key: cord-1018670-mrswtbyq
authors: Buonomo, B.; Della Marca, R.; d'Onofrio, A.; Groppi, M.
title: A behavioural modelling approach to assess the impact of COVID-19 vaccine hesitancy
date: 2021-06-25
journal: nan
DOI: 10.1101/2021.06.22.21259324
sha: 8a83560921ffed29439526a23f33df97d2fb9f42
doc_id: 1018670
cord_uid: mrswtbyq

In this paper we introduce a compartmental epidemic model describing the transmission of the COVID-19 disease in presence of non-mandatory vaccination. The model takes into account the hesitancy and refusal of vaccination. To this aim, we employ the information index, which mimics the idea that individuals take their decision on vaccination based not only on the present but also on the past information about the spread of the disease. Theoretical analysis and simulations show clearly as a voluntary vaccination can certainly reduce the impact of the disease but it is unable to eliminate it. We also show how the information-related parameters affect the dynamics of the disease. In particular, the hesitancy and refusal of vaccination is better contained in case of large information coverage and small memory characteristic time. Finally, the possible influence of seasonality is also investigated.

On 31 December 2019, the Chinese public health authorities reported to WHO the existence in Wuhan City of a cluster of cases of viral pneumonia [73] . The causal agent of the disease was shortly later identied as a new type of SARS, and named SARSCoV2. Although many governments undervalued the pandemic risks [55] , since 21 January 2020 WHO published on its website a daily situation reports. In respectively, in [26, 56] and in [23] by means of phenomenological models. In a recent paper [8] a model for the transmission of COVID19 disease has been introduced. The model considers the social distancing and quarantine as mitigation strategies by the Public Health System. The model is informationdependent, in the sense that contact rate and quarantine rate are assumed to depend on the available information and rumours about the disease status in the community. In [8] the model is applied to the case of the COVID19 epidemic in Italy. The paper estimates that citizen compliance with mitigation measures played a decisive role in curbing the epidemic curve, by preventing a duplication of deaths and about 46% more infections. The COVID19 pandemic caused a worldwide eort on the vaccine that resulted in the rapid development of new vaccines [45, 52] , some of which belongs to the new class of mRNA vaccines [2, 64] . In the light of the deep changes in the life of milliards of people and of the huge negative impact on world economics that the world has experienced, one could have expected that only a tiny proportion of people would really be hesitant towards vaccination. Unfortunately, this is not what occurred. As early as June 2020 NeumannBöhme and coworkers [62] investigated the attitudes about anti COVID19 vaccination of a representative sample of citizens of seven European countries. Amazingly, although the rst European epidemic wave had just ended, a large proportion of hesitancy and opposition to the vaccines were found in all class ages, and in both sex. In particular, in France the 38% of citizens were hesitant (28%) or strongly against (10%) anti COVID19 vaccines. Before mid December 2020 phase 3 of a number of vaccines ended, showing that they have a very outstanding eectiveness in preventing COVID19 [2, 52, 64] . Typically, drug regulatory agencies dened priority groups for the vaccination (elderly people with serious comorbidities, healthcare workers in senior residences, etc.). From a rational viewpoint there were all the premises to believe that the vaccine hesitancy would have been strongly reduced and that mandatory vaccination campaigns could have been conducted but this was not the case. As far as the mandatory nature of the vaccination campaign is concerned, in many countries the vaccines are no mandatory [48, 54, 69] . As for the vaccine hesitancy, an investigation conducted in October 2020 [38] suggests that 46% of French citizens are vaccine hesitant. Other countries have percentages of opposition and hesitancy that exceeds 30%: 36% in Spain and USA, 35% in Italy, 32% in South Africa, 31% in Japan and Germany. Globally, the hesitancy and objection area is as large as 27%. Given these large percentages of hesitance and opposition to the COVID19 vaccine, we think that applying the behavioural epidemiology approach to model the implementation of a vaccination campaign for COVID19 is appropriate. To this end, we adopt a strategy remindful of the one used in [26] . Namely, we assume that the vaccination rate is a phenomenological function of the present and past information that the citizens have on the spread of the epidemic. Note that, in the context of SIR and SEIR infectious diseases, more mechanistic models based on evolutionary game theories [3, 24, 25, 72] exist, but reduce to the approach of [10, 26] in case of volatile opinion switching [18, 24, 72] . In this paper, we consider a COVID19 aected population controlled by vaccination, where the nal choice to vaccinate or not is partially determined on a fully voluntary basis and depends on the publicly available information on both present and recent past spreading of the disease in the community. Our model is inspired by the compartmental epidemic model introduced in [8] , where the COVID19 transmission during the 2020 lockdown in Italy was studied. In some sense, compared with the model in [8] , the main dierence is that here the nonpharmaceutical interventions (social distancing and quarantine) are replaced by vaccination. An analogous situation was considered by Gumel and coworkers for SARS epidemic in 2003 when they studied a SARS model in [37] and then considered vaccination intervention in [36] . We perform a qualitative analysis based on stability theory and bifurcation theory. The analysis shows that, when the control reproduction number, R V , is less than 1, there exists only the diseasefree equilibrium (DFE) that is globally stable; otherwise, when R V > 1, the DFE is unstable and an endemic equilibrium arises. The model is then parametrized based on the COVID19 epidemic in Italy and on preliminary reports about anti COVID19 vaccines. In numerical simulations, we consider two possible starting times for a one yearlasting vaccination campaign. We assess the role of vaccine and information related parameters by evaluating how they aect suitable epidemiological indicators. Finally, the presence of seasonality eects is investigated by adding the assumption that the disease transmission and severity as well as the rate of vaccination, are lower during the warmer months. The paper is organized as follows. In Section 2 the model is introduced and in Section 3 the qualitative analysis is performed. Model parametrization and numerical solutions are given in Section 4 and Section 5, respectively. The case of seasonallyvarying parameter values is addressed in Section 6. Concluding remarks follow in Section 7. The paper is complemented by the Appendix A.

