F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.133768.1 Research Article Articles Hypertension and COVID-19 fractional derivative model with double dose vaccination

[version 1; peer review: 2 approved with reservations]

 Okyere Samuel Conceptualization Data Curation Formal Analysis Investigation Methodology Resources Software Validation Writing – Original Draft Preparation https://orcid.org/0000-0001-6262-1628 a 1 Ackora-Prah Joseph Conceptualization Resources Supervision Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0000-0001-9488-9610 1 Bonyah Ebenezer Conceptualization Formal Analysis Resources Software Supervision 2 Barnes Bennedict Conceptualization Resources Software Validation 1 Boateng Maxwell Akwasi Conceptualization Resources Software Validation https://orcid.org/0000-0003-4895-4754 1 Takyi Ishmael Conceptualization Resources Software Supervision Validation Writing – Review & Editing 1 Adarkwa Samuel Akwasi Conceptualization Data Curation Resources Software Writing – Review & Editing https://orcid.org/0000-0002-1610-9225 3 Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ashanti Region, Ghana Mathematics Education, Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development, Kumasi, Ashanti Region, Ghana Statistical Sciences, Kumasi Technical University, Kumasi, Ashanti Region, Ghana okyere2015@gmail.com

No competing interests were disclosed.

 15 5 2023 2023 12 495 21 4 2023 Copyright: © 2023 Okyere S et al. 2023 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The prevalence of at least one underlying medical condition, which increases the likelihood of developing the severe COVID-19 disease, is found in 22% of the world's population. The primary underlying medical condition that contributes to COVID-19 problems in Ghana is hypertension.This work investigate COVID-19 in a population with hypertension taking into account double dose vaccination of susceptible individuals. The study modifies a previous model proposed in the literature to include double dose vaccination and Atangana-Baleanu-Caputo fractional derivatives is used to solve the model. We give few definitions of the ABC operator and determine the existence and uniqueness of the solution. Using COVID-19 data for the period February 21, 2021 to July, 24 2021, the model is tested. The dynamics of the disease in the community were shown to be influenced by fractional-order derivatives. Contrary to the previous model proposed in the literature, the vulnerable group saw a significant reduction in the number, which may be attributed to the double dose vaccination. We recommend a cost-effective optimal control analysis in future work.

 Hypertension Fractional Derivative Basic Reproduction Number COVID-19 Double Vaccination The author(s) declared that no grants were involved in supporting this work. Introduction

It is now known that COVID-19 significantly worsens the health of people with primary medical condition of hypertension. 1 According to studies, those who have the difficulties are more likely to be given the COVID-19 and experience unfavourable effects. 2 – 4 The populations most at risk for infections include the elderly and those with underlying illnesses such as hypertension, diabetes, coronary artery etc. At least one of the underlying medical disorders is present in 22% of the world’s population, which raises their risk of contracting the severe COVID-19 disease. 5

A significant medical disease that increases the chance of developing heart, brain, kidney, and other disorders is hypertension, often known as high blood pressure. One of the “silent killers” is hypertension. Most persons with hypertension are unaware of the problem because the illness may not have any symptoms or warning signs. Some of the symptoms include early-morning headaches, nosebleeds, irregular heartbeats, changes in vision, and ear buzzing. 6

Since COVID-19 first appeared in China, it has spread worldwide, causing approximately 614 million illnesses and over 6.5 million fatalities. Ghana revealed its first certified case on March 12, 2020. In Ghana, there have been 171,412 confirmed cases of COVID-19 with 1,462 fatalities reported to WHO between March 12, 2020, to March 29, 2023. A total of 22,384,226 vaccine doses have been given as of March 26, 2023. It also has most of the COVID-19’s specified fundamental problems, making it particularly vulnerable to the disease. According to estimates, there are 5.27 million hypertensive individuals in the country. 7 In low- and middle-income countries, where the majority (two-thirds) of those with hypertension between the ages of 30 and 79 live, the condition affects 1.28 billion people worldwide, according to WHO.

Researchers have used mathematical models to investigate and analyze the dynamics of infectious illness transmission. The future outcomes of diseases and potential interventions can be predicted using models, which provide a condensed depiction of reality. Without considering the patients’ underlying medical conditions, studies on COVID-19 that have employed mathematical models have concentrated on the kinetics of transmission in a population. 8 – 22 Fractional derivatives have been suggested for use in mathematical models to study viral diseases. 18 , 23 – 30 Fractional-order models are more accurate and reliable than integer-order models because they have more degrees of freedom. Predictions can now include past data thanks to the memory property. 24 , 31

