A model for the outbreak of COVID-19: Vaccine effectiveness in a case study of Italy
AA model for the outbreak of COVID-19: Vaccineeffectiveness in a case study of Italy
Vasiliki Bitsouni, Nikolaos Gialelis and Ioannis G. Stratis
Abstract
We present a compartmental mathematical model with demography forthe spread of the COVID-19 disease, considering also asymptomatic infectiousindividuals. We compute the basic reproductive ratio of the model and study thelocal and global stability for it. We solve the model numerically based on the caseof Italy. We propose a vaccination model and we derive threshold conditions forpreventing infection spread in the case of imperfect vaccines.
Key words:
Mathematical modeling of COVID-19; SEIAR; SVEIAR; Asymp-tomatic cases; Endemic; Basic reproductive ratio; Stability analysis; Lyapunov func-tion; Vaccine effectiveness; Epidemic prediction; Italy case study; Numerical simu-lations
In late 2019, the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [1], a strain of coronavirus that causes coronavirus disease 2019 (COVID-19),appeared in Wuhan, China, and rapidly spread across the globe. In January 2020 thehuman-to-human transmission of SARS-CoV-2 was confirmed, during the COVID-19 pandemic, and SARS-CoV-2 was designated a Public Health Emergency of
Vasiliki BitsouniDepartment of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis,GR-15784 Zographou, Athens, Greece, e-mail: [email protected]
Nikolaos GialelisDepartment of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis,GR-15784 Zographou, Athens, Greece, e-mail: [email protected]
Ioannis G. StratisDepartment of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis,GR-15784 Zographou, Athens, Greece, e-mail: [email protected] a r X i v : . [ q - b i o . P E ] J u l Vasiliki Bitsouni, Nikolaos Gialelis and Ioannis G. Stratis
International Concern by the WHO, and in March 2020 the WHO declared it apandemic. Since December 31, 2019, and as of July 23, 2020, 15,455,848 cases ofCOVID-19 (in accordance with the applied case definitions and testing strategies inthe affected countries) are confirmed in more than 227 countries and 26 cruise ships[2]. Currently, there are 5,415,828 active cases and 631,775 deaths.Italy is one of the countries suffering the most with the COVID-19 outbreak. Oneof the most critical facts about COVID-19, that affected Italy too, is that a significantnumber of cases, mainly those of young age, has been reported as asymptotic [3],leading to fast spread of the infection. Fortunately, asymptomatic cases have a shorterduration of viral shedding and lower viral load [4, 5]. However, the proportionof asymptomatic cases can range from 4%-80% [6] and most of the time theyplay a key role in infection transmission, therefore it is very important to modelboth symptomatic and asymptomatic cases. The most crucial element of COVID-19 pandemic is demography. After the end of the quarantine and lock-down therehave been many examples of countries where the number of cases increased rapidly.Although, demography is a very important element in epidemics and especially in thecase of COVID-19, only a few models describe the dynamics of infection with SARS-CoV-2 considering demographic terms. To this end, here we develop an SEIARmodel, considering both symptomatic and asymptomatic cases, and demographicterms, with a constant influx of susceptible individuals. In this study we only considerhorizontal infection transmission .As the development of a vaccine against SARS-CoV-2 is an urgent