2 The model

We consider a population aected by COVID19 disease, where a vaccine is available and administered on voluntary basis and not mandatory. We assume that the vaccine provides only partial protection, so that the transmission of the disease due to contacts between vaccinated and infectious individual is still possible, although with reduced probability. We also assume that both the vaccineinduced immunity and the diseaseinduced immunity are not waning (see Remark 1 below for a discussion on this point). The total population at time t (say, N ) is divided into the following six disjoint compartments: -susceptibles, S: individuals who are healthy but can contract the disease; -exposed (or latent), E: individuals who are infected by SARSCoV2 but are not yet capable of transmitting the virus to others; -asymptomatic infectious, I a : this compartment includes two groups, namely the postlatent individuals, i.e. individuals who lie in the phase of incubation period following latency, where they are infectious and asymptomatic, and the truly asymptomatic individuals, i.e. who have no symptoms throughout the course of the disease; -symptomatic infectious, I s : infectious individuals who show mild or severe symptoms; -vaccinated, V : individuals who are vaccinated with at least one dose of COVID19 vaccine; -recovered, R: individuals who are recovered after the infectious period.

The size of each compartment at time t represents a state variable of the mathematical model, and

We assume that agents take their decision on vaccination based not only on the present but also on the past information they have on the spread of the disease, the past being weighted in an exponential way. Therefore the information on the status of the disease in the community is described by means of the information index [26, 72] :

Such index is an important tool of behavioural epidemiology [56] and is an extension of the idea of the prevalencedependent contact rate, developed by V. Capasso in the seventies, which describes the behavioural response of individuals to prevalence [11] . Here, the parameter a takes the meaning of inverse of the average time delay of the collected information on the disease (say, T a = a −1 ) and the parameter k is the information coverage, which summarises two opposite phenomena: the disease underreporting and the level of media coverage of the disease status, which tends to amplify the social alarm. It may be assumed that k ∈ (0, 1], see [9] . From (1), by applying the linear chain trick [53] , we obtain the dierential equationṀ = a (kI s − M ), ruling the dynamics of M . Remark 1. Together with the role of human behaviour in the vaccine decisions, the other major hypotheses of the above model are that the vaccine is not perfect and there is no waning eect of both natural and vaccineinduced immunity. The rst is related to the scientic results on the phase 3 clinical trials as well as general knowledge concerning vaccines. The second hypothesis is stronger, and some could read it as modelling an extreme optimistic case. Such assumption is based on some very recent experimental results [43, 71] and experimental review paper [44] on one of the most complex and intriguing topic concerning SARSCoV2: the immunological response associated to it. In particular, Iyer and colleagues [43] showed that the igG response has practically no conversion for a long period after the onset of symptoms, namely only 3 individuals over 90 had igG seroconversion. This very limited fraction of seroconversion can be taken into account (through a coecient σ, see Section 2.3) as some vaccinated individuals get infected because they had seroconversion of their vaccineinduced immune response. Moreover, in their review paper on T cell immunity to COVID19 [44] , Karlsson 

Global research on how SARSCoV2 is transmitted continues to be conducted at time of writing this paper. It is believed that infected people appear to be most infectious just before (around 12 days before) they develop symptoms (i.e. in the postlatency stage) and early in their illness [74] . Recent investigations conrmed that presymptomatic transmission was more frequent than symptomatic transmission [4] . The possibility of contagion from a truly asymptomatic COVID19 infected person (i.e. an infected individual who does not develop symptoms) is still a controversial matter. However it has been shown that little to no transmission may occur from truly asymptomatic patients [4] . In our model the routes of transmission from COVID19 patients are included in the Force of Infection (FoI) function, i.e. the per capita rate at which susceptibles contract the infection. As in [36] , the mass action incidence is considered:

where 0 ≤ ε a , ε s < 1.

The rationale for this choice is that during observed COVID19 outbreaks the total population has remained eectively constant. For instance, in Italy (one of the countries more hit by the epidemic [75] ), the drop in the total population (≈ 60 · 10 6 ) due to the diseaseinduced deaths (≈ 117 · 10 3 as of 19 April 2021 [41] ) is around 0.195%. In this case, we expect mass action and standard incidence to give similar results.

In (2) the parameters ε a and ε s are modication factors that represent the level of reduced infectiousness of compartments I a and I s when compared with the subgroup of I a given by postlatent individuals. Therefore, the baseline transmission rate β is the transmission rate of postlatent individuals (see also Section 4.2 where ε a and ε s are estimated). For the reasons discussed above we assume that the factor concerning the postlatent individuals is 1.

All the state variables decrease by natural death, with rate µ. The susceptible population S increases by the net inow Λ, incorporating both new births and immigration and decreases due to transmission and vaccination. For the time span covered in our simulations, demography could be neglected. However, including a net inow of susceptible individuals into the model allows one to consider not only new births, but also immigration, which plays an important role during COVID19 epidemics and can be well estimated in some cases [8] . Therefore, since the demography parameters can be easily obtained from data, we prefer to use an SEIRlike model with demography as successfully done for SARS models [37] . The exposed (or latent) individuals E arise as the result of new infections of susceptible and vaccinated individuals and decrease by development at the infectious stage (at rate ρ). We assume that after the end of the latency period, the individuals enter in the asymptomatic compartment I a , which includes postlatent and truly asymptomatic, as described in Section 2.1. Asymptomatic individuals I a diminish because they enter the compartment of symptomatic individuals I s (at a rate η) or they recover (at a rate ν a ). Mildly or severely symptomatic individuals I s come from the postlatency stage and get out due to recovery (at rate ν s ) or diseaseinduced death (at rate δ). Vaccinated individuals V come from the susceptible class after vaccination (at least one dose of COVID19 vaccine) and decrease due to infections (at a reduced rate σβ, where σ ∈ [0, 1)). Finally, recovered individuals come from the infectious compartments I a and I s and, as discussed in Remark 1, acquire long lasting immunity against the disease.

According to the description above, the time evolution of the state variables is ruled by the following system of balance equations:

with initial conditions

Since the equations (3) do not depend on R, the dynamics of the removed compartment can possibly be studied separately, by means of equationṘ = ν a I a + ν s I s − µR.

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The copyright holder for this preprint this version posted June 25, 2021. ; Figure 1 : Flow chart for the COVID19 model (3)(5). The population N (t) is divided into six disjoint compartments of individuals: susceptible S(t), exposed E(t), asymptomatic I a (t), symptomatic I s (t), vaccinated V (t) and recovered R(t). Blue colour indicates the informationdependent process in the model, with M (t) ruled by (3f).

