Citation: Baba, I.A.; Rihan, F.A.; Humphries, U.W.; Mikailu, B.B. A Fractional Order Model Studying the Role of Negative and Positive Attitudes towards Vaccination. Vaccines 2022, 10, 2135. https:// doi.org/10.3390/vaccines10122135 Academic Editor: Giuseppe La Torre Received: 19 November 2022 Accepted: 12 December 2022 Published: 13 December 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Article A Fractional Order Model Studying the Role of Negative and Positive Attitudes towards Vaccination Isa Abdullahi Baba 1,2 , Fathalla A. Rihan 3,4 , Usa Wannasingha Humphries 1, * and Badamasi Bashir Mikailu 2 1 Department of Mathematics, Faculty of Science, King Mongkuts University of Science and Technology Thonburi (KMUTT), Bangkok 10140, Thailand 2 Department of Mathematics, Bayero University Kano, Kano 700006, Nigeria 3 Department of Mathematical Sciences, College of Science, UAE University, Al Ain 15551, United Arab Emirates 4 Department of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt * Correspondence: usa.wan@kmutt.ac.th Abstract: 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. Keywords: mathematical model; fractional-order; Caputo–Fabrizio; existence and uniqueness; vaccination; awareness 1. Introduction Scientific discoveries and their applications are what define modern societies. Re- cently, 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 [2–4]. 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 or- thodox 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 pro- cess [8–11]. Most of these models are based on ordinary differential equations. In most Vaccines 2022, 10, 2135. https://doi.org/10.3390/vaccines10122135 https://www.mdpi.com/journal/vaccines