Adv Syst Sci Appl 2022; 03:36–48 Published online at https://ijassa.ipu.ru. Dynamics of a Generalized Fractional Epidemic Model of COVID-19 with Carrier Effect Khalid Hattaf1,2, Ahmed A. Mohsen3,4 1Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Casablanca, Morocco 2 Laboratory of Analysis, Modeling and Simulation (LAMS), Hassan II University of Casablanca, Casablanca, Morocco 3 Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad, Iraq 2 Ministry of Education, Baghdad, Iraq Abstract: Currently, coronavirus disease 2019 (COVID-19) continues to cause several new cases and deaths, and further new variants of the virus responsible this disease appear. The present paper proposes a new mathematical model to understand the mechanisms of the spread of COVID-19 and better describe its dynamics. The modes of infection spread of COVID-19 via asymptomatic and symptomatic individuals are modeled by two general nonlinear incidence functions in order to include several types of incidence rates existing in the literature. When a disease outbreak within a community, individuals acquire information about this disease. Therefore, the proposed model take into account the memory effect on the outbreaks of COVID-19. This effect is modeled by a fractional order derivative in Caputo sense. The mathematical analysis of the proposed model is rigorously investigated, including the computation of the basic reproduction number R0 and the stability of equilibria. Keywords: COVID-19, SARS-CoV-2, incidence rate, basic reproduction number, fractional differential equations, stability analysis. 1. INTRODUCTION Coronavirus disease 2019 (COVID-19) is a infectious disease caused by a new coronavirus called severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). World Health Organization (WHO) first learned of this new virus on 31 December 2019, following a report of a cluster of cases of viral pneumonia detected in Wuhan City, Hubei Province of China. Due to rapid worldwide outbreak of COVID-19, the WHO declared it a pandemic on 11 March 2020. Over 2.7 million new cases were reported on 7 March 2021 with 116 166 652 cumulative cases, 60 323 new deaths and 2 582 528 cumulative deaths [18]. Since the appearance of the first case in Morocco on 2 March 2020 in the city of Casablanca, the disease has caused 8 716 deaths on 12 March 2021 according to new statistics given by Moroccan Ministry of Health (MMH) [13]. However, a total of 13 671 deaths was confirmed in Iraq on the same day [19]. The outbreak COVID-19 impacted nearly every side of society worldwide. Therefore, the pandemic has required a substantial response by public health authorities at every level and considered aformidable global public health challenge. ∗Corresponding author: Corresponding author: aamuhseen@gmail.com DYNAMICS OF A GENERALIZED FRACTIONAL EPIDEMIC MODEL... 37 Despite the prevention measures, such as hand washing, social distancing, quarantine of suspected cases, isolation of confirmed cases, the closure of schools and non-essential businesses and services, and also the start of vaccination in some countries, COVID-19 continues to cause new cases and deaths, and new variants of SARS-CoV-2 appear. In fact, the transmission or outbreaks of the COVID-19 consists of direct and indirect contacts between infective and susceptible individuals, it seems that the several mathematical models well described the COVID-19. However, attention must be paid to the unique traits of COVID-19. These traits are (a) the presence of individuals who exhibit infectious symptoms even during the incubation period, such as such as High body temperature, body aches and pains, queasiness, or diarrhea typically (2 - 14) days after infective to the virus, (b) There are asymptomatic individuals called a carrier, who can move freely and transmit infection, which makes controlling the situation complicated such that, approximately half of the COVID-19 patients (40− 50%) in one study did not show any symptoms from above in (a). For example, there are several reports that have investigated the possibility of asymptomatic carriers of COVID-19. A report of Japanese travelers from Wuhan, China approximates the proportion of carriers of disease but asymptomatic at 30 %. In addition, studies of every person in an isolated Italian village revealed that 50-75 % of people were asymptomatic. A report of a nursing home in the USA found 30 % of patients were asymptomatic on the day of testing for COVID-19, with 4 % remaining asymptomatic upon follow-up a week later. A report in hospitalized patients in Beijing found that 5 % of patients testing positive for COVID-19 had asymptomatic infections see [3–5,10,15]. Recently, many mathematical models have been developed to describe the spread of infectious diseases, such as hepatitis B, HIV transmission and Ebola virus disease. For more information, we refer the reader to the recent book published by Hattaf and Dutta about this subject in [7]. For COVID-19, Moussaoui and Zerga [14] proposed a mathematical model that takes into account strategies against COVID-19 such as wearing masks and respecting safety distances. A model for the dynamics of COVID- 19 with quarantine strategy and media coverage effect was presented by Mohsen et al. in [12], while a mathematical model of COVID-19 pandemic involving the infective immigrants was also studied in [11]. A fractional order model with bilinear incidence rate was introduced in [2] to study transmission dynamics of COVID-19 in Japan 2020. Modeling of the dynamics of COVID-19 inside the human body was presented in [9]. The main objective of this study is to develop a new mathematical model in order to better describe the dynamics of the COVID-19 in human population by taking into account the effects of memory and carrier, and also others aspects such as the non- linearity of the incidence function, the death rate due to COVID-19 and the recovery rates of the asymptomatic and symptomatic individuals. To do this, the structure of this work is outlined as follows. The next section is devoted to the formulation of model, the existence of equilibria and the computation of the basic reproduction number R0. In Section 3, we use mathematical analysis to establish global stability results for the proposed model. Some applications of our main results are presented in Section 4. Eventually, the conclusion is given in Section 5. 2. MODEL FORMULATION AND EQUILIBRIA In this section, we first construct a mathematical model that takes into account the effects of memory and carrier. Then we divide the total population into four classes S(t), C(t), I(t) and R(t) that represent susceptible, carrier (asymptomatic infected individuals), infected and recovered individuals at time t, respectively. The dynamics Copyright© 2022 ASSA. Adv Syst Sci Appl (2022) 38 KHALID HATTAF, AHMED A. MOHSEN of the four classes is governed by the following nonlinear system of FDEs: DαS(t) = A− νS − Φ(S, C)C −Ψ(S, I)I, DαC(t) = Φ(S, C)C +Ψ(S, I)I − (ν + d+ γ + r1)C, DαI(t) = γC − (ν + d+ r2)I, DαR(t) = r1C + r2I − νR, (2.1) where the susceptible individuals are recruited at a rate A and become infected either by effective contact with carrier at rate Φ(S, C)C or by effective contact with infected individuals at rate ψ(S, I)I. So, the term Φ(S, C)C + ψ(S, I)I denotes the total asymptomatic infection rate of susceptible individuals. The natural death rate in all classes is denoted by ν, while d is the death rate due to COVID-19. The rate of transfer from the asymptomatic to symptomatic is denoted by γ. The parameters r1 and r2 are recovery rates of the asymptomatic and symptomatic individuals, respectively. Finally, Dα denotes the Caputo fractional derivative with α ∈ (0, 1] that describes the memory effect. A schematic diagram of model (2.1) is shown in Figure 1. Fig. 2.1. The diagram of model (2.1). Obviously, the first three equations of (2.1) do not depend on the variable R, model (2.1) can be rewrite by following format{ DαS(t) = A− νS − Φ(S, C)C −Ψ(S, I)I, DαC(t) = Φ(S, C)C +Ψ(S, I)I − d1C, DαI(t) = γC − d2I, (2.2) where d1 = ν + d+ γ + r1 and d2 = ν + d+ r2. Furthermore and according to [8], we assume that the infection rate is the general incidences Φ and Ψ are continuously differentiable in the interior of IR2 + and satisfy the two following conditions: (H1) Φ(0, C) = 0, ∂Φ ∂S (S, C) > 0, ∂Φ ∂C (S, C) ≤ 0 for all S, C ≥ 0. Copyright© 2022 ASSA. Adv Syst Sci Appl (2022) DYNAMICS OF A GENERALIZED FRACTIONAL EPIDEMIC MODEL... 39 (H2) Ψ(0, I) = 0, ∂Ψ ∂S (S, I) > 0, ∂Ψ ∂I (S, I) ≤ 0 for all S, I ≥ 0. Next, we study the existence of equilibria. Clearly, Ef (S0, 0, 0) is a disease-free equilibrium of (2.2), where S0 = A ν . Hence, the basic reproduction number of (2.2) can be calculated as follows R0 = d2Φ(S0, 0) + γg(S0, 0) d1d2 , (2.3) which can be rewritten as R01+R02, where � R01 = Φ(S0, 0) d1 is the basic reproduction number for asymptomatic mode of transmission. � R02 = γΨ(S0, 0) d1d2 is the basic reproduction number for symptomatic mode of transmission. The second equilibrium point of (2.2) satisfies the following system of equations A− νS − Φ(S, C)C −Ψ(S, I)I = 0, (2.4) Φ(S, C)C +Ψ(S, I)I − d1C = 0, (2.5) γC − d2I = 0. (2.6) Hence, C = A− νS d1 , I = γC d2 = γ(A− νS) d1d2 and d2Φ(S, A− νS d1 ) + γΨ(S, γ(A− νS) d1d2 ) = d1d2. Since C = A− νS d1 ≥ 0, we have S ≤ A ν . Then there is no epidemiological equilibrium when S > A ν . The function h can be define on the closed interval [0, A ν ] by h(S) = d2Φ(S, A− νS d1 ) + γΨ(S, γ(A− νS) d1d2 )− d1d2. We have h(0) = −d1d2 < 0, h( A ν ) = d1d2(R0 − 1) and h′(S) = d2 ( ∂Φ ∂S − ν d1 ∂Φ ∂C ) + γ ( ∂Ψ ∂S − γν d1d2 ∂Ψ ∂I ) > 0. Therefore, the equation h(S) = 0 has a unique root S∗ ∈ (0, A ν ) when R0 > 1. This implies that our model has a unique endemic equilibrium E∗(S∗, C∗, I∗) under the condition R0 > 1. Summarizing all the above cases in the following result. Theorem 2.1: Let R0 be defined by (2.3). Copyright© 2022 ASSA. Adv Syst Sci Appl (2022) 40 KHALID HATTAF, AHMED A. MOHSEN (i) If R0 ≤ 1, then system (2.2) has only disease-free equilibrium of the form Ef (S0, 0, 0), where S0 = A ν . (ii) If R0 > 1, then system (2.2) has a unique endemic equilibrium E∗(S∗, C∗, I∗), where S∗ ∈ (0, A ν ), C∗ = A− νS∗ d1 and I∗ = γ(A− νS∗) d1d2 . Also, the disease-free equilibrium exists. 