In (3) it is assumed that ϕ 0 > 0 and ϕ 1 (·) is a continuous increasing function of the information index M with ϕ 1 (0) = 0 and sup(ϕ 1 ) < 1 − ϕ 0 . The parameter ϕ 0 embeds: i) the fact that some categories of subjects such as patients and healthcare workers in senior care facilities will be strongly recommended to get the vaccine (and in some countries their vaccination will be even mandatory [48] ); ii) the fact that some people are strongly in favour of vaccines and act coherently by getting vaccinated. The ow chart in Fig. 1 illustrates all the processes included in the model; a description of each parameter together with their baseline values is given in Table 1 (see Section 4).

The following theorem ensures that the solutions of model (3) are epidemiologically and mathematically wellposed.

with initial conditions (4) is positively invariant for model (3) .

Proof. By standard procedure (see e.g. [66] ), from (3)(4) one can derive that

for all t ≥ 0.

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The copyright holder for this preprint this version posted June 25, 2021. Let us introduce the variableÑ = S + E + I a + I s + V , that is, at each time t, the total population devoid of the removed individuals. Adding the rst ve equations of the system (3), we obtaiṅ

where we use (7) . The solutionÑ of the dierential equation in (8) has the following property

implying that 0 <Ñ ≤ Λ/µ, as t → +∞. Specically, ifÑ (0) ≤ Λ/µ, then Λ/µ is the upper bound ofÑ ; ifÑ (0) > Λ/µ, thenÑ will decrease to Λ/µ. Similarly, from equation (3a) and property (7), it follows thatṠ ≤ Λ − (µ + ϕ 0 ) S, yielding

Then,

Inequalities (9) and (10), taking into account that σ ∈ [0, 1), imply that

Let us now prove that M ≤ kΛ/µ, as t → +∞. From the denition of M , as given in (1), it easily follows

This completes the proof that the region D, as dened in (6), is positively invariant under the ow induced by the system (3).

Thus, it is not restrictive to limit our analyses to the region D.

The model given by equations (3) has a unique diseasefree equilibrium (DFE), obtained by setting the r.h.s. of equations (3) to zero, given by

To establish the local and global stability of the DFE, suitable threshold quantities are computed: the basic and control reproduction numbers. The basic reproduction number, R 0 , is a frequently used indicator for measuring the potential spread of an infectious disease in a community. It is dened as the average number of secondary cases produced by one primary infection over the course of the infectious period in 8 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 25, 2021. ; a fully susceptible population. If the system incorporates vaccination strategies, then the corresponding quantity is named the control reproduction number and is usually denoted by R V . The reproduction number can be calculated as the spectral radius of the next generation matrix FV −1 , where F and V are dened as Jacobian matrices of the new infection appearance and the other rates of transfer, respectively, calculated for infected compartments at the diseasefree equilibrium [70] . In this specic case, if ϕ 0 + ϕ 1 (M ) = 0 in (3), namely when a vaccination program is not in place, we obtain the expression of R 0 ; otherwise, the corresponding R V can be computed. Theorem 2. The basic reproduction number of model (3) is given by (12) and the control reproduction number is given by

Proof. Following the procedure and the notations adopted by Diekmann et al. [21] and Van den Driessche & Watmough [70] , we derive the control reproduction number, R V . Let us consider the r.h.s. of equations (3b)(3c)(3d) (the balance equations for the infected compartments), and distinguish the new infections appearance from the other rates of transfer, by dening the vectors

The Jacobian matrices of F and V evaluated at model DFE (11) read, respectively,

and

As proved in [21, 70] , the control reproduction number is given by the spectral radius of the next generation matrix FV −1 . It is easy to check that FV −1 has positive elements on the rst row, being the other ones null. Thus,

Similarly one can prove that the basic reproduction number is given by (12) .

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From Theorem 2, it follows that [70] :

As far as the global stability of the DFE, we prove the following theorem Theorem 3. The DFE is globally asymptotically stable (GAS) if R V < 1.

Proof. To prove the global stability of the DFE, we adopt the approach developed by CastilloChavez et al. in [12] . We rewrite system (3) in the forṁ y =h(y, z) z =l(y, z), l(y, 0) = 0 where y = (S, V, M ) denotes the vector of uninfected compartments and z = (E, I a , I s ) that of infected compartments. The diseasefree equilibrium (11) is also rewritten as (ȳ, 0), withȳ = (S,V , 0) and 0 ∈ R 3 . Then, the DFE is globally asymptotically stable if R V < 1, provided that the two following conditions are satised [12] :

The matrixJ is given byJ =F-V, with F and V as computed in the proof of Theorem 2 and given in (14) and (15), respectively. It is easily follows thatJ is an Mmatrix. Further, in view of (6),

Hence, also condition C.2 is satised and the proof is completed.

For an alternative proof see Appendix A.

We remark that by introducing p =V S +V = ϕ 0 µ + ϕ 0 as the fraction of the population vaccinated at the diseasefree equilibrium (11) we can express

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The copyright holder for this preprint this version posted June 25, 2021. Note that R V ≤ R 0 with equality only if ϕ 0 = 0 (i.e., p = 0) or σ = 1. That is, despite being imperfect, the vaccine (characterized by ϕ 0 > 0 and 0 ≤ σ < 1) will always reduce the reproduction number of the disease. The expression (16) is the same as obtained by Gumel et al. [36] for the SARS epidemic control. In [36] , a detailed analysis is given, leading to the following main results: Proposition 2. The disease will be eliminated from the community if p ≥ p c , with p c given by

Proposition 3. Let us consider the following quantity:

We have that: if R 0 < 1/σ and ϕ 0 > ϕ 0c , then the disease will eliminate from the community. If R 0 ≥ 1/σ, then no amount of vaccination will prevent a disease outbreak in the community.

See also Fig. 5 in [36] , where the critical value, p c , is plotted as a function of 1 − σ for several values of R 0 .