To explore the propagation of the COVID-19, some writers have also suggested fractional-order derivatives. 18 , 29 , 30 , 32 – 36 The integer-order models of differential equations don’t seem to be as consistent with this illness as the fractional-order models do. This is so that the memory and heredity qualities that are inherent in different materials and processes can be described using fractional derivatives and integrals. 16 Therefore, understanding and applying fractional-order differential equations is a growing necessity. Using fractional derivative, Ogunrinde et al. 37 developed a dynamic model of COVID-19 and citizens’ responses from a portion of the population in Nigeria. Aba Oud et al. 34 considered the effects of isolation, containment, and ecologic factors to develop a fractional epidemic model in the Caputo sense to analyze the dynamics of the COVID-19 outbreak in Pakistan. Ahmed et al., 32 suggested a fractional-order derivative to investigate the COVID-19 transmission kinetics in Wuhan, China. A susceptible-asymptomatic-symptomatic-recovered-deceased model was put up by Giangreco 33 to study the COVID19 epidemic in Italy. They developed a mathematical model of the coronavirus illness 2019 (COVID-19) to examine the disease’s transmission and control mechanisms in the Nigerian population using the Atangana-Beleanu operator. A fractional-order epidemic model with the conventional Caputo operator and the Atangana-Baleanu-Caputo operator was suggested by authors in Ref. 29 for the propagation of the COVID-19 outbreak. To analyze the behavior of the COVID-19 in Ethiopia, Habenom et al. 18 developed a mathematical model of Caputo fractional-order. To account for both government action and individual response, authors in Ref. 36 used a fractional derivatives model to create a mathematical model for COVID-19 dynamics. Yadav et al. 6 investigated the fractional-order Covid-19 model using the powerful and successful analytical method known as q-homotopy analysis. A more thorough use of the fractional-order derivatives for simulating infectious diseases is provided in Refs. 10, 23– 25, 30, 38, 39.

Most countries are at war to halt the spread of COVID-19 and as such, have resulted to vaccinating susceptible individuals. According to the Ghana Health Service , about 41.7% of the total population have had at least one dose of vaccination and 32% of the population fully vaccinated against the COVID-19. As the globe seeks to find a cure for the disease, appropriate control measures remain one of the most efficient strategies to curb its spread. 15 Authors in Ref. 40 formulated an optimal control model, to study the transmission of the COVID-19. Authors in Ref. 14 created a fractional-order derivative model formulated in the Atangana-Baleanu in Caputo sense to study the COVID-19 disease’s propagation. They incorporated the best controls in the model to decrease infections and expand the population that was susceptible. Atangana-Baleanu-Caputo derivative-based fractional optimum control model was proposed by authors in Ref. 15. They considered COVID-19 treatment, public awareness campaigns on social barriers separating infected people from susceptible, and social distance between these groups of people.

In this work, we build on earlier work in Ref. 41 by introducing a segment of the population that have received first and second dose vaccination and a proportion of the population having hypertension. This work was inspired by the COVID-19 works that are currently available in this literature. 21 , 30 , 41 , 42 The classical model proposed in Ref. 41 does not adequately capture the dynamics of the virus’ non-local behavior. The model was built using the Atangana-Baleanu and Caputo derivatives, which have several desirable properties, such as a nonlocal and nonsingular kernel, which allows a better explanation of the crossover behavior in the model employing this operator. Other operators, including Caputo and Caputo-Fabrizio, which lack these characteristics, may or may not be able to accurately characterize the dynamics of the coronavirus. 9 In this inquiry, we make use of the fractional-order derivatives from Ref. 24. It is innovative in this case to simulate utilizing the fractional derivative while considering the population with first and second dose vaccination and the vulnerable with hypertension. As far as we are aware, this has never been investigated.

The other sections are the method, numerical analysis, and conclusion. The proposed model in Ref. 41 is modified to include first and second dose vaccination. Atangana-Baleanu-Caputo fractional-order derivative was used to describe the model in the method section. we also list some few definitions and determine the uniqueness of the solution. The model is solved numerically in the numerical analysis section.

 Methods Model formulation

The dynamics of COVID-19 transmission in a vulnerable population with first and second dose vaccination is examined in this part using fraction-order derivatives. Utilizing Atangana-Baleanu in the sense of Caputo stated in Refs. 24, 30 the fractional-order derivative is defined. The operator for fractional orders is defined as k, where 0 < k ≤ 1. Eight classes of individuals are distinguished in the model: Susceptible H S , individuals with hypertension H h , Exposed H E .

Infected with COVID-19 H I , COVID-19 patients with hypertension H C , Individuals with first dose vaccination H V 1 , Individuals with second dose vaccination H V 2 , and Recovered individuals H R. Figure 1 displays the model’s schematic.

Dynamic flow of the Hypentension-COVID-19 model with double dose vaccination.

The system is described below using the fractional-order derivatives. k ABC D t k H S = σ k + ξ 6 k H R − σ 2 k H I + H C N H S − σ 1 k + σ 7 k + ξ 1 k H S , k ABC D t k H h = σ 7 k H S + σ 6 k H C − σ 2 k H I + H C N H h − σ 8 k + σ 1 k + ξ 1 k H h , k ABC D t k H E = σ 2 k H I + H C N H S + H h + H V 1 + H V 2 − σ 4 k + σ 1 k H E , k ABC D t k H I = ξ σ 4 k H E − σ 5 k + σ 1 k + σ 3 k H I , k ABC D t k H C = 1 − ξ σ 4 k H E − σ 9 k + σ 6 k + σ 1 k H C , k ABC D t k H V 1 = ξ 1 k H S + H h − σ 2 k H I + H C N H V 1 − σ 1 k + ξ 3 k H V 1 , k ABC D t k H V 2 = ξ 3 k H V 1 − σ 2 k H I + H C N H V 2 − σ 1 k H V 2 , k ABC D t k H R = σ 3 k H I − ξ 6 k + σ 1 k H R . with initial conditions

H S (0) = H S 0 , H h (0) = H h 0 , H E (0) = H E 0 , H I (0) = H I 0 , H C (0) = H C 0 , H V 1 (0) = H V 1 , H V 2 (0) = H V 2 , H R (0) = H R 0 .

 Preliminaries

In this part, the general fractional derivatives definitions concerning Atangana-Baleanu-Caputo is given.

