key: cord-0904008-j6rwurtf
authors: Aguilar-Canto, Fernando Javier; de León, Ugo Avila-Ponce; Avila-Vales, Eric
title: Sensitivity theorems of a model of multiple imperfect vaccines for COVID-19.
date: 2022-01-31
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2022.111844
sha: 9383ef35d6492127b00a0c2a223fc8340d2d2797
doc_id: 904008
cord_uid: j6rwurtf

In response to the ongoing pandemic of COVID-19, several companies across the world have proposed a wide variety of vaccines of different mechanisms of action. As a consequence, a new scenario of multiple imperfect vaccines against the SARS-CoV-2 arose. Mathematical modeling needs to consider this complex situation with different vaccines, some of them with two required doses. Using compartmental models we can simplify, simulate and most importantly, answer questions related to the development of the outbreak and the vaccination campaign. We present a model that addresses the current situation of COVID-19 and vaccination. Two important questions were considered in this paper: are more vaccines useful to reduce the spread of the coronavirus? How can we know if the vaccination campaign is sufficient? Two sensitivity criteria are helpful to answer these questions. The first criterion is the Multiple Vaccination Theorem, which indicates whether a vaccine is giving a positive or negative impact on the reproduction number. The second result (Insufficiency Theorem) provides a condition to answer the second question. Finally, we fitted the parameters with data and discussed the empirical results of six countries: Israel, Germany, the Czech Republic, Portugal, Italy, and Lithuania.

The introduction of vaccines in the current pandemic of COVID-19 has been highlighted by many scholars as to the best opportunity to curtail the spread of the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) [1, 2] . In response to the outbreak, more than 50 organizations have started the Many compartmental models have been proposed since the early development of the pandemic [4, 5, 6, 7, 8, 9, 10, 11, 12] . Recent epidemiological models has added a compartment related to the vaccinated population [13, 14, 1, 15] . 10 Several papers in this topic are mainly focused on finding the optimal allocation of the vaccines in different partitions of the population, varying from age [16, 2, 3] , work [17, 18] , the presence of comorbidities [19] and the social contact network [20] .

The presence of multiple types of vaccines arises the necessity of considering 15 multiple compartments in the modeling. This problem is addressed in [21] , which presented the θ-ij-SVEIHQRD model including multiple vaccines and new variants. In addition, some articles have included specific compartments for one and two doses [3, 22, 23] , which is the case of some types of vaccines.

SEIARD model (which stands for Susceptible-Exposed-Infected-Asymptomatic-20 Recovered-Death) has been proposed in the context of the ongoing pandemic of COVID-19, due to the particular transmission dynamics of this disease [24, 25, 26] . It is worth mentioning that a similar model (SEIADR) was studied in papers such as [27] , which includes vaccination and antiviral treatment, but the Death compartment is considered as infectious, an aspect that fits in diseases 25 such as Ebola but not in the case of COVID-19. In [22] , the SEIARD model was extended to incorporate one vaccine compartment.

In this article, we extended the multiple imperfect vaccines preliminary model of [28] to develop a SEIARD-based system (instead of the original SIRbased model), to particularize it to the specific case of COVID-19, as an attempt 30 to correct some of the previous limitations. Similarly, some questions around the availability of multiple vaccines are to be addressed in this paper. Rather than considering how to distribute the vaccines, we are more interested in the complete campaign itself, and the presence of a variety of types of vaccine.

Is the vaccination campaign enough to eliminate or mitigate the global pan-35 demic? This question is the title of the paper of [14] , which was one of the first researches in studying the effects of vaccination as a key measure to curtail the outbreak. In the case of the United States of America, the mentioned article reported that a vaccine with an efficacy of 0.8 needs 82 % of the population to be vaccinated to reach herd immunity. However, the situation is more 40 complex with different vaccines with distinct "efficacies", and current data is required to answer this question a posteriori. Moreover, analytical conditions would be useful to state if the vaccine campaign is enough or a combination of non-pharmaceutical interventions is also necessary.