demand , astudy of the epidemiological consequences that an imperfect potential vaccine canhave is needed, and to the best of our knowledge, there is currently no relevant studyin the literature. Here, we develop a theoretical framework to assess the vaccineeffectiveness and the epidemiological consequences of a potential vaccine. Giventhat the second phase might lead to more asymptotic cases than the first phase weneed to investigate the asymptotic group for different coverage and efficacy. Themodel derived in this study provides new insights on the effect of different vaccinecoverage and efficacy on infection spread in a population with demographics.This study is organized as follows. In Section 2, we develop a new SEIAR modelfor COVID-19. We derive the R of the model and we study the local and globalstability of the steady states. Numerical result with a focus on data fitted to Italy caseare presented. In Section 3, we extend our model including also a vaccinated groupof individuals, we study the global stability of the steady states and we predict thevaccine effectiveness. We conclude in Section 4 with a summary and discussion offuture directions. Horizontal transmission is transmission by direct contact between infected and susceptibleindividuals or between disease vectors and susceptible individuals, that are not in a parent-progenyrelationship. Vertical transmission is the passage of the disease-causing agent from mother to babyduring the period immediately before and after birth. During the days of writing this paper, promising progress has been announced in this direction. model for the outbreak of COVID-19: Vaccine effectiveness in a case study of Italy 3
In this Section we model the transmission of SARS-CoV-2 among people causingCOVID-19. To derive our model we consider people who have been in contactwith an infectious individual, but remain uninfected for a latency period. Moreover,a significant number of individuals being infected remain asymptomatic, due tovarious factors such as age, health condition etc [3]. To this end, we classify thetotal population ( N ) into five subclasses: susceptible ( S ), latent ( E ), symptomaticinfectious ( I ), asymptomatic infectious ( A ), and recovered ( R ) individuals, hence wehave that S + E + I + A + R = N . We can consider that N is practically constant,since the time-span of the epidemiological phenomenon is relatively short and N is relatively large. We take into account demographic terms and we consider thetransmission to be exclusively horizontal. The SEIAR model is governed by thefollowing system of nonlinear ordinary differential equations:d S d t = µ N − β I SI − β A S A − µ S , (1a)d E d t = β I SI + β A S A − ( k + µ ) E , (1b)d I d t = k ( − q ) E − ( γ I + µ ) I , (1c)d A d t = kqE − ( γ A + µ ) A , (1d)d R d t = γ I I + γ A A − µ R , (1e)with initial conditions: ( S ( ) , E ( ) , I ( ) , A ( ) , R ( )) = ( S , E , I , A , R ) ∈ (cid:0) R + (cid:1) , (2)where µ is the birth/death rate, β I and β A the transmission rates of I and A , respec-tively , k the incubation rate, i.e. the rate of latent individuals becoming infectious, q the proportion of asymptotic infectious individuals, γ I and γ A the recovery ratesof infectious and asymptotic infectious individuals, respectively. A flow diagram ofthe model is illustrated in Fig. 1. A straightforward application of the classical ODEtheory yields that the above Cauchy problem is well-posed. β I , A ≡ c (cid:36) I , A N , where c is the average number of close contacts of an individual with otherindividuals and (cid:36) I , A is the probability of a contact to be effective in turning an S individual intoan I or A one, respectively. Vasiliki Bitsouni, Nikolaos Gialelis and Ioannis G. Stratis Fig. 1: Flow diagram of the SEIAR model (1).