Definition 1: Liouville and Caputo (LC) definition as in Refs. 24, 28 C t k h t = 1 Γ 1 − k ∫ 0 t τ − q − k h p dp , 0 < k ≤ 1 . where k is the fractional operator.

Definition 2: Let h ∈ T 1 x 1 x 2 , x 2 > x 1 ∈ 0 1 , then, the new Caputo fractional derivative as stated in Ref. 43 is: D t k h t = H k 1 − k ∫ x 2 t h ′ x exp − k t − x 1 − k dx , where H( k) denotes a normalized function such that H(0) = H(1) = 1. However, if the function h ∉ T 1 x 1 x 2 then equation (3) has the form D t k h t = kH k 1 − k ∫ x 2 t h t − h x exp − k t − x 1 − k dx ,

If we let ζ = 1 − k k ∈ 0 ∞ then = 1 1 − ζ ∈ 0 1 , the above equation assumes the form D t ζ h t = N ζ ζ ∫ x 2 t h ′ x exp − t − x ζ dx , N 0 = N ∞ = 1 .

Definition 3: Let h ∈ T 1 x 1 x 2 , x 2 > x 1 ∈ 0 1 , then, the new fractional derivative as defined by Ref. 43 is given as: ABC t k h t = H k 1 − k ∫ x 2 t h ′ x E k − k t − x k 1 − k dx ,

Laplace transform of equation (6) is given by L ABC k p = H k 1 − k L h t − p k − 1 h 0 p k + k 1 − k

When k = 0, we do not recover the original function except when at the origin, the function vanishes. To avoid this issue authors in Ref. 43 proposed the following definition

Definition 4: Let h ∈ T 1 x 1 x 2 , x 2 > x 1 ∈ 0 1 , then, the new fractional derivative as defined by Ref. 43 is given as: ABR t k h t = H k 1 − k d dt ∫ x 2 t h x E k − k t − x k 1 − k dx ,

When the function h( x) is constant, in equation (6), we get zero. The Laplace transform of equation (8) as given in Ref. 43 is: L ABR k p = H k 1 − k p k L h t p p k + k 1 − k

Existence and uniqueness of the solutions

Let G( T) denote a banach space, where T = [0, b], and M = G T × G T × G T × G T × G T × G T × G T × G T with the norm H S H h H E H I H C H V 1 H V 2 H R = H S + H h + H E + H I + H C + H V 1 + H V 2 + H R , where H S = Sup π ∈ T H S , H h = Sup π ∈ T H h , H E = Sup π ∈ T H E , H I = Sup π ∈ T H I , H C = Sup π ∈ T H C , H V 1 = Sup π ∈ T H V 1 , H V 2 = Sup π ∈ T H V 2 , H R = Sup π ∈ T H R . Using the Atangana-Baleanu-Caputo integral operator on the system (1), we have H S t − H S 0 = k ABC D t k H S t σ k + ξ 6 k H R − σ 2 k H I + H C N H S − σ 1 k + σ 7 k + ξ 1 k H S , H h t − H h 0 = k ABC D t k H h t σ 7 k H S + σ 6 k H C − σ 2 k H I + H C N H h − σ 1 k + σ 8 k + ξ 1 k H h , H E t − H E 0 = k ABC D t k H E t σ 2 k H I + H C N H h + H S + H V 1 + H V 2 − σ 4 k + σ 1 k H E t , H I t − H I 0 = k ABC D t k H I t ξ σ 4 k H E − σ 5 k + σ 1 k + σ 3 k H I , H C t − H C 0 = k ABC D t k H C t 1 − ξ σ 4 k H E − σ 9 k + σ 6 k + σ 1 k H C , H V 1 t − H V 1 0 = k ABC D t k H V 1 t ξ 1 k H S + H h − σ 2 k H I + H C N H V 1 − σ 1 k + ξ 3 k H V 1 , H V 2 t − H V 2 0 = k ABC D t k H V 2 t ξ 3 k H V 1 − σ 2 k H I + H C N H V 2 − σ 1 k H V 2 , H R t − H R 0 = k ABC D t k H R t σ 3 k H I − ξ 6 k + σ 1 k H R .