Another big question around this phenomenon is the following: is the addi- 45 tion of new types of vaccines in the campaign always useful to reduce the effects of the pandemic? Although, intuition might drive us to answer this question with yes, in the preliminary preprint, the Multiple Vaccination Theorem states some conditions where this addition of vaccines can increase the reproduction number. Does that theorem hold for a SEIARD-based model? 50 To answer the main section using a more specific model for COVID-19, we developed the model directly from SEIARD but taking the same procedure of the SIR-based system, and including compartments from one and two doses and it is presented in section 2. In section 3, we include all mathematical results, which are the basic propositions about the model (computation of the reproduc- 55 tive number, stability of the Disease-Free Equilibrium), and in the remaining section, we provide the sensitivity analysis of the reproduction number. An important result of the sensitivity is the mentioned Multiple Vaccine Theorem, which answers the second main question, whereas the first question is discussed in the Insufficiency Theorem. 60 

Mathematical models with imperfect vaccines consider different ways of vaccine failure, which are, the primary vaccine failure (probability of not reaching protection just after vaccination), the "leakiness" (probability of infection after exposure with an infectious individual), and the waning rate (rate of immunity loss) [29, 30, 31, 32, 33] . Most of the cited articles present a SVIR model focused on childhood vaccination, and thus, this situation might be adapted to consider the case of COVID-19.

Our model adds one vaccination compartment V j with its respectively fullvaccination compartment F j for each j-th vaccine. Let N the total number 70 of types of vaccines. Thus, j = 1, . . . , N . For each type, the primary vaccine failure is ε j,A , the waning rate is α j , the leakiness for the uncompleted vaccinated individuals (for instance, with one dose) is ε j,L1 , and ε j,L2 is the leakiness for complete vaccination. Additionally, r j and λ j , the vaccination rate of the first and second dose respectively, complete the vaccination parameters. The set 75 {ε j,A , ε j,L1 , ε j,L2 , α j } are the intrinsic vaccination parameters and the rates are extrinsic vaccination parameters because the intrinsic parameters cannot be changed but the rates depend on the campaign. The SEIARD base model is taking from [25] , with two differences: all the compartments C are directly normalized such that C ∈ [0, 1]. Another difference 80 is the supression of two recovered population R I and R A . In this manner, R = R I +R A . Transmission of disease happens from V j , F j , and S (Susceptible) to E (Exposed) with a force of infection β 1 I + β 2 A, with β 1 and β 2 the transmission rates of symptomatic and asymptomatic individuals. The final model is given by the following equations:

dD dt = δI.

A full description of the parameters and the compartments are provided in tables 1 and 2. A flowdiagram of the model is given in Figure 1 .

V j Vaccinated individuals with the label j-vaccine with one dose and whose protection has not wanned.

F j Vaccinated individuals with the label j-vaccine with two doses and whose protection has not wanned. 

1.2 The Disease-Free Equilibrium X 0 is given by

1.3 The Basic Reproduction Number R c associated with the model with vacci-95 nation is given by

where R 0 = wβ1 γ+δ + (1−w)β2 γ , is the Basic Reproduction Number of a SEIARD model without vaccination.

Proof. The proof of 1.1 will be omitted, since it is standard. In the case of item [34] . Let

Therefore, if x 0 is the DFE and

Thus,

On the other hand,

The spectral radious of F V −1 is the reproduction number, which is

where

impact, represented with ϕ. In [33] , this number is defined as "the relative reduction within a population per unit of vaccination". We will address this definition by parts. Since,

Thus, the factor (1 − ε 1,L1 ) + (1 − ε 1,L2 ) λj αj V 0 j represents the reduction on the reproductive number. Dividing this quantity by the vaccination rate r j we

Proposition 2. If R c < 1, the Disease-Free Equilibrium is locally stable. On the contrary, if R c > 1, the DFE is unstable.

Proof. The linearization matrix of the system evaluated in the DFE is (without 

The characteristic polynomial of the previous matrix is

where A is the following matrix

Using elemental operations Those solutions are also zeros for p(z) = 0. Finally, the second factor of p(z)

can be expressed as

where

By Routh-Hurwitz criterion, all the solutions of (25) have a negative real part provided that a, c > 0 and ab − c > 0. Since σ(γ + δ)γ, γwσβ 1 Σ, and

and therefore c > 0. Note that if R c > 1, the eigenvalues are positive.