3. GLOBAL DYNAMICS In this section, we analyze the stability of the disease-free equilibrium Ef and the endemic equilibrium E∗. Theorem 3.1: The disease-free equilibrium Ef is globally asymptotically stable if R0 ≤ 1 and unstable if R0 > 1. Proof We define the following Lyapunov functional F(t) = C + Ψ(S0, 0) d2 I. Then DαF(t) = DαC + Ψ(S0, 0) d2 DαI = Φ(S, C)C +Ψ(S, I)I − d1C + Ψ(S0, 0) d2 (γC − d2I). Using A = νS0, we get DαF(t) = d1 ( d2Φ(S, I) + γΨ(S0, 0) d1d2 − 1 ) C + ( Ψ(S, I)−Ψ(S0, 0) ) I. From the first equation of (2.2), we have DαS ≤ A− νS, which leads to S(t) ≤ S(0)Eα(−νtα) + A ν [1− Eα(−νtα)] , where Eα(z) = ∞∑ k=0 zα Γ(αk + 1) denotes the Mittag-Leffler function of one parameter α. Then lim sup t→∞ S(t) ≤ A ν , which involves that all omega limit points verify S(t) ≤ S0. Thus, it suffices to consider solutions for which S(t) ≤ S0. According to the explicit expression of R0 given in (2.3) and (H1)-(H2), we obtain DαF(t) ≤ d1 ( R0 − 1 ) C. Then DαF(t) ≤ 0 when R0 ≤ 1. Furthermore, DαF(t) = 0 if and only if S = S0, C = 0 and I = 0. Therefore, {Ef} is the largest invariant set in {(S, C, I) | DαF(t) = 0}. It Copyright© 2022 ASSA. Adv Syst Sci Appl (2022) DYNAMICS OF A GENERALIZED FRACTIONAL EPIDEMIC MODEL... 41 follows from the LaSalle’s invariance principale [16] that Ef is globally asymptotically stable when R0 ≤ 1. On the other hand, the characteristic equation of model (2.2) at Ef is given by (ν + λ)(α + ν + λ)Θ(λ) = 0, (3.7) where Θ(λ) = λ2 + ( d1 + d1 − Φ(S0, 0) ) λ+ d1d2(1−R0). We have lim λ→+∞ Θ(λ) = +∞ and Θ(0) = d1d2(1−R0) < 0 when R0 > 1. Then we have the eigenvalue λ0 ∈ (0,+∞) such that Θ(λ0) = 0, which implies that the equation (3.7) at Ef has at least one positive root if R0 > 1. Therefore, Ef is saddle (unstable) whenever R0 > 1. In the following, we assume that R0 > 1 and the incidence functions Φ and Ψ satisfy, for all S, C, I > 0, the following assumption( 1− Φ(S, C) Φ(S, C∗) )( Φ(S, C∗) Φ(S, C) − C C∗ ) ≤ 0,( 1− Φ(S∗, C∗)Ψ(S, I) Φ(S, C∗)Ψ(S∗, I∗) )( Φ(S, C∗)Ψ(S∗, I∗) Φ(S∗, C∗)Ψ(S, I) − I I∗ ) ≤ 0. (H4) To establish the global dynamics of system (2.2) when R0 > 1, we need the following Lemma. Lemma 3.1: Let k be a continuous function and u be a continuously differentiable function. For any constant c and α ∈ (0, 1], the Caputo fractional derivative of the function K defined by K(t) = ∫ u(t) c k(x)dx, (3.8) satisfies the following property DαK(t) = k(u(t))Dαu(t)− t−αv(0) Γ(1− α) − α Γ(1− α) ∫ t 0 (t− τ)−α−1v(τ)dτ, (3.9) where v(τ) = k(u(t)) ( u(t)− u(τ) ) + ∫ u(τ) u(t) k(x)dx. In particular, we have DαK(t) = k(u(t))Dαu(t) when α = 1. Moreover, (i) If k is a increasing function, then DαK(u(t)) ≤ k(u(t))Dαu(t). (3.10) (ii) If k is a decreasing function, then DαK(u(t)) ≥ k(u(t))Dαu(t). (3.11) Proof By using the definition of the Caputo fractional derivative, we have DαK(t)− k(u(t))Dαu(t) = 1 Γ(1− α) ∫ t 0 (t− τ)−αu′(τ) ( k(u(τ))− k(u(t)) ) dτ. Copyright© 2022 ASSA. Adv Syst Sci Appl (2022) 42 KHALID HATTAF, AHMED A. MOHSEN Clearly, v′(τ) = u′(τ) ( k(u(τ))− k(u(t)) ) and v(t) = 0. Integrating by parts the last integral, we get DαK(t)− k(u(t))Dαu(t) = 1 Γ(1− α) [ (t− τ)−αv(τ) ]τ=t τ=0 − α Γ(1− α) ∫ t 0 (t− τ)−α−1v(τ)dτ. From L’Hopital’s rule, we have lim τ→t (t− τ)−αv(τ) = lim τ→t v(τ) (t− τ)α = lim τ→t u′(τ) ( k(u(τ))− k(u(t)) ) −α(t− τ)α−1 = lim τ→t − 1 α (t− τ)1−αu′(τ) ( k(u(τ))− k(u(t)) ) = 0. Hence, DαK(t) = k(u(t))Dαu(t)− t−αv(0) Γ(1− α) − α Γ(1− α) ∫ t 0 (t− τ)−α−1v(τ)dτ. Now let us consider the following function ψc,k(τ) = k(c) ( c− τ) + ∫ τ c k(x)dx. It is obvious that ψ′ c,k(τ) = k(τ)− k(c). Assume that k is a increasing function. In this case, the function ψc,k(τ) is