Applying Liouville and Caputo (LC) definition of fractional-order derivatives gives, H S t − H S 0 = 1 − k B k π 1 k t H S t + k B k Γ k × ∫ 0 t t − τ k − 1 π 1 k τ H S τ d τ , H h t − H h 0 = 1 − k B k π 2 k t H h t + k B k Γ k × ∫ 0 t t − τ k − 1 π 2 k τ H h τ d τ , H E t − H E 0 = 1 − k B k π 3 k t H E t + k B k Γ k × ∫ 0 t t − τ k − 1 π 3 k τ H E τ d τ , H I t − H I 0 = 1 − k B k π 4 k t H I t + k B k Γ k × ∫ 0 t t − τ k − 1 π 4 k τ H I τ d τ , H C t − H C 0 = 1 − k B k π 5 k t H C t + k B k Γ k × ∫ 0 t t − τ k − 1 π 5 k τ H C τ d τ , H V 1 t − H V 1 0 = 1 − k B k π 6 k t H V 1 t + k B k Γ k × ∫ 0 t t − τ k − 1 π 6 k τ H V 1 τ d τ , H V 2 t − H V 2 0 = 1 − k B k π 7 k t H V 2 t + k B k Γ k × ∫ 0 t t − τ k − 1 π 7 k τ H V 2 τ d τ , H R t − H R 0 = 1 − k B k π 8 k t H R t + k B k Γ k × ∫ 0 t t − τ k − 1 π 8 k τ H R τ d τ , where π 1 k τ H S t = σ k + ξ 6 k H R − σ 2 k H I + H C N H S − σ 1 k + σ 7 k + ξ 1 k H S , π 2 k τ H h t = σ 7 k H S + σ 6 k H C − σ 2 k H I + H C N H h − σ 8 k + σ 1 k + ξ 1 k H h , π 3 k τ H E t = σ 2 k H I + H C N H S + H h + H V 1 + H V 2 − σ 4 k + σ 1 k H E , π 4 k τ H I t = ξ σ 4 k H E − σ 5 k + σ 1 k + σ 3 k H I , π 5 k τ H C t = 1 − ξ σ 4 k H E − σ 9 k + σ 6 k + σ 1 k H C , π 6 k τ H V 1 t = ξ 1 k H S + H h − σ 2 k H I + H C N H V 1 − σ 1 k + ξ 3 k H V 1 , π 7 k τ H V 2 t = ξ 3 k H V 1 − σ 2 k H I + H C N H V 2 − σ 1 k H V 2 , π 8 k τ H R t = σ 3 k H I − ξ 6 k + σ 1 k H R .

The ABC derivatives only meets the Lipschitz requirement 44 , 45 if

H S t , H h t , H E t , H I t , H C t , H V 1 t , H V 2 t , H R t , have an upper bound.

We assumed that H S t and H S ∗ t are paired functions, π 1 k , t , H S t − π 1 k , t , H S ∗ t = − σ 2 k H I + H C N H S + σ 1 k + σ 7 k + ξ 1 k H S t − H S ∗ t

Considering T 1 = − σ 2 k H I + H C N H S + σ 1 k + σ 7 k + ξ 1 k ,

Equation (13) simplifies to π 1 ( k t H S t ) − π 1 ( k t H S ∗ t ) ≤ T 1 ‖ H S t − H S ∗ t ‖

Similarly, ‖ π 2 k t H h t − π 2 k t H h ∗ t ‖ ≤ T 2 ‖ H h t − H h ∗ t ‖ , ‖ π 3 k t H E t − π 3 k t H E ∗ t ‖ ≤ T 3 ‖ H E t − H E ∗ t ‖ , ‖ π 4 k t H I t − π 4 k t H I ∗ t ‖ ≤ T 4 ‖ H I t − H I ∗ t ‖ , ‖ π 5 k t H C t − π 5 k t H C ∗ t ‖ ≤ T 5 ‖ H C t − H C ∗ t ‖ , ‖ π 6 k t H V 1 t − π 6 k t H V 1 ∗ t ‖ ≤ T 6 ‖ H V 1 t − H V 1 ∗ t ‖ , ‖ π 7 k t H V 2 t − π 7 k t H V 2 ∗ t ‖ ≤ T 7 ‖ H V 2 t − H V 2 ∗ t ‖ , ‖ π 8 k t H R t − π 8 k t H R ∗ t ‖ ≤ T 8 ‖ H R t − H R ∗ t ‖ . where T 2 = − σ 2 k H I + H C N − σ 8 k + σ 1 k + ξ 1 k , T 3 = − σ 4 k + σ 1 k , T 4 = − σ 5 k + σ 1 k + σ 3 k , T 5 = − σ 9 k + σ 6 k + σ 1 k , T 6 = − σ 2 k H I + H C N − σ 1 k + ξ 3 k , T 7 = − σ 2 k H I + H C N − σ 1 k , T 8 = − ξ 6 k + σ 1 k ,

Lipschitz condition is now valid. Now repeatedly applying system (10) gives, H S n t − H S 0 = 1 − k B k π 1 k t H S ( n − 1 ) t + k B k Γ k × ∫ 0 t t − τ k − 1 π 1 k τ H S ( n − 1 ) τ dτ , H h n t − H h 0 = 1 − k B k π 2 k t H h ( n − 1 ) t + k B k Γ k × ∫ 0 t t − τ k − 1 π 2 k τ H h ( n − 1 ) τ dτ , H E n t − H E 0 = 1 − k B k π 3 k t H E ( n − 1 ) t + k B k Γ k × ∫ 0 t t − τ k − 1 π 3 k τ H E n − 1 τ dτ , H I n t − H I 0 = 1 − k B k π 4 k t H I ( n − 1 ) t + k B k Γ k × ∫ 0 t t − τ k − 1 π 4 k τ H I ( n − 1 ) τ dτ , H C n t − H C 0 = 1 − k B k π 5 k t H C ( n − 1 ) t + k B k Γ k × ∫ 0 t t − τ k − 1 π 5 k τ H C ( n − 1 ) τ dτ , H V 1 n t − H V 1 0 = 1 − k B k π 6 k t H V 1 ( n − 1 ) t + k B k Γ k × ∫ 0 t t − τ k − 1 π 6 k τ H V 1 ( n − 1 ) τ dτ , H V 2 n t − H V 2 0 = 1 − k B k π 7 k t H V 2 ( n − 1 ) t + k B k Γ k × ∫ 0 t t − τ k − 1 π 7 k τ H V 2 ( n − 1 ) τ dτ , H R n t − H R 0 = 1 − k B k π 8 k t H R ( n − 1 ) t + k B k Γ k × ∫ 0 t t − τ k − 1 π 8 k τ H R ( n − 1 ) τ dτ ,