This procedure, yields the existence of solutions z 2N +2 , z 2N +3 , and z 2N +4 with negative real-part. Lastly, z 2N +5 = z 2N +6 = 0. This shows that the DFE is stable if R c < 1, and unstable if R c > 1. table 3 ). The incubation rate is set to σ = 1/5.2 following [37] , and w was set to 0.25.

In general, this procedure will be done by using gradient-based methods.

These algorithms are usually difficult to implement, since computing the exact 155 gradient requires the analytical solution of the model, which is remarkably difficult in SIR-based systems [38] . Some authors, such as [39] , fit the data with a polynomial and define a new loss with the derivatives of the polynomial and the model. In this case, we will use finite differences instead to approximate the gradient of the loss function Sum of Square Errors.

For some cases, we used a variation of Gradient Descend using finite differences and the backtracking algorithm. Instead of using a fixed learning η, we used a vector η = (η 1 , . . . , η k ), and hence, the Gradient Descend is given by

where represents the pointwise product of the two vectors. This change was introduced to consider the range of parameters: for instance: θ rj is usually 165 taken in a space [1, ∞) whereas other parameters are restricted in [0, 1]. Varying θ rj at the same rate as the rest of the parameters would only slow the process, and in practice, it could stack easily into a local minimum.

In the remaining cases, the optimization routine was done separately. For instance, r j (or its associate parameters, see the following subsection) can be 170 optimized by using the "reported compartment"Ṽ solely and the reported susceptible. However, at least the rates β 1 and β 2 need to be fitted with all the data.

Another problem related to this technique is that data is not fully provided in the sense of the model. A close look at the compartments shows that some 175 of them need to be redefined in order to make a proper estimation. This is the case of compartments V j , F j , and R, which will be redefined asṼ j ,F j ,R. We can redefine the equations to consider to implement the parameter estimation

The changes are motivated for various reasons. Reported vaccinated, for example, is an increasing function and it does not consider failure. Nevertheless, 180 we can define the dynamics of the reportedṼ j sharing parameters with the original model, which is the rate in this case. The situation ofF j is a bit complex:

(Ṽ j −F j ) represents the fraction of reported vaccinated individuals who has not been fully vaccinated. In this sense,Ṽ j are individuals that have been vaccinated at least one, and V j is the compartment of vaccinated individuals whose vaccine 185 has not failed. This establishes the difference between the two groups.

Finally,R are reported individuals that have been infected. This definition presents a controversy: reported recovered might have been asymptomatic individuals or not. We need to compare one compartment with the data, andR assumes that all the recovered that have been reported have been infected previ-190 ously, whereas the original compartment takes into account the asymptomatic.

As mentioned in [28] , one common problem for parameter optimization in the case of vaccination models was the fact that the vaccination rate is not the pandemic in the case of Italy and China (transmission, death, and recovery rates). Not always the addition of time-dependent parameters is useful to model 200 a particular situation (see [49] ). The simplest format is the constant function, but in some cases a change on the vaccination rate is induced. A simple model for this discrete change is given by the piecewise function:

Another possibility is the exponential growth of the vaccination, given by the expression:

A combination between the exponential and piecewise constant function is given by the piecewise exponential function:

These exponential function might not be the best options for inferences when t → ∞, and the vaccination rates might be, instead, logistic functions:

Transmission rates β 1 and β 2 were originally considered fixed but in some 210 cases was ideal to introduce some variations, following the structure of piecewise functions such as

for j = 1, 2. This function was used in the parameter estimation of Italy and Lithuania, whereas in the case of Germany and Czechia the function was set constant but we analyzed the possibility of introducing a modification. 215 

Six countries were selected to perform parameter optimization on their publicly available data: Israel, Germany, Czech Republic, Portugal, Italy, and Lithuania. Two strategies are used by the elected countries: one single type of vaccine In the following lines, we will describe the situation of the European countries and the parameter optimization. The case of Israel will be described in the 240 subsection of Reduction of vaccine efficacies, because further considerations are needed to estimate its parameters.