The average number of secondary cases arising from one infection when the entirepopulation is susceptible is defined as the basic reproductive ratio and denoted by R . Being the most important quantity on infectious disease epidemiology, the basicreproductive ratio is a dimensionless quantity, which is often used to reflect howinfectious a disease is. To define R we calculate the next-generation matrix of thesystem, see, e.g., [7, 8].First, we linearise model (1) around the disease-free steady state, ( N , , , , ) ,and we consider the infected states, i.e. E , I , A , to obtain the linearised infectionsubsystem d E d t = β I N I + β A N A − ( k + µ ) E , (3a)d I d t = k ( − q ) E − ( γ I + µ ) I , (3b)d A d t = kqE − ( γ A + µ ) A . (3c)Then, we set x = ( E , I , A ) tr , so that the system (3a)-(3c) can be written in the formd x d t = ( T + Σ ) x , where model for the outbreak of COVID-19: Vaccine effectiveness in a case study of Italy 5 T = β I N β A N is the transmission matrix , and Σ = − ( k + µ ) k ( − q ) − ( γ I + µ ) kq − ( γ A + µ ) is the transition matrix .Then, R is defined as the dominant eigenvalue of matrix K = − TΣ − as follows: K = − TΣ − = − β I N β A N · − ( k + µ ) − k ( − q )( k + µ ) ( γ I + µ ) − ( γ I + µ ) − kq ( k + µ ) ( γ A + µ ) − ( γ A + µ ) = β I N k ( − q )( k + µ ) ( γ I + µ ) + β A N kq ( k + µ ) ( γ A + µ ) β I N ( γ I + µ ) β A N ( γ A + µ ) , from which we obtain that R = β I N k ( − q )( k + µ ) ( γ I + µ ) + β A N kq ( k + µ ) ( γ A + µ ) . (4) We proceed with the local stability analysis of the model. The Jacobian matrix ofsystem (1) is J = − β I I − β A A − µ − β I S − β A S β I I + β A A − ( k + µ ) β I S β A S k ( − q ) − ( γ I + µ ) kq − ( γ A + µ )
00 0 γ I γ A − µ . (5) Theorem 1 If R < , the disease-free steady state, ( N , , , , ) , of system (1) islocally stable. Proof
For the disease-free steady state ( N , , , , ) we obtain a double negativeeigenvalue, λ = − µ , and the characteristic equation of the reduced 3x3 matrix Vasiliki Bitsouni, Nikolaos Gialelis and Ioannis G. Stratis λ + (cid:18) γ I + γ A + k + µ (cid:19) λ + (cid:20) ( k + µ ) ( γ I + γ A + µ ) + ( γ I + µ ) ( γ A + µ )− N k ( β I ( − q ) + β A q ) (cid:21) λ + ( k + µ ) ( γ I + µ ) ( γ A + µ )− N k (cid:18) β I ( − q ) ( γ A + µ ) + β A q ( γ I + µ ) (cid:19) = . We prove the stability of this steady state using the Routh-Hurwitz criterion [9].The disease-free steady state is stable if and only if γ I + γ A + k + µ > , (6) ( k + µ ) ( γ I + µ ) ( γ A + µ ) − β I N k ( − q ) ( γ A + µ ) − β A N kq ( γ I + µ ) > , (7) ( γ I + γ A + k + µ ) (cid:20) ( k + µ ) ( γ I + γ A + µ ) + ( γ I + µ ) ( γ A + µ )− N k (cid:18) β I ( − q ) + β A q (cid:19)(cid:21) − ( k + µ ) ( γ I + µ ) ( γ A + µ ) + N k (cid:18) β I ( − q ) ( γ A + µ ) + β A q ( γ I + µ ) (cid:19) > . (8)The inequality (6) holds always. The inequality (7) can be equivalently written as R = β I N k ( − q )( k + µ ) ( γ I + µ ) + β A N kq ( k + µ ) ( γ A + µ ) < . (9)By using Mathematica [10] we can confirm that the inequality (8) holds for R < (cid:3) Theorem 2 If R > , the endemic steady state, ( S ∗ , E ∗ , I ∗ , A ∗ , R ∗ ) , of system (1) with S ∗ = ( γ I + µ ) ( γ A + µ ) ( k + µ ) k ( β I ( γ A + µ ) + q ( β A ( γ I + µ ) − β I ( γ A + µ ))) = N R , E ∗ = N µ ( k + µ ) (cid:18) − R (cid:19) , I ∗ = N k ( − q ) µ ( γ I + µ ) ( k + µ ) (cid:18) − R (cid:19) , A ∗ = N kq µ ( γ A + µ ) ( k + µ ) (cid:18) − R (cid:19) , R ∗ = N k (cid:18) q γ A µ + γ I ( γ A + µ − q µ ) (cid:19) ( γ I + µ ) ( γ A + µ ) ( k + µ ) (cid:18) − R (cid:19) , (10) is locally stable. Proof
