)(fa(eξ1)+fs(eξ2))eξ1(γ+m+σa),˙ξ2=h2(ξ1,ξ2):=p(Λeξ1eξ2)(fa(eξ1)+fs(eξ2))eξ2)+γeξ1eξ2(m+σs). (3.8)

    We have

    h1ξ1+h2ξ2=(fa(eξ1)+fs(eξ2))γeξ1eξ2(Λeξ1eξ2)((1p)fs(eξ2)eξ1+pfa(eξ1)eξ2)(1p)(Λeξ1eξ2)(fa(eξ1)fa(eξ1)eξ1)eξ1p(Λeξ1eξ2)(fs(eξ2)fs(eξ2)eξ2)eξ2<0ξ1,ξ2R.

    Thus using Dulac criterion [17,18], system (3.8) has no periodic trajectory. Therefore system (3.2) has no periodic orbit inside Ω.

    Theorem 4. The solution of (3.2) converges asymptotically to :

    E0 if R<1.

    E if R>1.

    Proof. We restrict the proof to the case where R>1. The other case can be done similarly. The system (3.2) admits two equilibrium points E0 and E. E0 is an unstable node, and only E is a stable node. We aim to prove that E is globally asymptotically stable. Let Ia(0)>0,Is(0)>0 and ω the ω-limit set of (Ia(0),Is(0)). ω is an invariant compact set and ωˉΩ. M can't be E0 because E0 is an unstable node and can't be a part of the ω-limit set of (Ia(0),Is(0)). System (3.2) has no periodic orbit inside Ω. Using the Poincaré-Bendixon Theorem [17,18], E is a globally asymptotically stable equilibrium point for system (3.2).

    The necessary conditions that an optimal pair must satisfy come from Pontryagin's Maximum Principle. Our goal is to minimize the number of infected individuals either asymptomatic or symptomatic while keeping those corresponding cost of the quarantine strategies low during the epidemic. Thus, we define a control function set as V={v1,v2} where v1(t) and v1(t) are the control variables for the quarantine strategies measuring the effort to reduce the contact between susceptible individuals and both kind of infected individuals (asymptomatic and symptomatic), respectively.

    Therefore the model (2.1) is modified to the following new model:

    {˙S=mΛS((1v1)fa(Ia)+(1v2)fs(Is))+σaIa+σsIsmS,˙Ia=(1p)S((1v1)fa(Ia)+(1v2)fs(Is))γIamIaσaIa,˙Is=pS((1v1)fa(Ia)+(1v2)fs(Is))+γIamIsσsIs, (4.1)

    Several optimal strategies for epidemic models were proposed in several previous works [19,20,21,22]. In our case, we discussed an optimal strategy that reduces the contact between infected and uninfected individuals to optimise the number of infective individuals.

    Recall that the set Ω={(S,Ia,Is)R3+/S+Ia+Is=Λ} is a positively invariant compact set for (4.1).

    Assume that fa and fs are globally Lipschitz with Lipschitz constants La and Ls where ˉfa=supI>0fa(I) and ˉfs=supI>0fs(I). Define the space

    vad={(v1,v2)|v1,v2Lebesque measurable,0<ˉv1v11,0<ˉv2v21,0tT}

    where ˉv1 and ˉv1 are two non-negative constants.

    Our goal is to control the number infected individuals and minimize the cost of the effort to reduce the contact between susceptible and infected individuals (v1,v2). The optimal control problem considered here is by minimizing an objective functional J(v1,v2) as following

    minv1,v2vadJ(v1,v2)=minv1,v2vadT0(α12v21(t)+α22v22(t)+β1Ia(t)+β2Is(t))dtsubject to (4.1). (4.2)

    β1 and β2 are two constants.

    Let φ=(S,Ia,Is)t, then we express (4.1) as

    ˙φ=Bφ+ψ(φ)=Z(φ) (4.3)

    where B=(mσaσs0(m+σa+γ)00γ(m+σs)) and ψ(φ)=(mΛS((1v1)fa(Ia)+(1v2)fs(Is))(1p)S((1v1)fa(Ia)+(1v2)fs(Is))pS((1v1)fa(Ia)+(1v2)fs(Is))).

    Proposition 3. Z(φ) is a Lipschitz continuous function.

