A Fractional Order Model Studying the Role of Negative and Positive Attitudes towards Vaccination

A fractional-order model consisting of a system of four equations in a Caputo–Fabrizio sense is constructed. This paper investigates the role of negative and positive attitudes towards vaccination in relation to infectious disease proliferation. Two equilibrium points, i.e., disease-free and endemic, are computed. Basic reproduction ratio is also deducted. The existence and uniqueness properties of the model are established. Stability analysis of the solutions of the model is carried out. Numerical simulations are carried out and the effects of negative and positive attitudes towards vaccination areclearly shown; the significance of the fractional-order from the biological point of view is also established. The positive effect of increasing awareness, which in turn increases positive attitudes towards vaccination, is also shown numerically.The results show that negative attitudes towards vaccination increase infectious disease proliferation and this can only be limited by mounting awareness campaigns in the population. It is also clear from our findings that the high vaccine hesitancy during the COVID-19 pandemicisan important problem, and further efforts should be madeto support people and give them correct information about vaccines.


Introduction
Scientific discoveries and their applications are what define modern societies. Recently, the emergence of anti-scientific attitudeshas led to a decrease in public trust in science [1]. Vaccines areamong the most significant discoveries in science, and have saved many lives. However, the increases anti-vaccine groups leads to vaccine rejection [2][3][4]. Hence, theseanti-vaccine groups increase the danger of infectious disease proliferation to themselves and to the entire society. Since the emergence ofthe COVID-19 pandemic and the serious problems it has caused, the study of the problemsleading to vaccine rejection is of paramount significance. Many people from different backgrounds are against vaccines, which consequently leads to reductions inpre-existing immunity [5].
Studying the causes of both negative and positive attitudes towards vaccination is therefore very significant as the purely scientific and applied perspectives are concerned. Several studies have investigated the causes of the increases in anti-vaccine groups and their focus has beengeared towards individual differencesmost of the time.For example, in [6] they claim that anti-vaccine attitudes are related tomoral purity concerns, and orthodox religiousness. In [7] they claim anti-vaccine attitudes have direct relationships withindividualistic/hierarchical worldviews and conspiratorial thinking.
Many models in theliterature have considered the vaccination decision-making process [8][9][10][11]. Most of these models are based on ordinary differential equations. In most compartment, the infected compartment, and the recovered compartment, respectively. The model is given below: with the following initial conditions: Define N = F + A + I + R, to be the total population. The meaning of the parameters involved in the model is given in Table 1 below.

Analysis of the Model
Here, existence and uniqueness analysis of the solution of the model iscarried out. Moreover, equilibria, basic reproduction number, and local stability analysis of the solution of the model are studied.

Existence and Uniqueness of a Solution of the Model
In this paper, a fixed-point result is applied to check the existence and uniqueness of the solution of the model. Let the system be re-written as Applying the Caputo-Fabrizio operator, the system becomes: Now, we need to prove F 1 , . . . , F 4 satisfy Lipschitz continuity and contraction. See the theorem below: it is a contraction. Proof.
In the same way, we show the Lipschitz continuity and contraction for F 2 , . . . , F 4 , where we obtain L 2 , . . . , L 4 , respectively, as their Lipschitz constants.
In recursive form, let with initial conditions: Taking norm of q 1n , we have: Vaccines 2022, 10, 2135

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Applying triangular inequality, we have: This implies: Similarly, Subsequently, we have: To show the existence of the solution, we prove the following theorem: The solution exists if there exist t 1 such that the following inequality is true, Proof. Recursively, we have Hence, solutions exist and are continuous. To show that the functions above construct the solutions, consider: Vaccines 2022, 10, 2135 6 of 13 Hence, Carrying out the procedure, we get Taking limit as n → ∞, we get Finally, to show uniqueness, assume there exists some solutions say, F 1 (t), A 1 (t), I 1 (t) and R 1 (t), then The following theorem completes the result.
then the solution is unique.

Proof.
Consider Since, This implies, This applies to the remaining functions.
Since R = N − (F + A + I), we can limit our analysis tothree compartments, {F, A, I}.

Equilibria and Basic Reproduction Number
The equilibrium solutions are obtained by equating the equations in the model to zero and solving the system simultaneously. We obtain two equilibrium solutions: i.
Disease-free equilibrium (E 0 ) and I 1 is obtained by solving the following quadratic equation, Clearly, we can see that the endemic equilibrium exists only if, where R 0 is the basic reproduction number.