Germany presented a reduction in the cases from April to June, but this situation seems to change in July. The first three vaccines were set constant, but 

Portugal has particular dynamics on the development of its pandemic. In order to estimate correctly the behaviour of I(t), it was needed to add three 285 changes on the transmission parameters β 1 and β 2 , following the function

for j = 1, 2. In this case, θ β,0 = 78, and θ β,1 = 125, specifying the changes on the dynamics of the transmission rates. This behavior can be summarized as follows: in January 2021, Portugal reaches its peak of new cases with 16432 new reported cases in one single day (January 28). In February, a rapid descend of 290 the cases was presented, followed by a period of a relatively low number of cases (around 500 new daily cases), which was ended in the last two months (June and July), when a new increment is presented. In the next month, we forecast that the cases would continue growing until 37 days, reaching a peak of 71770 active infected. On the other hand, the vaccine parameters of the first three 295 vaccines were logistic, but in the case of the last vaccine, we used the piecewise exponential function.

64.01 % of the total population has been vaccinated, but only 47.66 % are estimated to be in the vaccinated compartments. It was also found that S 0 = 0.2464, which means the DFE includes roughly 75 % of the total population in 300 the vaccination compartments. 

In the case of Lithuania, all the vaccination rates were set as logistic functions, since the alternative options did not fit appropriately with the data. On 

In order to study the influence of each parameter on the reproduction number, we can use the normalized forward sensitivity index, introduced in [50].

This index, also called elasticity index [51] , measures the relative change of the reproduction number R c when a specific parameter is changed.

Definition 1. If R c is differenciable with respecto to a parameter θ, the normalized forward sensitivity index of R c is given by

Using the above definition, we can calculate the forward sensitivity of the

Note that Γ Rc β1 , Γ Rc β2 are always positive and Γ Rc γ , Γ Rc δ are always negative.

The fact that the death rate has a negativity sensitivity should not be unexpected, since a very deadly infectious disease cannot spread easily. Just to provide one example, the MERS, an infection caused by another coronavirus, has a higher death rate, but a lower reproduction number [52] .

Nevertheless, a natural assumption is to expect β 1 > δ, which yields |Γ Rc β1 | > 345 |Γ Rc δ |, indicating that if β 1 is larger than δ, the recovery rate γ contributes more to the reduction of the reproductive number. Another possibility is that the transmission rate from infected is lower than the transmision rate from asymptomatic β 1 < β 2 , in such case, any increment on the percentage of asymptomatic would increase the reproduction number. In general, this situation happens if and only if β 1 γ < (γ + δ)β 2 .

In the case of vaccine parameters, let v = 1+ N j=1 ε j,L1 +ε j,L2 λj αj

Thus, R c = R 0 (vS 0 ) and

Index of vaccine parameters ε i,A , ε i,L1 , ε i,L2 are always positive, which means that they must be as lower as possible. A much complex situation arises when 355 analyzing key parameters α i , r i , and λ i , since, against common sense, they do not show a clear tendency. This topic will be covered in the subsection of Multiple Vaccination Criteria, as part of the Multiple Vaccination Theorem.

The sensitivity analysis of the studied countries presents the values of elas- This dissimilarity might be explained by the higher vaccination rates of Pfizer vaccines.

An interested situation is given in the positive elasticity of the AstraZeneca vaccine of Portugal. This seems to be counterintuitive, but we shall see that this rare situation can be explained. In addition, it is worth mentioning that the recovery rate provides the most negative elasticity in all cases, with a value of almost -1. 

In this subsection, we study variations of vaccine efficacies and transmission In addition, we studied the situation of other countries (Germany, Czech

Republic, Italy, Lithuania) by analyzing different panoramas of parameter variation. Portugal was not included in this analysis, because we performed a pa-390 rameter estimation by introducing changes in the behavior of the transmission rates, and that was not the case for the remaining countries.