decreasing on the interval (−∞, c] and increasing on [c,+∞) with ψc,k(c) = 0. Then ψc,k(τ) has the global minimum at τ = c. Thus, ψc,k(τ) ≥ 0, for all (c, τ) ∈ IR2. Since v(τ) = ψu(t),k(u(τ)), we deduce that v(τ) ≥ 0 and then (i). By using the same technique, we can easily prove (ii). By applying of the fundamental theorem of analysis, we obtain DαK(t) = k(u(t))Dαu(t), when α = 1. (3.12) This completes the proof. Remark 3.1: Lemma 3.1 extends the lemma presented in [1] that allows to find Lyapunov candidate functions for demonstrating the stability of many fractional order systems, and also the elementary lemma given in [17] that estimates fractional derivatives of Volterra-type Lyapunov functions in the sense Caputo for any monotone function. In fact, we have 1 2 Dαu2(t) = Dα ∫ u(t) c xdx ≤ u(t)Dαu(t), (3.13) and Dα [ u(t)− u∗ − u∗ ln u(t) u∗ ] = Dαu(t)− u∗Dα ∫ u(t) u∗ 1 x dx ≤ ( 1− u∗ u(t) ) Dαu(t). (3.14) Copyright© 2022 ASSA. Adv Syst Sci Appl (2022) DYNAMICS OF A GENERALIZED FRACTIONAL EPIDEMIC MODEL... 43 Theorem 3.2: Assume that R0 > 1 and (H4) holds. Then the endemic equilibrium E∗ is globally asymptotically stable. Proof We define a Lyapunov functional as follows G(t) =S − S∗ − ∫ S S∗ Φ(S∗, C∗) Φ(X, C∗) dX + C∗χ ( C C∗ ) + Ψ(S∗, I∗) d2 I∗χ ( I I∗ ) , (3.15) where χ(x̂) = x̂− 1− ln x̂ for x̂ > 0. Since S 7→ 1 Φ(X,C∗) is a decreasing function, it follows from (ii) of Lemma 3.1 and (3.14) that DαG(t) ≤ ( 1− Φ(S∗, C∗) Φ(S, C∗) ) DαS(t) + ( 1− C∗ C ) DαC(t) + Ψ(S∗, I∗) d2 ( 1− I∗ I ) DαI(t) = ( 1− Φ(S∗, C∗) Φ(S, C∗) )( A− νS − Φ(S, C)C −Ψ(S, I)I ) + ( 1− C∗ C )( Φ(S, C)C +Ψ(S, I)I − d1C ) + Ψ(S∗, I∗) d2 ( 1− I∗ I )( γC − d2I ) . Since A = νS∗ + Φ(S∗, C∗)C∗ +Ψ(S∗, I∗)I∗, Φ(S∗, C∗)C∗ +Ψ(S∗, I∗)I∗ = d1C∗ and γC∗ = d2I∗, we have DαG(t) ≤ νS∗ ( 1− S S∗ )( 1− Φ(S∗, C∗) Φ(S, C∗) ) +Φ(S∗, C∗)C∗ ( 2− Φ(S∗, C∗) Φ(S, C∗) + Φ(S, C)C Φ(S, C∗)C∗ − C C∗ ) +Ψ(S∗, I∗)I∗ ( 3− Φ(S∗, C∗) Φ(S, C∗) + Φ(S∗, C∗)Ψ(S, I)I Φ(S, C∗)Ψ(S∗, I∗)I∗ − I I∗ − Ψ(S, I)IC∗ Ψ(S∗, I∗)I∗C − I∗C IC∗ ) . Copyright© 2022 ASSA. Adv Syst Sci Appl (2022) 44 KHALID HATTAF, AHMED A. MOHSEN Hence, DαG(t) ≤ νS∗ ( 1− S S∗ )( 1− Φ(S∗, C∗) Φ(S, C∗) ) +Φ(S∗, C∗)C∗ ( − 1− C C∗ + Φ(S, C∗) Φ(S, C) + Φ(S, C)C Φ(S, C∗)C∗ ) +Ψ(S∗, I∗)I∗ ( − 1− I I∗ + Φ(S, C∗)Ψ(S∗, I∗) Φ(S∗, C∗)Ψ(S, I) + Φ(S∗, C∗)Ψ(S, I)I Φ(S, C∗)Ψ(S∗, I∗)I∗ ) +Φ(S∗, C∗)C∗ ( 3− Φ(S, C∗) Φ(S, C) − Φ(S∗, C∗) Φ(S, C∗) − Φ(S, C) Φ(S∗, C∗) ) +Ψ(S∗, I∗)I∗ ( 4− Φ(S∗, C∗) Φ(S, C∗) − Φ(S, C∗)Ψ(S∗, I∗) Φ(S∗, C∗)Ψ(S, I) − Ψ(S, I)IC∗ Ψ(S∗, I∗)I∗C − CI∗ C∗I ) . From above inequality, we get that 3− Φ(S, C∗) Φ(S, C) − Φ(S∗, C∗) Φ(S, C∗) − Φ(S, C) Φ(S∗, C∗) ≤ 0, and 4− Φ(S∗, C∗) Φ(S, C∗) − Φ(S, C∗)Ψ(S∗, I∗) Φ(S∗, C∗)Ψ(S, I) − Ψ(S, I)IC∗ Ψ(S∗, I∗)I∗C − CI∗ C∗I ≤ 0. The hypothesis (H1) leads to( 1− S S∗ )( 1− Φ(S∗, C∗) Φ(S, C∗) ) ≤ 0. By (H4), we get −1− C C∗ + Φ(S, C∗) Φ(S, C) + Φ(S, C)C Φ(S, C∗)C∗ = ( 1− Φ(S, C) Φ(S, C∗) )( Φ(S, C∗) Φ(S, C) − C C∗ ) ≤ 0 and −1− I I∗ + Φ(S, C∗)Ψ(S∗, I∗) Φ(S∗, C∗)Ψ(S, I) + Φ(S∗, C∗)Ψ(S, I)I Φ(S, C∗)Ψ(S∗, I∗)I∗ = ( 1− Φ(S∗, C∗)Ψ(S, I) Φ(S, C∗)Ψ(S∗, I∗) )( Φ(S, C∗)Ψ(S∗, I∗) Φ(S∗, C∗)Ψ(S, I) − I I∗ ) ≤ 0. Therefore, DαG(t) ≤ 0 with equality holds if and only if S = S∗, C = C∗ and I = I∗. From LaSalle’s invariance principle, we conclude that E∗ is globally asymptotically stable. 4. APPLICATIONS AND NUMERICAL SIMULATIONS In this section, we first apply our above analytical results to the following fractional COVID-19 model  DαS(t) = A− νS − β1SC 1 + ϵ1C − β2SI 1 + ϵ2I , DαC(t) = β1SC 1 + ϵ1C + β2SI 1 + ϵ2I − d1C, DαI(t) = γC − d2I. (4.16) Copyright© 2022 ASSA. Adv Syst Sci Appl (2022) DYNAMICS OF A GENERALIZED FRACTIONAL EPIDEMIC MODEL... 