Difference of consecutive terms yields Ω H S n t = H S n t − H S n − 1 t = 1 − k B k π 1 k t H S n − 1 t − π 1 k t H S n − 2 t + k B k Γ k ∫ 0 t t − τ k − 1 π 1 k τ H S n − 1 τ − π 1 k τ H S n − 2 τ dτ , Ω H h n t = H h n t − H h n − 1 t = 1 − k B k π 2 k t H h n − 1 t − π 2 k t H h n − 2 t + k B k Γ k ∫ 0 t t − τ k − 1 π 2 k τ H h n − 1 τ − π 2 k τ H h n − 2 τ dτ , Ω H E n t = H E n t − H E n − 1 t = 1 − k B k π 3 k t H E n − 1 t − π 3 k t H E n − 2 t + k B k Γ k ∫ 0 t t − τ k − 1 π 3 k τ H E n − 1 τ − π 3 k τ H E n − 2 τ dτ , Ω H I n t = H I n t − H I n − 1 t = 1 − k B k π 4 k t H I n − 1 t − π 4 k t H I n − 2 t + k B k Γ k ∫ 0 t t − τ k − 1 π 4 k τ H I n − 1 τ − π 4 k τ H I n − 2 τ dτ , Ω H C n t = H C n t − H C n − 1 t = 1 − k B k π 5 k t H C n − 1 t − π 4 k t H C n − 2 t + k B k Γ k ∫ 0 t t − τ k − 1 π 5 k τ H C n − 1 τ − π 5 k τ H C n − 2 τ dτ , Ω H V 1 n t = H V 1 n t − H V 1 n − 1 t = 1 − k B k π 6 k t H V 1 n − 1 t − π 6 k t H V 1 n − 2 t + k B k Γ k ∫ 0 t t − τ k − 1 π 6 k τ H V 1 n − 1 τ − π 6 k τ H V 1 n − 2 τ dτ , Ω H V 2 n t = H V 2 n t − H V 2 n − 1 t = 1 − k B k π 7 k t H V 2 n − 1 t − π 7 k t H V 2 n − 2 t + k B k Γ k ∫ 0 t t − τ k − 1 π 7 k τ H V 2 n − 1 τ − π 7 k τ H V 2 n − 2 τ dτ , Ω H R n t = H R n t − H R n − 1 t = 1 − k B k π 8 k t H R n − 1 t − π 8 k t H R n − 2 t + k B k Γ k ∫ 0 t t − τ k − 1 π 8 k τ H R n − 1 τ − π 8 k τ H R n − 2 τ dτ , where

H S n t = ∑ i = 0 n Ω H S n t , H h n t = ∑ i = 0 n Ω H h n t , H E n t = ∑ i = 0 n Ω H E n t , H I n t = ∑ i = 0 n Ω H I n t , H C n t = ∑ i = 0 n Ω H C n t , H V 1 n t = ∑ i = 0 n Ω H V 1 n t , H V 2 n t = ∑ i = 0 n Ω H V 2 n t , H R n t = ∑ i = 0 n Ω H R n t . Taking into consideration equations (15)– (16) and considering Ω H S n − 1 t = H S n − 1 t − H S n − 2 t , Ω H h n − 1 t = H h n − 1 t − H h n − 2 t , Ω H E n − 1 t = H E n − 1 t − H E n − 2 t , Ω H I n − 1 t = H I n − 1 t − H I n − 2 t , Ω H C n − 1 t = H C n − 1 t − H C n − 2 t , Ω H V 1 n − 1 t = H V 1 n − 1 t − H V 1 n − 2 t , Ω H V 2 n − 1 t = H V 2 n − 1 t − H V 2 n − 2 t , Ω H R n − 1 t = H R n − 1 t − H R n − 2 t , Ω H S n t ≤ 1 − k B k T 1 Ω H S n − 1 t k B k Γ k T 1 × ∫ 0 t t − τ k − 1 Ω H S n − 1 τ dτ , Ω H h n t ≤ 1 − k B k T 2 Ω H h n − 1 t k B k Γ k T 2 × ∫ 0 t t − τ k − 1 Ω H h n − 1 τ dτ , Ω H E n t ≤ 1 − k B k T 3 Ω H E n − 1 t k B k Γ k T 3 × ∫ 0 t t − τ k − 1 Ω H E n − 1 τ dτ , Ω H I n t ≤ 1 − k B k T 4 Ω H I n − 1 t k B k Γ k T 4 × ∫ 0 t t − τ k − 1 Ω H I n − 1 τ dτ , Ω H C n t ≤ 1 − k B k T 5 Ω H C n − 1 t k B k Γ k T 5 × ∫ 0 t t − τ k − 1 Ω H C n − 1 τ dτ , Ω H V 1 n t ≤ 1 − k B k T 6 Ω H V 1 n − 1 t k B k Γ k T 6 × ∫ 0 t t − τ k − 1 Ω H V 1 n − 1 τ dτ , Ω H V 2 n t ≤ 1 − k B k T 7 Ω H V 2 n − 1 t k B k Γ k T 7 × ∫ 0 t t − τ k − 1 Ω H V 2 n − 1 τ dτ , Ω H R n t ≤ 1 − k B k T 8 Ω H R n − 1 t k B k Γ k T 8 × ∫ 0 t t − τ k − 1 Ω H R n n − 1 τ dτ ,