    Proof. First we shall prove that the function ψ is uniformly Lipschitz and continuous since

    |ψ(φ1)ψ(φ2)|=2(1v1)|fa(Ia1)S1fa(Ia2)S2|+2(1v2)|fs(Is1)S1fs(Is2)S2|=2(1v1)|S1(fa(Ia1)fa(Ia2))+fa(Ia2)(S1S2)|+2(1v2)|S1(fs(Is1)fs(Is2))+fs(Is2)(S1S2)|2(1ˉv1)Λ|fa(Ia1)fa(Ia2)|+2(1ˉv1)ˉfa|(S1S2)|+2(1ˉv2)Λ|fs(Is1)fs(Is2)|+2(1ˉv2)ˉfs|(S1S2)|2(1ˉv1)ΛLa|Ia1Ia2|+2(1ˉv1)ˉfa|(S1S2)|+2(1ˉv2)ΛLs|Is1Is2|+2(1ˉv2)ˉfs|(S1S2)|M|φ1φ2|

    where M=2max(2(1ˉv1)ΛLa,2(1ˉv1)ˉfa,2(1ˉv2)ΛLs,2(1ˉv2)ˉfs).

    |Bφ1Bφ2|B|φ1φ2| (4.4)

    where . is the matrix norm. Then |Z(φ1)Z(φ2)|H|φ1φ2| where the constant H=max(M,B). Therefore Z is Lipschitz continuous function. As the function ψ is uniformly Lipschitz and continuous then there exists a unique solution for the system (4.3). First, introduce the Lagrangian for the optimal problem (4.1) and (4.2)

    L(S,Ia,Is,v1,v2)=α12v21+α22v22+β1Ia+β2Is. (4.5)

    We derive necessary conditions on the optimal control by applying Pontryagin's Maximum Principle [23]. Let the Hamiltonian H for the control problem (4.1) and (4.2):

    H(S,Ia,Is,v1,v2,λ1,λ2,λ3)=α12v21+α22v22+β1Ia+β2Is+λ1˙S+λ2˙Ia+λ3˙Is=α12v21+α22v22+β1Ia+β2Is+λ1(mΛS((1v1)fa(Ia)+(1v2)fs(Is))+σaIa+σsIsmS)+λ2((1p)S((1v1)fa(Ia)+(1v2)fs(Is))γIamIaσaIa)+λ3(pS((1v1)fa(Ia)+(1v2)fs(Is))+γIamIsσsIs) (4.6)

    where λ1,λ2 and λ3 are the adjoint variables satisfying the following adjoint equations

    {˙λ1=HS=mλ1+((1v1)fa(Ia)+(1v2)fs(Is))(λ1(1p)λ2pλ3),˙λ2=HIa=β1σaλ1+(γ+m+σa)λ2γλ3+S(1v1)fa(Ia)(λ1(1p)λ2pλ3),˙λ3=HIs=β2+S((1v2)fs(Is)(λ1(1p)λ2pλ3)σsλ1+λ3(m+σs). (4.7)

    Final conditions are given as follows: λ1(T)=0,λ2(T)=0,λ3(T)=0.

    The Hamiltonian is minimized with respect to the control variables as following:

    minv1,v2,λ1,λ2,λ3H(S,Ia,Is,v1,v2,λ1,λ2,λ3,t). (4.8)

    The derivative of the Hamiltonian H with respect to v1, and v2 is given by

    Hv1=α1v1+Sfa(Ia)(λ1(1p)λ2pλ3),Hv2=α2v2+Sfs(Is)(λ1(1p)λ2pλ3). (4.9)

    The optimal control is obtained by resolving the necessary conditions: Hv1=0 and Hv2=0 on some non-trivial intervals. In this case, the controls are expressed as

    v1(t)=Sfa(Ia)(λ1+(1p)λ2+pλ3)α1 and v2(t)=Sfs(Is)(λ1+(1p)λ2+pλ3)α2

    once

    ˉv1Sfa(Ia)(λ1+(1p)λ2+pλ3)α11 and ˉv2Sfs(Is)(λ1+(1p)λ2+pλ3)α21.

    For all numerical simulations, we used the Monod function as a non-linear incidence rates satisfying the assumption 1 fa(I)=ηaIκa+I and fs(I)=ηsIκs+I. Here ηa, κa, ηs and κs are non-negative constants. The parameters used for the numerical simulations are given in Table 1.