The characteristic equation of the Jacobian matrix (5) at the endemic steadystate is model for the outbreak of COVID-19: Vaccine effectiveness in a case study of Italy 7 λ + α λ + α λ + α λ + α = , with α = γ I + γ A + k + µ + µ R > ,α = µ R ( γ I + γ A + k + µ ) + ( γ I + γ A + µ ) ( k + µ ) + ( γ I + µ ) ( γ A + µ )− N R k ( β I ( − q ) + β A q ) ,α = µ R ( γ I + γ A + µ ) ( k + µ ) + ( γ I + µ ) ( γ A + µ ) ( µ R + k + µ )− N R k ( β I ( − q ) ( γ A + µ + ) + β A q ( γ I + µ + )) ,α = µ R ( k + µ ) ( γ I + µ ) ( γ A + µ ) − µ N R k ( β I ( − q ) ( γ A + µ ) + β A q ( γ I + µ )) . From the Routh-Hurwitz criterion, the endemic steady (10) is locally stable if andonly if α > , α > , α > α α α − α − α α > . We have that α > a > R >
1. By usingMathematica [10] we can confirm that the rest of the above relations hold for R > (cid:3) Theorem 3 If R ≤ , then the disease-free steady state, ( N , , , , ) , of system (1) is globally asymptotic stable. Proof
We prove the global stability of the disease-free steady state ( N , , , , ) byconstructing a Lyapunov function. We consider the function V : ( R + ) → R + with V ( S ( t ) , E ( t ) , I ( t ) , A ( t )) = (cid:18) S − N − N ln SN (cid:19) + E + N β I γ I + µ I + N β A γ A + µ A . We take the derivative of V with respect to t : V (cid:48) = S (cid:48) (cid:18) − NS (cid:19) + E (cid:48) + N β I γ I + µ I (cid:48) + N β A γ A + µ A (cid:48) = µ N − µ S − µ N S + E ( k + µ ) (cid:18) β I N k ( − q )( k + µ ) ( γ I + µ ) + β A N kq ( k + µ ) ( γ A + µ ) − (cid:19) = − µ N (cid:18) SN + NS − (cid:19) + E ( k + µ ) (R − ) . Thus, if R ≤ V (cid:48) ≤ t ≥ ( S , E , I , A ) ∈ ( R + ) sufficientlyclose to ( N , , , ) , and V (cid:48) ( t ) = ( S , E , I , A ) = ( N , , , ) . Hence, Vasiliki Bitsouni, Nikolaos Gialelis and Ioannis G. Stratis the singleton {( N , , , )} is the largest invariant set for which V (cid:48) =
0. Then,from LaSalle’s Invariant Principle [11] it follows that the disease-free steady state isglobally asymptotic stable. (cid:3)
Theorem 4 If R > , then the endemic steady state, ( S ∗ , E ∗ , I ∗ , A ∗ , R ∗ ) , of system (1) is globally asymptotic stable. Proof
We consider the function V : ( R + ) → R + with V ( S ( t ) , E ( t ) , I ( t ) , A ( t )) = (cid:18) S − S ∗ − S ∗ ln SS ∗ (cid:19) + (cid:18) E − E ∗ − E ∗ ln EE ∗ (cid:19) + β I S ∗ γ I + µ (cid:18) I − I ∗ − I ∗ ln II ∗ (cid:19) + β A S ∗ γ A + µ (cid:18) A − A ∗ − A ∗ ln AA ∗ (cid:19) . We take the derivative of V with respect to t : V (cid:48) = S (cid:48) (cid:18) − S ∗ S (cid:19) + E (cid:48) (cid:18) − E ∗ E (cid:19) + β I S ∗ γ I + µ I (cid:48) (cid:18) − I ∗ I (cid:19) + β A S ∗ γ A + µ A (cid:48) (cid:18) − A ∗ A (cid:19) = ( µ N − β I SI − β A S A − µ S ) (cid:18) − S ∗ S (cid:19) + ( β I SI + β A S A − ( k + µ ) E ) (cid:18) − E ∗ E (cid:19) + β I S ∗ γ I + µ ( k ( − q ) E − ( γ I + µ ) I ) (cid:18) − I ∗ I (cid:19) + β A S ∗ γ A + µ ( kqE − ( γ A + µ ) A ) (cid:18) − A ∗ A (cid:19) . After using the relations µ N = β I S ∗ I ∗ + β A S ∗ A ∗ + µ S ∗ , and β I S ∗ I ∗ + β A S ∗ A ∗ = ( k + µ ) E ∗ , k ( − q ) E ∗ = ( γ I + µ ) I ∗ , kqE ∗ = ( γ A + µ ) A ∗ , and adding and subtracting the terms β I S ∗ I ∗ EIE ∗ and β A S ∗ A ∗ EAE ∗ we have V (cid:48) = − µ S ∗ (cid:18) SS ∗ + S ∗ S − (cid:19) − β I S ∗ I ∗ (cid:18) S ∗ S + SS ∗ II ∗ E ∗ E + I ∗ I EE ∗ − (cid:19) − β A S ∗ A ∗ (cid:18) S ∗ S + SS ∗ AA ∗ E ∗ E + A ∗ A EE ∗ − (cid:19) . From the arithmeticâĂŞgeometric mean inequality we have that13 (cid:18) S ∗ S + SS ∗ II ∗ E ∗ E + I ∗ I EE ∗ (cid:19) ≥ (cid:114) S ∗ S SS ∗ II ∗ E ∗ E I ∗ I EE ∗ = , and model for the outbreak of COVID-19: Vaccine effectiveness in a case study of Italy 9 (cid:18) S ∗ S + SS ∗ AA ∗ E ∗ E + A ∗ A EE ∗ (cid:19) ≥ (cid:114) S ∗ S SS ∗ AA ∗ E ∗ E A ∗ A EE ∗ = , hence V (cid:48) ≤ ( S , E , I , A ) ∈ ( R + ) , and the equality holds only for the endemicsteady state ( S ∗ , E ∗ , I ∗ , A ∗ ) . We conclude again from LaSalle’s Invariance Principlethat the endemic steady state is globally asymptotic stable. We proceed to the estimation of the already known epidemic curve of the disease inItaly, as obtained from the data set [12], by d R d t (see, e.g., [13] and [14]). We plottogether the two functions in Fig. 2. The total population of Italy is 60,456,999. Oncethe restriction of movement (quarantine) during the manifestation of COVID-19 wasapplied, it limited the spread of the disease. To this end we follow the approach in[15] and we consider as the total population of N = 60,456,999/250.Fig. 2: Number of confirmed cases per day in Italy. The blue dots represents thedata obtained from [12] and the red curve the graph of d R d t , as obtained by (1). Theparameters used here are: β I = . / N , β A = . / N , k = . , µ = . , γ I = . , γ A = . , q = . S = N − , I = , A = , E = R = I ∗ + A ∗ . Fig. 3: The dynamics of the proportion of the values of model (1). (a) For allsubgroups; (b) For the groups of both symptomatic and asymptomatic infectiousindividuals, I + A , towards the steady state I ∗ + A ∗ . The parameters used are thesame as in Fig. 2 (see also Table 1). The initial conditions are: S = − . , I = . , A = . , E = R = In this section we consider the subclass of the vaccinated-with-a-prophylactic-vaccine ( V ) individuals. We set 0 ≤ p ≤ ≤ (cid:15) < S d t = ( − p ) µ N − β I SI − β A S A − µ S , (11a)d V d t = p µ N − ( − (cid:15) ) ( β I V I + β A V A ) − µ V , (11b)d E d t = ( S + ( − (cid:15) ) V ) ( β I I + β A A ) − ( k + µ ) E , (11c)d I d t = k ( − q ) E − ( γ I + µ ) I , (11d)d A d t = kqE − ( γ A + µ ) A , (11e)d R d t = γ I I + γ A A − µ R , (11f)along with the initial condition: ( S ( ) , V ( ) , E ( ) , I ( ) , A ( ) , R ( )) = ( S , V , E , I , A , R ) ∈ (cid:0) R + (cid:1) . (12) model for the outbreak of COVID-19: Vaccine effectiveness in a case study of Italy 11 Following the same steps as before and using the disease-free steady state of themodel, (( − p ) N , pN , , , , ) , we have that the basic reproductive ratio for themodel where vaccination is applied is R V = ( − (cid:15) p ) (cid:18) β I N k ( − q )( k + µ ) ( γ I + µ ) + β A N kq ( k + µ ) ( γ A + µ ) (cid:19) = ( − (cid:15) p ) R . (13)The endemic steady state, ( S ∗ , V ∗ , E ∗ , I ∗ , A ∗ , R ∗ ) , of model (11) is (cid:18) ( − p ) N R V , pN R V , N µ ( k + µ ) (cid:32) − R V (cid:33) , ( − (cid:15) p ) N k ( − q ) µ ( γ I + µ ) ( k + µ ) (cid:32) − R V (cid:33) , ( − (cid:15) p ) N kq µ ( γ A + µ ) ( k + µ ) (cid:32) − R V (cid:33) , ( − (cid:15) p ) N k (cid:18) q γ A µ + γ I ( γ A + µ − q µ ) (cid:19) ( γ I + µ ) ( γ A + µ ) ( k + µ ) (cid:32) − R V (cid:33)(cid:19) . Fig. 4: Flow diagram of the SVEIAR model (11).We prove the global stability of the model following the same steps as before.