Theorem: The system (1) has a unique solution for t ∈ 0 b subject to the condition 1 − k B k T i + k B k Γ k b k n i < 1 , i = 1 , 2 , 3 , . … … , 8 hold. 15

Proof: Since H S t , H h t , H E t , H I t , H C t , H V 1 t , H V 2 t , H R t , are bounded functions and Equation (15) − (16) holds. Ω H S n t ≤ H S o t 1 − k B k T 1 + k b k B k Γ k T 1 n , Ω H h n t ≤ H h o t 1 − k B k T 2 + k b k B k Γ k T 2 n , Ω H E n t ≤ H E O t 1 − k B k T 3 + k b k B k Γ k T 3 n , Ω I n t ≤ H I o t 1 − k B k T 4 + k b k B k Γ k T 4 n , Ω H C n t ≤ H C O t 1 − k B k T 5 + k b k B k Γ k T 5 n , Ω H V 1 n t ≤ H V 1 o t 1 − k B k T 6 + k b k B k Γ k T 6 n , Ω H V 2 n t ≤ H V 2 o t 1 − k B k T 7 + k b k B k Γ k T 7 n , Ω H R n ( t ) ≤ H R O ( t ) 1 - k B ( k   ) T 8 + k b k B ( k ) Γ ( k ) T 8 n , and

Ω H S t → 0 , Ω H h t → 0 , Ω H E t → 0 , Ω H I t → 0 , Ω H C t → 0 , Ω H V 1 t → 0 , Ω H V 2 t → 0 Ω H R t → 0 as n → ∞ . Incorporating the triangular inequality and for any j , system (21) yields H S n + j t − H S n t ≤ ∑ i = n + 1 n + j F 1 j = F 1 n + 1 − F 1 n + k + 1 1 − F 1 , H h n + j t − H h n t ≤ ∑ i = n + 1 n + j F 2 j = F 2 n + 1 − F 2 n + k + 1 1 − F 2 , H E n + j t − H E n t ≤ ∑ i = n + 1 n + j F 3 j = F 3 n + 1 − F 3 n + k + 1 1 − F 3 , H I n + j t − H I n t ≤ ∑ i = n + 1 n + j F 4 j = F 4 n + 1 − F 4 n + k + 1 1 − F 4 , H C ( n + j ) t − H C n t ≤ ∑ i = n + 1 n + j F 5 j = F 5 n + 1 − F 5 n + k + 1 1 − T 5 , H V 1 n + j t − H V 1 n t ≤ ∑ i = n + 1 n + j F 6 j = F 6 n + 1 − F 6 n + k + 1 1 − F 6 , H V 2 n + j t − H V 2 n t ≤ ∑ i = n + 1 n + j F 7 j = F 7 n + 1 − F 7 n + k + 1 1 − F 7 H R n + j t − H R n t ≤ ∑ i = n + 1 n + j F 8 j = F 8 n + 1 − F 8 n + k + 1 1 − F 8 . where F i = 1 − k B k T i + k B k Γ k b k T i < 1 . Hence there exists a unique solution for system (1).

 Numerical analysis

In this section, we numerically investigate the behaviour of (1) using the Matlab program ODE45 Runge Kutta Method. The model is parameterized to investigate disease burden in a vulnerable population (specifically a population with hypertension) with double dose vaccination within the period February 24, 2021, to July 24, 2021 (from the beginning of the vaccination program in Ghana) (UNICEF). The 2021 Population and housing census revealed Ghana’s population is 30.8 million (Ghana Statistical Service) , and 5.27 million people estimated to have hypertension. 7 On this day, February 24, 2021, Ghana had 81,245 cumulative cases of COVID-19, with total deaths being 584 and 6,614 active cases. The recoveries stood at 74,047 (Worldometer). First and second doses had not started yet after receiving the vaccines. The first dose was rollout on March 2, 2021 (WHO) and second dose on May 7, 2021 (Ghana Health Service) which falls within the stipulated time. We assumed that the number of individuals exposed to the COVID-19 is twice the number of active cases. The number of hypertensive patients infected with COVID-19 is not known at the period, so we assume to be zero. Using the initial conditions N(0) = 30,800,000, H h (0) = 5,270,000, H E (0) = 2 ×6614 = 13228, H I (0) = 6614, H C (0) = 0, H V 1 (0) = 0, H V 2 (0) = 0, H R (0) = 74047, H S (0) = N(0) − H h (0) − H E (0) − H I (0) − H V1 − H V 2 − H R (0) = 25436111, and the parameter values given in Table 1, the results of the simulation are displayed in Figures 2– 9 for the period of 150 days within the stipulated period.