    Table 1.  Parameters and variables of system (3.2).
    Notation Definition
    S(t) Number of susceptible individuals at time t
    State variables Ia(t) Number of asymptomatic infected individuals at time t
    Is(t) Number of symptomatic infected individuals at time t
    Functions fa Saturated incidence rate in the asymptomatic compartment
    fs Saturated incidence rate in the symptomatic compartment
    Λ Recruitment rate
    γ Rate at which an individual enters the symptomatic compartment from the asymptomatic compartment.
    1/γ Duration time spent in compartment Ia
    σa Per-capita recovery rate of the infected individual in the asymptomatic compartment
    σs Per-capita recovery rate of the infected individual in the symptomatic compartment
    Parameters m Per-capita natural mortality rate
    0<p<1 Probability at which those newly infected individuals enter the symptomatic compartment
    (1p) Probability at which those newly infected individuals enter the asymptomatic compartment

     | Show Table
    DownLoad: CSV
    Table 2.  Used parameters for numerical simulations.
    Parameter κa κs σa σs m γ Λ α1 α2 β1 β2
    Value 2 1 0.4 0.5 1.5 0.1 8 1 1 1 1

     | Show Table
    DownLoad: CSV

    We begin by some numerical results that confirm the stability of the equilibrium points of (3.2). In Figure 2, we give the results for the case where R<1. The numerical solution of the given model (3.2) approaches to the DFE=(0,0), which confirms that DFE is GAS when R<1.

    Figure 1.  Dynamical system taking into account the asymptomatic infection. Circles describe the compartments S,Ia and Is and the arrows (and labels) correspond to rates of transition between them.
    Figure 2.  For ηa=0.1 and ηs=0.2 then R=0.58<1.

    In Figures 3 and 4, we give the results for the case where R>1. The numerical solution of given model (3.2) approaches asymptotically to the EE, which confirms that EE is GAS when R>1.

    Figure 3.  For ηa=1 and ηs=2 then R=5.8>1.
    Figure 4.  For ηa=1 and ηs=0.2 then R=1.48>1.

    Sensitivity analysis can be helpful as to how the variability in the output of a mathematical model can be allocated to different sources of uncertainty in its input parameters [24,25]. Sensitivity analysis has several purposes; one is to determine the input parameters that most contribute to the system's dynamics. The other is to detect the impacts of each parameter on the other parameters and then determine the potential to simplify the model. In this study, the sensitivity study can be helpful as it will inform us how essential each parameter is to the transmission of the disease. We must discover the highest effect on the R. Therefore, these input parameters will be critical targets for future intervention strategies. A fundamental approach expresses a relative change of a variable by a relative change of a parameter. Consequently, if the sensitivity index is positive, increasing the parameter value will cause an increase in the R value. Further, if the result is negative, then the parameter value and the R are inversely proportional.

    Figure 5 shows that Λ,p and γ are the only three parameters with a positive sign. Therefore, as the value of Λ,p and γ increase, the value of R increases. Furthermore, since the remaining parameters increases, the value of R decreases. Figure 5 shows the behaviour of the R with respect to the model parameters.

    Figure 5.  Sensitivity of the reproduction number R with respect to the model parameters.

    The numerical results of the control problem were obtained using the an association between the Gauss-Seidel-like implicit finite-difference scheme and the first-order backward-difference. We used the same parameters for systems (4.1) and (4.2) as for the direct problem (3.2) with ηa=1 and ηs=2 (left) and with ηa=1 and ηs=0.2 (where EE is GAS) with variables v1 and v2 such that the initial condition v1(0)=v2(0)=0.5. We plot in Figure 6 the behaviours S(t),Ia(t) and Is(t) with respect to time.

    Figure 6.  Behaviours for ηa=1 and ηs=2 (left) and for ηa=1 and ηs=0.2 (right).

    As seen in Figure 2, susceptible compartment increases about 21%, however, the infected compartment decreases about 14% for the asymptomatic infected compartment and 14% for the asymptomatic infected compartment of their values at steady state in Figure 2. Note that there is no considerable influence of the values of α1,α2,β1 and β2 on the final values of susceptible and both infected compartments.