45 where β1 and β2 are the infection rates caused by carrier and infected individuals, respectively. The nonnegative constants ϵ1 and ϵ2 are the saturation rates. Model (4.16) is a special case of system (2.2) with Φ(S, C) = β1S 1 + ϵ1C and Ψ(S, I) = β2S 1 + ϵ2I . It easy that see the basic reproduction number R0 is given by R0 = A(d2β1 + γβ2) νd1d2 . (4.17) Evidently, the infected rates Φ and Ψ satisfy the conditions (H1)− (H3). According to Theorems 3.1 and 3.2, we get the following result. Corollary 4.1: (i) When R0 ≤ 1, the disease-free equilibrium Ef of model (4.16) is globally asymptotically stable. (ii) When R0 > 1, the equilibrium Ef gets unstable and the endemic equilibrium E∗ of model (4.16) is globally asymptotically stable. The second application of our results concerns a fractional model proposed in 2020 by Das and Samanta [2] to study the transmission dynamics of COVID-19 in Japan. This recent fractional model is a particular case of system (4.16), it suffices to take ϵ1 = ϵ2 = 0 and r1 = r2 = d = 0. In [2], the authors just discussed the local stability of the equilibria and calculated epidemic peak in Japan scenario. They proved that the disease-free equilibrium is locally asymptotically stable if R0 < 1 and the endemic equilibrium is locally asymptotically stable under R0 > 1 and others three conditions (see, Theorem 2 [2]). By applying Corollary 4.1, we deduce that when R0 ≤ 1, the disease-free equilibrium of model [2] is globally asymptotically stable. However, this equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable under only R0 > 1. This improve the asymptotic stability results presented recently in [2]. For numerical simulations, we take the some parameters of model (4.16) accordingly to the World Health Organization reports as well as adopted by many references as stated in Table 4.1 below. Parameter Definition Value A Recruitment rate 50 β1 Transmission contact rate between S and C 1.2× 10−5 β2 Transmission contact rate between S and I Varied γ Symptoms period 1/7 day−1 ϵ1 The measure of inhibition effect for carrier 0.04 ϵ2 The measure of inhibition effect for infected 0.01 ν Natural death rate 0.01 day−1 r1 Recovery rate from carrier 1/21 day−1 r2 Recovery rate from infected 1/15 day−1 d Death due to disease rate 0.1 day−1 Table 4.1. Parameter values of model (4.16). For β2 = 5.5× 10−5, we have R0 = 0.9532 ≤ 1. Then model (4.16) has a disease- free equilibrium Ef (5000, 0, 0, 0). By Corollary 4.1 (i), we know that Ef is globally asymptotically stable. Figure 2 demonstrates this result. Copyright© 2022 ASSA. Adv Syst Sci Appl (2022) 46 KHALID HATTAF, AHMED A. MOHSEN 0 100 200 300 400 500 3500 4000 4500 5000 Days S u s c e p ti b le ( S ) α=0.3 α=0.5 α=0.8 α=1 0 100 200 300 400 500 0 200 400 600 Days C a rr ie r (C ) α=0.3 α=0.5 α=0.8 α=1 0 100 200 300 400 500 0 100 200 300 Days In fe c te d ( I) α=0.3 α=0.5 α=0.8 α=1 3500 4000 4500 5000 0 500 0 200 400 SC I Fig. 4.2. Dynamics of the model (4.16) when R0 = 0.9532 ≤ 1. 0 50 100 150 200 3400 3600 3800 4000 Days S u s c e p ti b le ( S ) α=0.3 α=0.5 α=0.8 α=1 0 50 100 150 200 0 200 400 600 Days C a rr ie r (C ) α=0.3 α=0.5 α=0.8 α=1 0 50 100 150 200 0 100 200 300 Days In fe c te d ( I) α=0.3 α=0.5 α=0.8 α=1 3400 3600 3800 4000 0 500 0 200 400 SC I Fig. 4.3. Dynamics of the model (4.16) when R0 = 1.9489 > 1. For β2 = 1.3× 10−4, we have R0 = 1.9489 > 1. Figure 3 shows that the trajectories of model (4.16) converge to E∗(3670.8515, 35.1197, 43.4314) from different values of order fractional derivative α. This confirms the global stability result given by Corollary 4.1 (ii). 