Description of the variables [Author's own table].
Parameter Description Estimate (per day) Source
σ New susceptible recruitment 29.08 41 , 46
σ 2 COVID-19 transmission rate 0.9 41
σ 4 Infectious rate 0.25 17 , 41
σ 1 natural mortality rate 0.4252912 ×10 −4 41
σ 5 COVID-19 disease-induced death rate 1.6728 ×10 −5 32 , 41
σ 6 Recovery rate of hypertension patients from COVID-19 1/14 41
σ 8 Hypertension disease-induced death rate 0.05 41 , 47
σ 7 hypertension prevalence rate 0.01 Assume
σ 3 Recovery rate of COVID-19 patients 1/14 41
σ 9 COVID-19 disease-induced death of hypertension patients 0.0144 48 , 49
ζ 1 First dose vaccination rate 0.12 21
ζ 3 Second dose vaccination rate 0.34 42
ζ 6 Loss of immunity 0.0167 21
Dynamics of the susceptible compartment. Dynamics of exposed compartment. Behaviour of the infected compartment. Dynamics of the COVID-19 patients with hypertension compartment. Dynamics of individuals with hypertension. Dynamics of the recovered compartment. Dynamics of first dose compartment. Dynamics of second dose compartment. Discussion

The resulting solutions of the system (1) for 5 different values of k ∈ [0,1] at step-size 0.2 are shown in Figures 2 through 9. The figures denote the susceptible, susceptible with hypertension, exposed, infected, COVID-19 infected individuals with hypertension, recovered, first- and second dose vaccination compartments, respectively. With the introduction of vaccines into the model, it can be observed that the number of vulnerable individuals drastically reduces within the first 50 days for all the operator values (see Figures 2 and 6). The integer model ( k = 1.0), shows the number of exposed, infected, and COVID-19 infected patients with hypertension reaching their highest peak, i.e., 150,000, 110,000, and 220,000 respectively, in the first 25 days (see Figures 3– 5). On the other hand, a reduction in the fractional operator value revealed the number of exposed individuals reaching an early peak and lowest minimum even before the first ten days. The dynamics of the first and second dose vaccinations are depicted in Figures 8 and 9 respectively. A lower fractional operator value reveals a higher number of individuals receiving the first dose in the first five days. This number declines steadily thereafter. There is an increase in the number of recovered individuals and those receiving the second dose for the first fifty days (see Figures 7 and 9). This accounts for the reduction in the vulnerable population. The fall in the number of recoveries and individuals with the second dose may be due to the loss of immunity in those compartments.

 Conclusion

In this study, the fractional-order derivative defined in the Atangana-Baleanu in the Caputo sense has been used to investigate a COVID-19 model that considers a population with hypertension as a primary condition and with double dose vaccination. We investigated, the solutions’ existence and distinctiveness. The numerical simulation showed different compartment dynamics for both integer and non-integer values of the fractional operator. The dynamics of the disease in the community were shown to be influenced by fractional-order derivatives. Contrary to the previous model proposed in Ref. 41, the vulnerable group (susceptible and susceptible with hypertension) saw a significant reduction in the number, which may be attributed to the double dose vaccination. We recommend a cost-effective analysis and optimal control model in future work.

 Data availability Statement Underlying data

The Ghana Health Service website, https://www.ghs.gov.gh/covid19, provided the data and information used to develop the mathematical model in this paper. This has been referenced in the text as. 50

Extended data

OSF: Output_data DOI:10.17605 / OSF.IO / K5PJ7 This project contains the following extended data:

MATLAB Command Window.pdf (Matlab output data that accompanied the numerical simulation figures)

Data are available under the terms of the Creative Commons Zero “No rights reserved” data waiver (CC0 1.0 Public domain dedication).

 Software availability

Source code available from: https://github.com/okyere2015/Matlab_codes/releases/tag/v2.0.1 Archived source code at the time of publication: https://doi.org/10.5281/zenodo.7671815.

License: Apache 2.0.

 Acknowledgement

The reference 51 has an old version of this work which was submitted as pre-print.

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Article ID 1943410. 10.1155/2020/1943410 Stokes EK Zambrano LD Anderson KN : Coronavirus disease 2019 case surveillance – United States, January 22–May 30, 2020. Morb. Mortal. Wkly. Rep. 69(2020):759–765. 32555134 10.15585/mmwr.mm6924e2 PMC7302472 Yang C Wang J : COVID-19 and underlying health conditions: A modelling investigation’. Math. Biosci. Eng. 2021;18(4):3790–3812. 34198413 10.3934/mbe.2021191 PMC8359646 Ghana health service: COVID-19 cases.(Accessed: 20th November, 2021). Reference Source Okyere S Prah JA Bonyah E : A model of COVID-19 with underlying health condition using fraction order derivative. arXiv:2201.11659 [q-bio.PE]. Reference Source 10.5256/f1000research.146781.r214799 Reviewer response for version 1 B Jeelani Shaikh Mohammadi 1 Referee https://orcid.org/0000-0002-8812-2859 Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia

Competing interests: No competing interests were disclosed.

 27 10 2023 Copyright: © 2023 B Jeelani Shaikh M 2023 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve-with-reservations

The author(s) make a research article on“Hypertension and COVID-19 fractional derivative model with double dose vaccination”. The article yields interesting findings, and the material is appropriate for this journal. The list below includes a few observations. I can support its indexing, however I advise considerable revisions before it is indexed.

The entire document needs to have its grammatical and typographical errors fixed. Authors ought to edit it.

Could you all provide a more specific research question or hypothesis that your study aims to address regarding the dynamics and treatment strategies with double dose vaccination. (specifically)

The writers need to remark on the results more effectively. The observed patterns should have a plausible physical explanation in addition to reporting what can be seen graphically in the figures. There needs to be more physical understanding of the discussion part.

What is the aim of this review? The novelty is not mentioned in abstract. 

How does the use of Atangana-Baleanu-Caputo fractional derivative enhance the model's ability to capture the complexity of covid model dynamics compared to conventional derivatives?

 The text lacks any validation of the numerical results. Include a comparison of the results to earlier publications. Additionally, it needs to be emphasized in the abstract.

At the end of introduction section, research questions that highlight the research gap and novelty of the proposed problem.