    In the present paper, a generalized "SIS" model with a non-linear incidence rate including asymptomatic and symptomatic infection was presented and analyzed. The basic reproduction number (R) was calculated for the proposed system using the next generation matrix method. The system admits at most two equilibria: the disease-free equilibrium (E0) and the endemic equilibrium (E). Local and global stability was carried. If R<1, the disease-free equilibrium E0 is locally and globally asymptotically stable; if R>1, the endemic equilibrium E is locally and globally asymptotically stable. Furthermore, we find several reasonable optimal control strategies to the prevention and control the disease. Finally, numerical simulations are presented to verify the above theoretical results are given. It is concluded that asymptomatic infections have a greater contribution to the diseases spread even in the case where the asymptomatic infections are less infectious.

    One of the compartmental models' limitations is the assumption of a homogeneous population. However, variations in individuals in epidemiological characteristics are more realistic to consider. Further, the influences of time delay and intrinsic fluctuations are considered limitations to the compartmental models.

    The author declares no conflict of interest.

    The author is grateful to the unknown referees for many constructive suggestions, which helped to improve the presentation of the paper.

    Used numerical scheme (control problem)

    Subdividing the interval [0,T] as the following [0,T]=N1n=0[tn,tn+1] where tn=ndt and dt=T/N. Let Sn, Ina, Ins, λn1, λn2, λn3, vn1 and vn2 approximate S(t),Ia(t),Is(t),λ1(t),λ2(t),λ3(t), v1(t) and v2(t), respectively at the time tn. S0,I0a,I0s, λ01,λ02,λ03,v01 and v02 be their values at initial time (t=0). SN,INa,INs, λN1,λN2,λN3,vN1 and vN2 be the values at t=T. Then, we use the following scheme [6,8,21,22]

    {Sn+1Snδt=mΛSn+1((1vn1)fa(Ina)+(1vn2)fs(Ins))+σaIna+σsInsmSn+1,In+1aInaδt=(1p)Sn+1((1vn1)fa(Ina)+(1vn2)fs(Ins))γIn+1amIn+1aσaIn+1a,In+1sInsδt=pSn+1((1vn1)fa(In+1a)+(1vn2)fs(Ins))+γIn+1amIn+1sσsIn+1s,λNn1λNn11δt=mλNn11+((1vn1)fa(In+1a)+(1vn2)fs(In+1s))(λNn11(1p)λNn2pλNn3),λNn2λNn12δt=β1σaλNn11+(γ+m+σa)λNn12γλNn3+Sn+1(1vn1)fa(In+1a)(λNn11(1p)λNn12pλNn3),λNn3λNn13δt=β2+Sn+1(1vn2)fs(In+1s)(λNn11(1p)λNn12pλNn13)σsλNn11+λNn13(m+σs).

    Therefore the following algorithm will be applied.

    Table Algorithm 1.  Optimal control resolution.
    S0S0,I0aIa0, I0sIs0, λN10,λN20,λN30,
    for n=0 to N1 do
    [Sn+1Sn+δt(mΛ+σaIna+σsIns)1+δt(m+(1vn1)fa(Ina)+(1vn2)fs(Ins)),In+1aIna+δt(1p)Sn+1((1vn1)fa(Ina)+(1vn2)fs(Ins))1+δt(γ+m+σa),In+1sIns+δt(pSn+1((1vn1)fa(In+1a)+(1vn2)fs(Ins))+γIn+1a)1+δt(m+σs),λNn11λNn1+δt((1vn1)fa(In+1a)+(1vn2)fs(In+1s))((1p)λNn2+pλNn3)1+δt(m+(1vn1)fa(In+1a)+(1vn2)fs(In+1s)),λNn12λNn2+δt(β1+σaλNn11+γλNn3Sn+1(1vn1)fa(In+1a)(λNn11pλNn3)1+δt(γ+m+σa(1p)Sn+1(1vn1)fa(In+1a)),λNn13λNn3+δt(β2Sn+1(1vn2)fs(In+1s)(λNn11(1p)λNn12)+σsλNn111+δt(m+σspSn+1(1vn2)fs(In+1s)),vn+11max(min(Sn+1fa(In+1a)(λNn11+(1p)λNn12+pλNn13)α1,1),0),vn+12max(min(Sn+1fs(In+1s)(λNn11+(1p)λNn12+pλNn13)α2,1),0).
    end

     | Show Table
    DownLoad: CSV


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