5. CONCLUSION In this work, we have presented a new fractional-order model of COVID-19 that takes into modes of infection spread of COVID-19 due to the direct contact between the susceptible individuals, with symptomatic and asymptomatic individuals. The incidence of infection is described by general nonlinear functions. It is proved that the proposed model has two steady states namely disease-free equilibrium and endemic equilibrium. The stability analysis of the model shows that the disease-free equilibrium is globally asymptotically stable if the basic reproduction numberR0 ≤ 1, which biologically means that the COVID-19 is cleared. While the endemic equilibrium is globally asymptotically stable when R0 > 1. In this case, the COVID-19 persists in the world and the infection Copyright© 2022 ASSA. Adv Syst Sci Appl (2022) DYNAMICS OF A GENERALIZED FRACTIONAL EPIDEMIC MODEL... 47 becomes similar to many endemic viral diseases to date, such as influenza, AIDS and viral hepatitis. On the other hand, we have established a new generalized lemma to analyze the global stability of the endemic equilibrium. This lemma presents a new property for Caputo fractional derivative when α ∈ (0, 1] and extends the results presented in [1, 17] for any monotone function. Also, the new generalized lemma can be an important tool to study the global dynamics of many fractional order systems modeling infectious diseases such as COVID-19. Furthermore, the asymptotic stability results of the recent model of COVID-19 in Japan [2] are improved. Several countries around the world have started vaccination against COVID-19 in order to reduce the prevalence of this dangerous disease. We will extend our model to study the impact of vaccination on the propagation of COVID-19. Also, our model used a fractional derivative with singular kernel. It will be more interesting to extend the model by using a fractional derivative with non-singular kernel as in [6]. DATA AVAILABILITY The data used to support the findings of this study are available from the corresponding author upon request. The articles used to support the findings of this study are included within the article and are cited at relevant places within the text as references. CONFLICTS OF INTEREST The authors declare that they have no conflicts of interest. ACKNOWLEDGEMENTS The authors would like to thank the editor and anonymous reviewers for their constructive comments and suggestions, which have improved the quality of the manuscript. REFERENCES 1. Aguila-Camacho, N., Duarte-Mermoud, M. A., & Gallegos, J. A. (2014) A Lyapunov functions for fractional order systems. Commun. Nonlinear. Sci. Numer. Simulat. 19, 2951–2957. 2. Das, M., Samanta, G. P. (2020) A Fractional Order COVID-19 Epidemic Transmission Model: Stability Analysis and Optimal Control. [Online]. Available https://ssrn.com/abstract=3635938. 3. Dehingia, K., Mohsen, A.A., Alharbi, S.A., Alsemiry, R.D., & Rezapour, S. 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Worldometer, COVID-19 Coronavirus pandemic. [Online]. Available https://www.worldometers.info/coronavirus/country/iraq/. Copyright© 2022 ASSA. Adv Syst Sci Appl (2022) Introduction Model formulation and equilibria Global dynamics Applications and numerical simulations Conclusion