The variation of parameters can also be used to determine how can we increment the vaccination rates, to reach R c < 1. In particular, it was observed that R c > 1 in the case of Portugal and Israel. In the first situation, if we increase implies that the increments on the vaccination rate are unable to reach R c < 1.

In this case, reduction of the transmission rates seems to be the most effective option to reduce the impact of the pandemic.

Israel is a remarkable case where only one type of vaccination was primarily used, which is the Pfizer (BNT162B2) vaccine [55] . figure   11 .

In general, this change in the transmission rates has some consequences on the behavior of I(t), which is increasing. We estimate that a new peak might occur in 50 days, taken 309212 infected (see figure 12 ). 63.6 % of the population have received at least one dose, whereas 58.4 % have received two doses, but only 33.81 % of the population belongs to the vaccine compartments. which is the situation of Israel. A complex panorama appears when N > 1, but for visuallization purposes, we will consider only this case. Let x = V 0 j , y = F 0 j . Therefore,

R c (x, y) can be understood as the control reproduction number as a function of the final proportions of the vaccinated population with one and two doses respectively. These proportions must satisfy the constraint x + y ≤ 1.

In figure 13 , three contour plots of R c (x, y) are depicted, considering the confidence interval of the vaccine with the reduction of efficacies established 430 at the beginning of this subsection of Israel. The isosceles triangle in the right upper corner is represented with the value 0 but is not included in the domain x+y ≤ 1. Higher proportions of F 0 j seems to be better to reach the value R c < 1 (more than 80 %, with 20 % of people vaccinated with one dose). Nevertheless, it is important to mention that in this specific case, R 0 = 4.6905 is particularly 435 high and increasing the vaccination rate solely is unable to find R c < 1, as shown previously.

Germany. An interesting situation is modeled when we change the values of vaccine efficacies, and double the values of the transmission rates β 1 and β 2 , in 440 order to observe the reduction of efficacies with incoming SARS-CoV-2 variants.

Four different panoramas are expressed in Figure 14 : In contrast to Germany, Czech Republic required time-dependent vaccine parameters, as stated previously, specifically the logistic function. This particular situation motivates to study of the variation of the function across 465 time and influences the behavior of R c (t). In the four cases it was found that lim t→∞ R c (t) = 0. A consequence of this limit is observed in figure 15 , when all the four cases yield the same panorama.

Italy. In order to simulate possible changes on the behaviour of the parameters, we considered the following panoramas: Simulation of the referred panoramas is presented in figure 16 . Case 3 seems to be more related to the local trend of the data, which means that the parameters might have already changed.

Lithuania. The following cases were considered to simulate different panoramas: Let

The following properties are satisfied:

1. η is constant over any chances of the vaccine parameters of the N -th vaccine.

Therefore,

which does not include any vaccine parameter of the N -th vaccine. In addition, the line given by

equals R 0 u provided = η, and if < η, R c < R 0 u, and if η < , R 0 u < R c .

Finally, recall that

where x 2 > M 2 and x 2 + · · · + x 2 > M 2 , which yields |g(x) − L| < ε.

For the Insufficiency Theorem, the notion of the best vaccine is needed to use it to evaluate the possibility of reaching R c < 1 with the extrinsic vaccination parameters, in particular, r j . This notion is derived from the following definition.

Definition 2. A vaccine with label i is said to be with the best better parameters than the j-vaccine if for each vaccine parameter θ i (excluding the rates),

Vaccines are said to be equivalent if for all i, j, θ i = θ j .

The sensitivity analysis gives prior conditions about when the parameters of a vaccine are better. For instance, less values of ε i,A , ε i,L1 , ε i,L2 are better, but a more complex situation is observed with α i . The next proposition permits 565 to study of a lower bound for inequivalent vaccines, and the limit when all the rates tend to infinite.

Proposition 3. For equivalent vaccines,

(60)

Proof. This a straightful computation. Using Lemma 1, 

As a consequence, for equivalents vaccines, there exist the possibility of not reaching the DFE, depending on the parameters of the model. This includes R 0 , which means that for a greater reproductive number without vaccination, the combined leakiness must be set as lower as possible to prevent the undesired situation of unstability of the DFE. If a best vaccine exists, we can generalize the last result in a form of criteria.