Let us denote the generic endemic equilibrium (EE) of model (3) with

By setting the r.h.s. of equations (3b)(3c)(3d)(3e)(3f) to zero, one can derive the relationships

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and

In view of (6), we can limit ourselves to seek I e a in the interval (0, Λ/µ). Firstly, let us list some proprieties of the functions ψ(I a ) and χ(I a ), that can be easily veried:

is a concave quadratic function;

(ii) χ(I a ) is the product of a linearane increasing function and a positive increasing function (ϕ e (·));

Then, we distinguish three cases:

implying that a 0 = ψ(0) ≤ 0, a 1 = ψ (0) < 0 and χ(I a ) is increasing and positive ∀I a > 0. From (i)(iii) it follows that ψ(I a ) and χ(I a ) cannot intersect for I a > 0, namely no endemic equilibria exist.

is a positive increasing function ∀I a > 0. From (i)(iii)(iv) it follows that ψ(I a ) and χ(I a ) have one positive intersection point and it is in (0, Λ/µ), namely an unique endemic equilibrium exists.

Then, a 0 = ψ(0) > 0 and χ(I a ) is negative for 0 < I a < I * a and it is positive and increasing for I a > I * a . Further,

From (i)(iii)(iv)(vi) it follows that ψ(I a ) and χ(I a ) have one positive intersection point and it is in (I * a , Λ/µ), namely an unique endemic equilibrium exists.

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The copyright holder for this preprint this version posted June 25, 2021. ;  Hence, EE exists if and only if R V > 1 and the endemic number of asymptomatic individuals I e a is characterized by ψ(I e a ) = χ(I e a ) > 0, ψ (I e a ) < 0 < χ (I e a ) and

where the last term in (20) is the (unique) positive root of ψ(I a ).

The results are summarized in the following theorem. (3) admits an unique endemic equilibrium, dened in (17)(18), with I e a such that

and a i , i = 0, . . . , 2, given in (19).

To derive a sucient condition for the occurrence of a transcritical bifurcation at R V = 1, we can use a bifurcation theory approach. We adopt the approach developed in [27, 70] , which is based on the general center manifold theory [35] . In short, it establishes that the normal form representing the dynamics of the system on the central manifold is given by:

and

Note that in (21) and (22) β has been chosen as bifurcation parameter, β c is the critical value of β, x = (S, E, I a , I s , V, M ) is the state variables vector, f is the righthand side of system (3), and v and w denote, respectively, the left and right eigenvectors corresponding to the null eigenvalue of the Jacobian matrix evaluated at criticality (i.e. at DFE and β = β c ).

Observe that R V = 1 is equivalent to:

so that the diseasefree equilibrium is stable if β < β c , and it is unstable when β > β c . The direction of the bifurcation occurring at β = β c can be derived from the sign of coecients (21) and (22) . More precisely, if A > 0 [resp. A < 0] and B > 0, then at β = β c there is a backward [resp. forward] bifurcation. For our model, we have the following:

13 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Proof. The Jacobian of system (3) is

. J evaluated at DFE (11) for β = β c becomes:

Its spectrum is:

where λ ± are given by

As expected, it admits a simple zero eigenvalue and the other eigenvalues have negative real part. Hence, when β = β c (or, equivalently, when R V = 1), the DFE is a nonhyperbolic equilibrium. It can be easily checked that a left and a right eigenvector associated with the zero eigenvalue so that v·w = 1 are:

The coecients A and B may be now explicitly computed. Considering only the nonzero components of the eigenvectors and computing the corresponding second derivative of f , it follows that:

where v 2 > 0. Then, A < 0 < B. Namely, when β − β c changes from negative to positive, DFE changes its stability from stable to unstable; correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable. This completes the proof.

Demographic and epidemiological parameter values are based on the COVID19 epidemic in Italy reported since the end of February 2020 [41] . Vaccinerelated parameter values are mainly inferred by preliminary reports about anti COVID19 vaccines and by the initial trend of the Italian immunization campaign. A detailed derivation of such quantities is reported in the following.

In order to provide appropriate initial conditions that mark the beginning of an epidemic wave, we make the following considerations. After the rst dramatic epidemic wave (FebruaryMay 2020) Italy experiences the socalled`living with the virus' period, characterized by a relatively low level of prevalence and loosening of restrictions. But this breathing space ends towards the second half of August 2020, when the virus regained strength and progressively grew its prevalence, marking the arrival of the second wave.

Since data available at the beginning of the second wave are reasonably more accurate than those at the epidemic starting time, we consider them as initial data. More specically, we take the ocial national data for infectious (I a + I s ) and recovered (R) people at 16 August 2020, that is estimated as the rst time after the end of the rst wave that the eective reproduction number exceeds the threshold 1 [42] . For that period, the Italian National Institute of Health estimates the fraction of asymptomatic individuals w.r.t. the total case as 49.25% about, namely I a (0) = 0.4925(I a (0) + I s (0)) [39] . As far as the initial values of exposed individuals E and the information index M are concerned, in the absence of exact data, we infer them by the corresponding expressions at the endemic state, as given in (18) . Hence, one yields E(0) = I a (0)(η + ν a + µ)/ρ and M (0) = kI s (0). Finally, the initial value of susceptible individuals S is obtained by subtracting from the total initial population (say, N 0 ), as given in [8] , namely

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The copyright holder for this preprint this version posted June 25, 2021. 

In the absence of empirical data about vaccinating attitudes, we follow the approach of [7, 8, 26] and assume that ϕ 1 (M ) is a MichaelisMenten function [61] ϕ 1 (M ) = CM 1 + DM ,

Similarly to what done in [7, 8, 26] , we set C = D (ϕ max − ϕ 0 ), where ϕ max > ϕ 0 . This reparametrisation means an asymptotic overall rate of ϕ max days −1 . The ensuing vaccination function is:

As of April 2021, the rate of anti COVID19 vaccination in Italy was less than 400,000 administrations per day in a population of N 0 ≈ 60 millions of inhabitants [39] , but acceleration plans have been laid out.

Here, we take ϕ max = 0.02 days −1 potentially implying a ceiling of 0.02 days −1 in vaccination rate under 16 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 25, 2021. ; Table 1 .

circumstances of high perceived risk. This value is in line with data concerning the 2009 H1N1 pandemic inuenza, whose daily rate of vaccine administration has been largely investigated and it was below 2% of the total population (see [49] and references therein). Furthermore, threshold values of 12% per day were also considered in epidemic models of dengue [67] and cholera diseases [30] .

In order to obtain a baseline value for D, we observe that in [6, 26] it was set D = 500, where M varied in [0, k]. Here M varies in [0, kΛ/µ] (see (6)), hence we expect that D = 500µ/Λ could be a good starting point.