The range of defined parameters needs to be added.

 Write the conclusion more precise, It should highlight the novel finding in the current study. It should be point wise.

 In practical terms, how might the insights from your model influence clinical decision-making or the development of new therapeutic approaches 

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

No

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Partly

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Mathematical modeling, Fractional calculus, Differential equation, Numerical Analysis, Approximation theory,

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

 10.5256/f1000research.146781.r200955 Reviewer response for version 1 Kubra Khadija Tul 1 Referee Government College University Faisalabad, Faisalabad, Punjab, Pakistan

Competing interests: No competing interests were disclosed.

 5 10 2023 Copyright: © 2023 Kubra KT 2023 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve-with-reservations

This article's abstract reads well. The research, however, does not support the viewpoint emphasized in the abstract. Upon reading the abstract, one may find it interesting and persuasive. However, upon delving into the research itself, it becomes evident that the evidence presented does not align with the viewpoint highlighted in the abstract. The abstract may provide a concise and enticing overview of the topic, drawing readers in with its intriguing claims. Conversely, a closer examination of the research reveals that the data and analysis presented do not substantiate the assertions made in the abstract. It is important to critically evaluate the research itself rather than solely relying on the initial impression created by the abstract. This disconnect raises questions about the accuracy and validity of the claims made in the article. 

In the abstract, the researcher made this claim  "The study modifies a previous model proposed in the literature to include double dose vaccination and Atangana-Baleanu-Caputo fractional derivatives is used to solve the model", however, this research work shows no discernible modification. The lack of modification in the research work raises concerns about the credibility of the researcher's claim in the abstract. Without any visible changes to the model or inclusion of Atangana-Baleanu-Caputo fractional derivatives, it is unclear how the study supports or contributes to existing literature. This discrepancy calls into question the overall reliability and integrity of the article's findings. 

In the abstract, the researcher made this claim  "By using COVID-19 data for the period February 21, 2021 to July, 24 2021, the model is tested". The research, however, does not provide any such verified findings. This lack of tested results undermines the credibility of the study and raises doubts about the validity of the model. Without empirical evidence to support the claims made in the abstract, it becomes difficult to assess the practical implications and significance of the research. 

In the KEYWORDS section, the researcher uses the term  "Basic Reproduction Number". However, the proposed model does not include a calculation for the Basic Reproduction Number. Not even the researcher attempts any sort of qualitative analysis of the model's results. This work would be worthless without the inclusion of a qualitative analysis of the epidemic model. Without the inclusion of a calculation for the Basic Reproduction Number and the absence of any qualitative analysis of the model's results, the researcher's claim regarding the testing of the model using COVID-19 data becomes questionable. The lack of these crucial components significantly diminishes the value and reliability of the research findings. 

The introduction is written very poorly. The introduction lacks clarity and fails to provide a concise overview of the research topic. It lacks a clear article's statement and fails to establish the significance of the study. Additionally, the introduction does not effectively capture the reader's attention or provide a compelling rationale for why the research is important. This weak start may undermine the reader's confidence in the overall quality of the article. 

Not even the researcher makes any attempt at a qualitative analysis for the model. Not equilibria, basic reproduction number and their stability, senstivity of parameters any of these. This lack of qualitative analysis is a significant limitation as it prevents a deeper understanding of the model's behavior and its implications. Without this analysis, it becomes difficult to assess the model's applicability and relevance to real-world scenarios. Additionally, the absence of sensitivity analysis limits the researchers' ability to explore the robustness of their findings and evaluate the potential impact of different parameter values on their results. There is no value to the work done if only the symbols are changed, so the qualitative analysis is essential. 

In the Numerical Analysis section, the researcher claims that  "They numerically investigate the behaviour of (1) using the Matlab program ODE45 Runge Kutta Method". In that case, why do they propose the ABC scheme for the model if numerical investigations are performed using the ode45 function of Matlab? In other words, this directly contradicts the researcher's abstract's main statement. It is important to note that while the researcher claims to numerically investigate the behavior of (1) using the ode45 function, it does not necessarily mean that this method is the most suitable or accurate for the given model. The proposed ABC scheme may offer alternative advantages such as improved accuracy, efficiency, or robustness compared to ode45. Therefore, it is crucial for the researcher to provide a clear justification for why they propose the ABC scheme despite conducting numerical investigations with ode45. 

No any clear connection between discussion, conclusion with the point of view offered in abstract. It is important for the researcher to establish a clear connection between their discussion, conclusion, and the point of view offered in the abstract. This ensures that the research findings are coherent and align with the overall objectives of the study. It is important for the researcher to address this discrepancy and provide a comprehensive explanation that ties all aspects of the study together.

 Due to the aforementioned gaps, I cannot recommend accepting your article for indexing. The researcher should revise their work to ensure a seamless flow of information and a strong connection between all sections. Additionally, they should provide a comprehensive explanation that effectively ties together all aspects of the study, addressing any discrepancies that may arise. Therefore, it is crucial for the researcher to revise and strengthen these aspects before resubmitting the article for indexing consideration.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.

Are all the source data underlying the results available to ensure full reproducibility?

No

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Partly

Are sufficient details of methods and analysis provided to allow replication by others?

No

Reviewer Expertise:

Mathematical Modelling, Heat and Mass transfer, Numerical and analytical solutions of ODEs and PDEs. Fluid mechanics, Numerical Simulations, Geophysics

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.