Local Stability Analysis of the Solution of the Model
Consider the following Jacobian matrix from (1), Theorem 4. The disease-free equilibrium is locally asymptotically stable if R 0 < 1. Proof. Consider (29) at E 0 , we get The characteristics polynomial of (30) is Therefore, .
Substituting the values of F 0 and A 0 , we get Using the Routh-Hurwitz criterion, P(s) = s 2 + a 1 s + a 2 has both roots with negative real parts iff both coefficeints, a i > 0, i = 0, 1, 2. Here, Hence, E 0 is locally asymptotically stable if R 0 < 1.

Numerical Simulation
The numerical method used in this paper is similar to that of [21] and numerical simulations are carried out. Parameter values are given as, The dynamics of the model are depicted in Figure 1. It is clear that none of the populations go to zero. The infected population and the recovered population simultaneously reach their peak at around 50 h, which is approximately two days. The numerical method used in this paper is similar to that of [21] and numerical simulations are carried out. Parameter values are given as, ᴧ 0.6day , β 0.5day , 0.001day , 0.0195day , 0.0005day , 0.1day , 0.05day , 0.2-1.0 dimentioneless , р 0.5 dimentioneless . The dynamics of the model are depicted in Figure 1. It is clear that none of the populations go to zero. The infected population and the recovered population simultaneously reach their peak at around 50 h, which is approximately two days.     It can be seen that, in the absence of the pro-vaccine population, the infected population increases. This is because the remaining people in the population are against the vaccine and hence a large portion of the population will not be vaccinated. This leads to the proliferation of the disease. Figure 2. compares the population of infected individuals with thepro-vaccine population. It can be seen that, in the absence of the pro -vaccine population, the infectedpopulation increases. This is because the remaining people in the population are against the vaccine and hence a large portion of the population will not be vaccinated. This leads to the proliferation of the disease.         Figure 4 shows that increase in the level of awareness lead to decreases in the population of infected individuals. This is because as the level of awareness increases, the number of pro-vaccine individuals increases. This leads to increases in the number of vaccinated individuals, which in turn leads to decreases in the population of infected individuals.

Conclusions
In this paper, we studied a fractional-order model consisting of a system of four equations in the Caputo-Fabrizio sense. Our aim wasto study the role of negative and positive attitudes towards vaccination in relation to infectious disease proliferation. The compartments of the model were thepro-vaccine susceptible compartment, the anti-vaccine susceptible compartment, theinfected compartment, and the recovered compartment. We obtained two equilibrium solutions, i.e., disease free and endemic. We were also able to obtain the basic reproduction ratio. This paper studied the existence and uniqueness properties of the model in detail. Numerical simulations werecarried out to support the analytic results. The effect of negative and positive attitudes towards vaccination wasclearly shown. Furthermore,the significance of the fractional-order from the biological point of view wasestablished. It was shown that increases in the level of awareness lead to decreases in the population of infected individuals. This is because as the level of awareness increases, the number of pro-vaccine individuals increases. This leads to increases in the number of vaccinated individuals, which in turn leads to decreases in the population of infected individuals.
The limitation of this study is that there is a need for real data collection to validate themodel, and people's opinions need to be heard and incorporated into the model for

Conclusions
In this paper, we studied a fractional-order model consisting of a system of four equations in the Caputo-Fabrizio sense. Our aim was to study the role of negative and positive attitudes towards vaccination in relation to infectious disease proliferation. The compartments of the model were the pro-vaccine susceptible compartment, the anti-vaccine susceptible compartment, the infected compartment, and the recovered compartment. We obtained two equilibrium solutions, i.e., disease free and endemic. We were also able to obtain the basic reproduction ratio. This paper studied the existence and uniqueness properties of the model in detail. Numerical simulations were carried out to support the analytic results. The effect of negative and positive attitudes towards vaccination was clearly shown. Furthermore, the significance of the fractional-order from the biological point of view was established. It was shown that increases in the level of awareness lead to decreases in the population of infected individuals. This is because as the level of awareness increases, the number of pro-vaccine individuals increases. This leads to increases in the number of vaccinated individuals, which in turn leads to decreases in the population of infected individuals.
The limitation of this study is that there is a need for real data collection to validate the model, and people's opinions need to be heard and incorporated into the model for further analysis.