As far as the factor of vaccine ineectiveness, σ, and the informationindependent constant vaccination rate, ϕ 0 , are concerned, in Section 5 numerical solutions by varying both σ ∈ [0, 1) and ϕ 0 ∈ [0, ϕ max ] are given. Anyway, for illustrative purposes, a corresponding baseline value is selected: σ = 0.2, meaning that the vaccine oers 80% protection against infection, and ϕ 0 = 0.002 days −1 , that is the 10% of the ceiling vaccination rate ϕ max (ϕ 0 = 0.1ϕ max ). Specically, 80% is the estimated eectiveness of partial immunization (14 days after rst dose but before second dose) of some authorized mRNA COVID19 vaccines [13] . We estimate the rate at which symptoms onset as η = qγ, where q = 0.15 represents the fraction of infected people that develops symptoms after the incubation period and γ = 1/1.25 days −1 is the post latency rate, as given in [8] . The fraction q is also used to infer ε a , the modication factor concerning transmission from I a , namely we set ε a = q + (1 − q)0.033, where 1 [resp. 0.033] is the modication factor concerning transmission from postlatent [resp. truly asymptomatic] individuals, as considered in the models [8, 33] . Following the approach adopted by Gumel et al. [37] , based on the formula given by Day [17] , we estimate the diseaseinduced death rate as

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The copyright holder for this preprint this version posted June 25, 2021. ;  where C F is the fatality rate and Θ is the expected time from the onset of symptoms until death. We compute C F by the ocial national data from 16 August to 13 October 2020 [41] (the same period considered for the estimation of the transmission rate β, as explained below), yielding C F = 0.75%. As far as Θ is concerned, from [39] we get Θ = 12 days, providing δ ≈ 6.248 ·10 −4 days −1 . Similarly, the recovery rates ν j with j ∈ {a, s} are estimated as

where Θ a [resp. Θ s ] is the expected time until recovery for asymptomatic [resp. symptomatic] individuals. We assume Θ a = 6, Θ s = 18 days on the basis of the considerations made in [8] .

Values for Λ, µ, ε s , ρ, a and k are based on the estimates given in [8] . Like as for σ and ϕ 0 , numerical solutions by varying both k ∈ [0.2, 1] and a ∈ [1/60, 1] days −1 are given in Section 5 (for a detailed motivation about the ranges of values of the information parameters see [8] ). Finally, in order to obtain an appropriate value for the baseline transmission rate β, we consider model (3)(23) in absence of vaccination strategies (ϕ 0 = 0 days −1 , D = 0) and search for the value that best ts with the initial`uncontrolled' phase of the second Italian epidemic wave. More precisely, we consider the number of COVID19induced deaths in Italy from 16 August, assumed as the starting date of the second wave (see Section 4.1), and 13 October 2020, the last day of loose restrictions. Indeed, on 13 October the Council of Ministers approved a decree to reintroduce stricter rules to limit the spread of the disease [40] . The choice of the curve to t is motivated by the fact that data about deaths seem to be more accurate with respect to other ones, e.g. the number of infected people, who are not always identied, especially if asymptomatic or with very mild symptoms. Anyway, by setting β = 2.699 · 10 −8 days −1 , we obtain a good t not only with the cumulative deaths (see Fig. 2B ) but also with the total infectious cases, I a + I s (see Fig. 2A ).

All the parameters of the model as well as their baseline values are reported in Table 1 .

Numerical simulations are performed in MATLAB [57] . We use the 4th order RungeKutta method with constant step size for integrating the system and the platformintegrated functions for getting the plots. First, we numerically investigate the impact of two vaccinerelated parameters, namely the information independent constant vaccination rate, ϕ 0 , and the factor of vaccine ineectiveness, σ, on the control reproduction number R V of formula (13) . The corresponding contour plot of R V (ϕ 0 , σ) is shown in Fig.  3A . This gure shows that: i) for very small values of ϕ 0 this parameter impacts on R V but ϕ 0 > 0.002 days −1 about yields that R V depends practically only on σ in a linearane manner as shown in Fig.  3B ; ii) for small values of σ (as those declared for some of the vaccines) the R V is small, for example for σ = 0.05 it is R V < 0.1; iii) for values of σ ≈ 1/3, comparable with those observed often for vaccine against the seasonal u, it is R V ≈ 0.5; iv) if we dene as threshold of noneectiveness the curve R V = 1 we observe that for ϕ 0 > 0.002 days −1 this threshold is reached for values of σ between around 0.6 and 0.7.

Let us consider the time frame [0, t], where 0 ≤ t ≤ t f . We introduce four relevant cumulative quantities that will be used in the following: the cumulative vaccinated individuals CV(t), i.e. the total number 18 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 

β(S(τ ) + σV (τ )) (ε a I a (τ ) + ε s I s (τ )) dτ,

We also consider two possibilities for the time at which vaccines administration starts, namely VAX-0, that is the baseline case that the vaccination campaign starts at day t = 0;

VAX-30, that is the case that the vaccination campaign starts at day t = 30.

We assume that in both cases the vaccination campaign lasts 1 year, namely t f = 365 [resp. t f = 395] days in the case VAX-0 [resp. VAX-30].

Numerical simulations for the case VAX-0 are displayed in Fig. 4 . Namely, we report the temporal dynamics of three relevant state variables: susceptible individuals S (Fig. 4A) , vaccinated individuals 19 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 25, 2021. ; https://doi.org/10.1101/2021.06.22.21259324 doi: medRxiv preprint V (Fig. 4B) and symptomatic infectious individuals I s (Fig. 4C) , as well as the cumulative number of deaths CD (Fig. 4D) . We consider the following four signicant scenarios (for each of them we also report the observed results):

Constant vaccination (D = 0), with baseline rate ϕ 0 = 0.002 days −1 (blue lines). We observe at t = 202 days the occurrence of a large peak of symptomatic cases I s (225,025) and at the end of simulation a large cumulative number of deaths (28,343); Informationdependent vaccination: ϕ 0 = 0.002 days −1 , D = 500µ/Λ (black lines). This case is characterized by a time of I s peak that is halved w.r.t. the constant baseline case, namely at t = 105 days about, and a much lower prevalence: 57, 588, i.e. one quarter about w.r.t. the constant baseline case. This could be an excellent performance, but it is not the case since better performance could have been reached appropriately higher vaccination rate levels;

Constant vaccination (D = 0), with rate ϕ 0 = ϕ p1 0 = 4.25 · 10 −3 days −1 (red lines), which is such that the peak value of I s is equal to the peak value observed in the case of informationdependent vaccination. One can observe that in this case the epidemic peak occurs earlier, at t = 119 days, and the nal cumulative number of death is smaller: CD(t f ) = 5, 948;

Constant vaccination (D = 0), with rate ϕ 0 = ϕ p2 0 = 7.87 · 10 −3 days −1 (green lines), where the peak of I s is halved w.r.t. the case of informationdependent vaccination. The epidemic peak occurs very early, at t = 72 days, and the nal cumulative number of death is relatively modest: CD(t f ) = 2, 203.