Theorem 2 (Insufficiency Theorem). Let us consider the existence of the vaccine with the best parameters (labeled with i). The inequality

holds if and only if for any values R c (r 1 , . . . , r N ) ≥ 1. 

The 

which proves that inf{(r 1 , . . . , r M ) : R c (r 1 , . . . , r M ))} ≥ 1.

In other words, the Insufficiency Theorem states that if the best vaccine exists, the reproduction number is greater or equal than one if and only if the best vaccine with a theoretically infinite vaccination rate is at least one. If there is no a "best vaccine", a weak result can be stated if we construct an idealistic best vaccine choosing the best parameters. In such a case, the converse of the 595 theorem does not hold but provides us a condition when the desired R c < 1 cannot be reached. Due to its importance, the quantity

R 0 will be named as insufficiency parameter, and this name will be refered in the empirical results.

Both the Multiple Vaccination Theorem and Insufficiency Theorem can be used for specific purposes in this study. For instance, the η-analysis is an alternative method to evaluate the effect on a particular vaccine. In this case, for

European countries (where multiple vaccines were applied) we can see that the inequality (k) < η k holds for most cases, except for k = 3 (AstraZeneca) in 605 Italy, which is confirmed by the positive value in both sensitivity and elasticity, which means that the particular type is not helping to reduce the reproduction number. The Insufficiency Theorem is also useful in this context, in particular when R c > 1. This is observed in the analysis of Portugal, where R c > 1, but the 615 Insufficiency parameter is 0.2255. In other countries, the Insufficiency Theorem can be used for the cases of variation of parameters. This is Case 3 of Germany, when the Insufficiency parameter is 0.2137 (using Moderna vaccine), which implies that even in the most dramatic and considered situation, it is possible to reach R c < 1. This situation is also observed in Case 4 of Lithuania because 620 the insufficiency parameter is lower than 1, (0.8679), which means that even in a dramatic situation vaccines could still be useful.

The "Sensitivity Theorems" (Multiple Vaccination and Insufficiency Theorems) are a useful tool to analyze the behavior of a system of multiple 

In this paper, we presented a model with multiple types of vaccines adapting 665 some of the structure of the VSIR model of Magpantay [31] for adult vaccination and including a general number of imperfect vaccines. This model was specifically for the case of the ongoing pandemic of COVID-19, which motivates the developed theory.

Rather than focusing on the distribution of vaccines across special compart-670 ments or the effects of the honeymoon period described by [30] . This special interest follows the current situation of the pandemic: while the first papers of vaccination tried to optimize the distribution by minimizing deaths or other metrics, nowadays is important to decide how can the vaccination campaign be improved by incrementing or reducing vaccination rates. In further research, it 675 would be useful to see the effects of the referred honeymoon period and the potential risk associated with the new variants.

Two relevant results of the Sensitivity Analysis are the Multiple Vaccination

Theorem and Insufficiency Theorem. The first one states that the reduction of the reproduction number only holds if one derived vaccination parameter ( (k) ) 680 is less than a threshold called η k . Although it contradicts common sense, if this condition is not met, the reproduction number would increase, and more intriguing, any increment in the vaccination rate results in an increment in the reproduction number. This is odd since we could think that more vaccines always improves the situation, but theory shows evidence against our beliefs. It was 685 also observed that such believes were considered in the vaccination campaigns of some European countries.

Insufficiency Theorem establishes another counterintuitive proposition: vaccines might not be sufficient to curtail an outbreak, especially if the R 0 is high enough. Luckily, in all the cases where this theorem applies, it was found that 690 R c < 1 is possible if the rate of the vaccine with better parameters is increased.

We expect that this research could bring some mathematical tools to study the vaccination campaigns as objects of research and improve the public decisions in this matter.

Funding Not applicable.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work

Availability of data and materials Data was collected from two different sources: Records from reported recovered and cumulative infected were taken from the John Hopkins repository [35] and for vaccination data, we used the repository of Our World in Data

Code availability All implementations of the simulations and symbolic computations were written in Python and are

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