Theorem 5 If R V ≤ , then the disease-free steady state, (( − p ) N , pN , , , , ) ,of system (11) is globally asymptotic stable. Proof
We prove that the disease-free steady state of system (11) is globally asymp-totic stable by applying again the LaSalle’s Invariant Principle for the Lyapunovfunction V V : ( R + ) → R + with V V ( S ( t ) , V ( t ) , E ( t ) , I ( t ) , A ( t )) = (cid:18) S − ( − p ) N − ( − p ) N ln S ( − p ) N (cid:19) + (cid:18) V − pN − pN ln VpN (cid:19) + E + ( − (cid:15) p ) N β I γ I + µ I + ( − (cid:15) p ) N β A γ A + µ A , and following the steps corresponding to the proof of Theorem 3. (cid:3) Theorem 6 If R V > , then the endemic steady state, ( S ∗ , V ∗ , E ∗ , I ∗ , A ∗ , R ∗ ) , ofsystem (11) is globally asymptotic stable. Proof
We prove that the endemic steady state of system (11) is globally asymptoticstable by applying again LaSalle’s Invariant Principle for the Lyapunov function V V : ( R + ) → R + , with V V ( S ( t ) , V ( t ) , E ( t ) , I ( t ) , A ( t )) = (cid:18) S − S ∗ − S ∗ ln SS ∗ (cid:19) + (cid:18) V − V ∗ − V ∗ ln VV ∗ (cid:19) + (cid:18) E − E ∗ − E ∗ ln EE ∗ (cid:19) + β I S ∗ + ( − (cid:15) ) β I V ∗ γ I + µ (cid:18) I − I ∗ − I ∗ ln II ∗ (cid:19) + β A S ∗ + ( − (cid:15) ) β A V ∗ γ A + µ (cid:18) A − A ∗ − A ∗ ln AA ∗ (cid:19) , and following the steps corresponding to the proof of Theorem 4. (cid:3) To assess vaccine effectiveness we focus on three important epidemiological mea-sures [17]: (i) the risk of infection spread, represented by R V ; (ii) the peak prevalenceof infection; (iii) the time at which the peak prevalence occurs. Relation (13) showsthat the vaccine coverage, p , and vaccine efficacy, (cid:15) , act multiplicatively on R . Asthe proportion of asymptotic cases is still unknown, in Fig 5a we present a contourplot of the dependence of R V on the vaccine coverage and vaccine efficacy, for dif-ferent proportion of asymptotic cases. The coloured curves represent the threshold R V = R V >
1) or not (represented by the area above the threshold; R V < model for the outbreak of COVID-19: Vaccine effectiveness in a case study of Italy 13 where the severity of COVID-19 was not yet known and the average number ofclose contacts between individuals was very high due to occasions and events, thetransmission rate, β I , as obtained by the data is not the most appropriate index topredict vaccine effectiveness, as the situation has changed dramatically and closecontacts have been significantly reduced. Hence, in Fig 5b we present a correspond-ing contour plot for a lower β I . We see that in the case of a reduced transmission ratethe vaccine can prevent the infection spread, even for imperfect vaccines and smallvaccine coverage.In Fig 6 we see the effect of vaccine efficacy on the proportion of the infectiondynamics for high (Fig 6a) and low (Fig 6b) transmission rates. Higher vaccineefficacy leads to milder, but prolonged epidemics due to the slower rate of infectiontransmission. Moreover, it causes later occurrence of the first infection incidence andpeak prevalence, and a slower rate of postpeak prevalence decline. (a) (b) Fig. 5: Assessing the vaccine effectiveness: A contour plot showing the dependenceof R V on the vaccine efficacy, vaccine coverage and proportion of asymptotic casesfor (a) β I = .
55; (b) β I = .
2. The rest of the model parameters are given in Table 1.
We presented an ad hoc
SEIAR model with horizontal transmission and demo-graphic terms for the epidemic spread of COVID-19, and we extended the model toinclude vaccination. The stability of both models is proved by implementing suitableLyapunov functions; the model is fitted to real data from the epidemic in Italy. Westudied the condition under which a vaccine can prevent disease spread. We accessed
Fig. 6: Assessing the vaccine effectiveness: The effect of vaccination on the preva-lence of infection, with ICs: S = V = . , I = A = . β I = .
55; (b) β I = .
2. The rest of the model parameters are given in Table 1.Table 1: Model parameters, values, units and relevant references.
Param. Description Value Unit Reference µ Birth/Death rate 0 . ( · − − . ) days − [18, 19] β I Transmission rate of infectious individuals 2 . ( . − . ) individuals − · days − [5, 15, 20, 21,22] β A Transmission rate of asymptotic infectious individ-uals β I − · days − [5, 15, 20, 21, 22,23] k Incubation rate (rate of latent individuals becominginfectious) 0 . ( , − . ) days − [24, 25] q Proportion of the asymptotic infectious individuals 0 . ( − ) - [3] γ I Recovery rate of the infectious individuals 0 . ( . − . ) days − [4, 26, 27] γ A Recovery rate of the asymptomatic infectious indi-viduals 0 . ( . − . ) days − [4, 27] p Proportion of vaccinated individuals 0 . ( − ) - Estimated (cid:15) Vaccine efficacy 0 . ( − ) - Estimated the vaccine effectiveness focusing on the risk of infection spread, the peak prevalenceof infection and the time at which the peak prevalence occurs.Future work includes further investigation of the vaccine model, by incorporatingdifferent vaccination strategies, and if possible the comparison with biological data.An extension of the model will also include additional important factors of COVID-19 spread, such as the age and the waning immunity gained by infected individuals,as well as vertical transmission and migration terms for the infected individuals. model for the outbreak of COVID-19: Vaccine effectiveness in a case study of Italy 15 References
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