Simulations for the case VAX-30 are, of course, graphically similar to those in Fig. 4 , hence corresponding plots are here omitted. From a quantitative point of view, in order to compare the results in the case VAX-30 w.r.t. the case VAX-0, we focus on the scenario of informationdependent vaccination and report in Table 2 the value of the following epidemiological indicators (not necessarily in this order): the number of susceptible and vaccinated individuals, and the cumulative quantities (24) at the end of the time horizon t f , the peak of symptomatic cases and its occurrence time. Comparison between the cases VAX-0 and VAX-30 is given though the dierence operator: Table 2 ). Observe that, in both VAX-0 and VAX-30 case, cumulative asymptomatic people at the nal time t f (that is the dierence CI(t f )−CY(t f )) account for approximately 57% of cumulative SARSCoV2 infections. This result is in line with the current estimates (as of April 2021) reported by the Italian National Institute of Health [39] . We also investigate the temporal dynamics of the ratio ϕ 1 (M )/ϕ 0 in the case of informationdependent vaccination. Numerical solutions are displayed in Fig. 5 for both the case VAX-0 (black line) and the case VAX-30 (blue line). We note that in the case VAX-30 the ratio is larger than in the case VAX-0 since the delay in the start of the vaccination campaign induces a larger epidemic peak. Namely, in the case VAX-0, the maximum value reached by ϕ 1 (M )/ϕ 0 is 2.49 and the time at it is reached is approximately t = 108 days. In the case of VAX-30 this peak is reached at t = 114, i.e. 84 days after the start of VAX-30, but the peak value is much larger: it is 3.4.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Table 1 and in Section 5.1.

1.33 · 10 7 1.09 · 10 7 −2.39 · 10 6 V (t f ) 4.58 · 10 7 4.77 · 10 7 1.92 · 10 6 CV(t f ) 4.62 · 10 7 4.82 · 10 7 2.00 · 10 6 max(I s ) 5.76 · 10 4 9.14 · 10 4 3.38 · 10 4 arg max(I s ) 105.14 110.61 5.47 CY(t f ) 4.42 · 10 5 6.44 · 10 5 2.02 · 10 5 CI(t f ) 1.03 · 10 6 1.51 · 10 6 4.80 · 10 5 CD(t f ) 5.04 · 10 3 7.30 · 10 3 2.26 · 10 3 

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Table 1 .

Here, we focus on the VAX-0 case and evaluate the sensitivity of some relevant epidemiological indicators to variations of critical parameter values. Note that for the case VAX-30 we obtain similar results, which we omit. Specically, we assess how changing suitable information and vaccinerelated parameters aects the cumulative quantities (24) evaluated at the nal time t f , the peak of symptomatic cases and its occurrence time. We anticipate here that the nal cumulative incidence, CI(t f ), the nal cumulative symptomatic cases, CY(t f ) and the peak of symptomatic cases, max(I s ), have in all cases contour plots qualitatively similar to the nal cumulative deaths CD(t f ), thus we do not plot them. Hence, the following gures display the counter plots of just three quantities: the cumulative vaccinated individuals at t f = 365 days, CV(t f ); the occurrence time of the symptomatic prevalence peak, argmax(I s ); the cumulative diseaseinduced deaths at t f = 365 days, CD(t f ).

We start by investigating how the information parameters, namely the information coverage, k, and the information delay, T a = a −1 , may aect the epidemic course, see Fig. 6 . We observe that for argmax(I s ) and CD(t f ) (as well as max(I s ), CI(t f ) and CY(t f )) the patterns of the contour plots are similar, and in particular: for small k = 0.2 the range of the simulated variable when T a increases is large, whereas for k = 1 the range is restricted and low. The inverse phenomenon is observed for CV(t f ): the range is restricted and small for low k = 0.2 whereas it is larger for k = 1. Then, we investigate how the factor of vaccine ineectiveness, σ, and the informationindependent constant vaccination rate, ϕ 0 , aect the same quantities considered above. The results are shown in the 22 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Table 1 .

contour plots in Fig. 7 for the case of constant baseline vaccination (ϕ 0 = 0.002 days −1 , D = 0) and in Fig. 8 for the case of informationdependent vaccination (ϕ 0 = 0.002 days −1 , D = 500µ/Λ). We may observe that the quantitative impact of the informationdependent vaccination is remarkable (but this was expected). As far as the shapes of the plots, we note that the plots for CV(t f ) (panels A) and for the time at symptomatic prevalence peaks (panels B) are remarkably dierent from the other plots. Moreover the plot for CV(t f ) is qualitatively dierent in the informationdependent vaccination case w.r.t. the case of constant vaccination.

There is an ongoing debate on possible seasonality eects on the transmission and global burden of COVID19 [1, 50, 59, 65] . Thus, for the sake of the completeness, we consider here the case of information dependent vaccination and simulate the presence of seasonality on three key parameters: not only the transmission rate, β, but also the rate of symptoms onset, η, and the total rate of vaccination, ϕ(M ) = ϕ 0 + ϕ 1 (M ), with ϕ 1 (M ) given in (23) . For the latter, the seasonality could be determined by a lower vaccination rate due to the summer vacations.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Table 1 .

Namely, we use in our simulations par(t) = par b χ(t), par = β, η, ϕ 0 , ϕ max where: par b are the baseline values and χ(t) is simply two states switch, i.e. similar to the one proposed in [28] for the transmission rate:

χ(t) = 0.75, t ∈ (July and August) 1, t ∈ (September to June)

Since we used initial conditions corresponding to COVID19 data at 16 August 2020, as ocially communicated by Italian health authorities (see Section 4.1), we consider:

We will denote this simulation scenario with VAX-0S.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Table 1 .

Numerical simulations are displayed in Fig. 9 and compared with the baseline scenario, VAX-0. Corresponding relevant quantities are reported in Table 3 . Our simulation suggests that: i) the impact of the summer vacation on the vaccine delivery and on S(t) is minimal (and they are omitted from Fig. 9) ; ii) the peak of symptomatic cases decreases many months after the summer decrease of the transmission and symptoms onset w.r.t. the no seasonality scenario, and it is delayed (Fig. 9A) ; iii) the cumulative number of deaths decreases a little bit (Fig. 9B ).

In this paper we introduced a mathematical model describing the transmission of the COVID19 disease in presence of non mandatory vaccination. The main novelty is that the hesitancy and refusal of vaccination is taken into account. To this aim, we used the information index, which mimics the idea that individuals take their decision on vaccination based not only on the present but also on the past information they have on the spread of the disease. Theoretical analysis and simulations show clearly as a voluntary vaccination can of course reduce the impact of the disease but it is unable to eliminate it. The qualitative path of the disease remains the same 25 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Table 1 and in Section 6.

4.47 · 10 7 −1.09 · 10 6 CV(t f ) 4.51 · 10 7 −1.12 · 10 6 max(I s ) 5.03 · 10 4 −7.34 · 10 3 arg max(I s ) 115. 66 10.52 CY(t f ) 4.02 · 10 5 −3.97 · 10 4 CI(t f ) 9.44 · 10 5 −8.91 · 10 4 CD(t f ) 4.59 · 10 3 −444.88 Table 3 : Informationdependent vaccination case (ϕ 0 = 0.002 days −1 , D = 500µ/Λ). Relevant quantities as predicted by model (3)(23) in the scenario including seasonality VAX-0S (rst column). The second column reports the dierences between the values corresponding to the VAX-0S case w.r.t. the case VAX-0 (see also Table 2 ). Initial conditions and other parameter values are given in Table 1 and in Section 6.

but the quantitative results are strongly dierent: an epidemic outbreak (a new epidemic wave) occurs, even if (as we observed in our simulations) the informationdependent vaccination rate is, at its peak, more than three times larger than the constant baseline vaccination rate.

A key result is in particular the fact that the informationrelated parameters deeply aect the dynamics of the disease: large information coverage and small memory characteristic time are needed to have the best results. The dierent impact of behaviour and information with respect to the scenario of mandatory constant vaccination can be further appreciated by examining the contour plots in Figs. 68.

As it is reasonable, the parameter σ, i.e. the risk of infection for vaccinated people, has a major impact.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted June 25, 2021. ; https://doi.org/10.1101/2021.06.22.21259324 doi: medRxiv preprint Namely, the control reproduction number R V (σ, ϕ 0 ) essentially depends on σ in a linearane manner. This suggest to stick to vaccines that have very low σ, where R V (σ, ϕ 0 ) is tiny. A very positive result is that the threshold of nonecacy of the vaccine, which can roughly be delineated as the curve (σ, ϕ 0 ) where R V (σ, ϕ 0 ) = 1 is located for values σ ∈ (0.6, 0.7), i.e. for very large values of σ ( Figure 3A ). As far as the impact of human behaviour w.r.t. scenarios with constant vaccination rates is concerned, we obtained that that the performances were better only w.r.t. a constant vaccination rate as low as ϕ 0 , whereas the scenario where ϕ 0 = ϕ p2 0 (see Section 5.1) would lead to excellent result and a substantially smaller number of deaths. As far as the comparison of the VAX-0 vs VAX-30 scenarios is concerned, we also measured its impact on the ratio between the informationdependent and the constant components of the vaccination rate, namely ϕ 1 (M )/ϕ 0 . As expected, the peak was considerably larger in the scenario VAX-30. The peaks occur in the same week if measured in the absolute time, i.e. the peak for VAX-30 occurs one month before the peaks of VAX-0 if measured in time since the start of the vaccination (see Figure 5 ). Finally, seasonality has a relative but non neglectable relevance. For example, although the decrease of the transmission rate and of the onset of symptoms occur in the summer, the predicted winter epidemic peak of symptomatic cases is decreased and delayed w.r.t. the one in the noseasonality scenario. A small but not neglectable decrease and delay of the cumulative deaths is also observed. This overall suggests that a decrease of the transmission and of the onset of symptoms has positive impact even many months after their end (see Figure 9 ). An apparent limitation of this study is the absence of modelling for the dynamics of the transmission rate. In other words, neither spontaneous changes of the parameter β and imposed changes due to social distancing laws and partial/full lockdowns are taken into the account. However, these aspects are intentionally neglected here since our goal is to assess the impact of a possible voluntary vaccination campaign. As far as future research is concerned, we plan: i) to explore (mainly numerically) a realistic model of the COVID19 spread that includes the timechanges of the transmission rate; ii) to explore the possibility that eradication of the COVID19 is not reached and the disease stays endemic.

A Alternative proof of Theorem 3 Consider the following function L = E + (ρ + µ) [(ε a (ν s + δ + µ) + ε s η) I a + ε s (η + ν a + µ) I s ] ρ (ε a (ν s + δ + µ) + ε s η) .

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted June 25, 2021. ; https://doi.org/10.1101/2021.06. 22.21259324 doi: medRxiv preprint It is easily seen that the L is nonnegative in D (see (6) ) and also L = 0 if and only if E = I a = I s = 0. The time derivative of L along the solutions of system (3) in D readṡ L =Ė + (ρ + µ) (ε a (ν s + δ + µ) + ε s η)İ a + ε s (η + ν a + µ)İ s ρ (ε a (ν s + δ + µ) + ε s η) =β(S + σV )(ε a I a + ε s I s ) − (ρ + µ)E+ + (ρ + µ) [(ε a (ν s + δ + µ) + ε s η) (ρE − (η + ν a + µ)I a ) + ε s (η + ν a + µ) (ηI a − (ν s + δ + µ)I s )] ρ (ε a (ν s + δ + µ) + ε s η) =(ε a I a + ε s I s ) β(S + σV ) − (ρ + µ)(η + ν a + µ)(ν s + δ + µ) ρ (ε a (ν s + δ + µ) + ε s η) ≤(ε a I a + ε s I s ) β Λ(µ + σϕ 0 ) µ(µ + ϕ 0 ) − (ρ + µ)(η + ν a + µ)(ν s + δ + µ) ρ (ε a (ν s + δ + µ) + ε s η)

= − (ε a I a + ε s I s ) (ρ + µ)(η + ν a + µ)(ν s + δ + µ) ρ (ε a (ν s + δ + µ) + ε s η) (1 − R V ).

It follows thatL ≤ 0 for R V < 1 withL = 0 only if I a = I s = 0. Hence, L is a Lyapunov function on D and the largest compact invariant set in {(S, E, I a , I s , V, M ) ∈ D :L = 0} is the singleton {DFE}. Therefore, from the La Salle's invariance principle [47] , every solution to system (3) with initial conditions (4) approaches the DFE, as t → +∞.

Seasonality of respiratory viral infections: Will COVID19 follow suit? Frontiers in Public Health

Ecacy and safety of the mRNA1273 SARSCoV2 vaccine

Imitation dynamics predict vaccinating behaviour

Analysis of asymptomatic and presymptomatic transmission in SARSCoV2 outbreak

Optimal dynamic prioritization of scarce COVID 19 vaccines. medRxiv

Eects of informationdependent vaccination behavior on coronavirus outbreak: insights from a SIRI model

Oscillations and hysteresis in an epidemic model with information dependent imperfect vaccination

Eects of informationinduced behavioural changes during the COVID19 lockdowns: the case of Italy

Global stability of an SIR epidemic model with information dependent vaccination

Modeling of pseudorational exemption to vaccination for SEIR diseases

A generalization of the KermackMcKendrick deterministic epidemic model

On the computation of R 0 and its role on global stability

Interim estimates of vaccine eectiveness of BNT162b2 and mRNA1273 COVID19 vaccines in preventing SARSCoV2 infection among health care personnel, rst responders, and other essential and frontline workers Eight U.S. locations

COVID19 Global Map

Optimal strategies for vaccination and social distancing in a gametheoretic epidemiologic model

and on behalf of the Centre for the Mathematical Modelling of Infectious Diseases COVID19 working group. Eects of non pharmaceutical interventions on COVID-19 cases, deaths, and demand for hospital services in the UK: a modelling study

On the evolution of virulence and the relationship between various measures of mortality

Volatile opinions and optimal control of vaccine awareness campaigns: chaotic behaviour of the forwardbackward Sweep algorithm vs. heuristic direct optimization

A network model of Italy shows that intermittent regional strategies can alleviate the COVID19 epidemic

Joint impacts of media, vaccination and treatment on an epidemic lippov model with application to COVID19

On the denition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations

Heterogeneous social interactions and the COVID19 lockdown outcome in a multigroup SEIR model

Informationrelated changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases

The impact of vaccine side eects on the natural history of immunization programmes: an imitationgame approach

The interplay of public intervention and private choices in determining the outcome of vaccination programmes

Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases

Backwards bifurcations and catastrophe in simple models of fatal diseases

A simple model for complex dynamical transitions in epidemics

Contact rate epidemic control of COVID19: an equilibrium view

Optimal control of vaccination in an agestructured cholera model

Estimating the eects of nonpharmaceutical interventions on COVID19 in Europe

Données hospitalières relatives á l'èpidèmie de COVID19

Spread and dynamics of the COVID19 epidemic in Italy: Eects of emergency containment measures

Modelling the COVID19 epidemic and implementation of populationwide interventions in Italy

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

An SVEIR model for assessing potential impact of an imperfect antiSARS vaccine

Modelling strategies for controlling SARS outbreaks

IPSOS. Global attitudes on a COVID-19 vaccineIpsos survey for The World Economic Forum

Italian Ministry of Health. Covid19, rmato il nuovo Dpcm

Monitoraggio settimanale Covid19, report 31 agosto 6 settembre

Persistence and decay of human antibody responses to the receptor binding domain of

The known unknowns of T cell immunity to COVID 19

OxfordAstraZeneca COVID19 vaccine ecacy

and on behalf of the Centre for the Mathematical Modelling of Infectious Diseases COVID19 working group. Early dynamics of transmission and control of COVID19: a mathematical modelling study. The Lancet Infectious Diseases

Stability by Liapunov's Direct Method with Applications

Il vaccino contro il Covid sarà obbligatorio solo in casi estremi

Modeling optimal agespecic vaccination strategies against pandemic inuenza

The role of seasonality in the spread of COVID19 pandemic

Risk Management in PostTrust Societies

A. Gintsburg, and Gam-COVID-Vac Vaccine Trial Group. Safety and ecacy of an rAd26 and rAd5 vectorbased heterologous prime boost COVID19 vaccine: an interim analysis of a randomised

Biological Delay Systems: Linear Stability Theory

Adresse aux francais

Deteriorated Covid19 control due to delayed lockdown resulting from strategic interactions between Governments and oppositions. medRxiv

Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases

MATLAB. Matlab release 2020a. The MathWorks

Seasonality and uncertainty in global COVID19 growth rates. Proceedings of the National Academy of

Quantifying early COVID19 outbreak transmission in South Africa and exploring vaccine ecacy scenarios

Mathematical Biology

Once we have it, will we use it? A European survey on willingness to be vaccinated against COVID19

Mathematical assessment of the impact of nonpharmaceutical interventions on curtailing the 2019 novel coronavirus

Safety and ecacy of the BNT162b2 mRNA Covid19 vaccine

Temperature, humidity, and latitude analysis to estimate potential spread and seasonality of coronavirus disease 2019 (COVID19)

Analysis of a drinking epidemic model

Optimal dengue vaccination strategies of seropositive individuals

World governments should protect their population from COVID19 pandemic using Italy and Lombardy as precursor. medRxiv

Covid19 vaccine: Boris Johnson says jab`will not be compulsory' but he rejects wrong' antivaxxers

Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission

Robust neutralizing antibodies to SARSCoV2 infection persist for months

Statistical physics of vaccination

Situation Report1

Coronavirus disease (COVID19): How is it transmit

Reported cases and deaths by country, territory, or conveyance