Hypertension and COVID-19 fractional derivative model with double dose vaccination

Samuel Okyere; Joseph Ackora-Prah; Ebenezer Bonyah; Bennedict Barnes; Maxwell Akwasi Boateng; Ishmael Takyi; Samuel Akwasi Adarkwa

doi:10.12688/f1000research.133768.1

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Research Article

Hypertension and COVID-19 fractional derivative model with double dose vaccination

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

Samuel Okyere ¹, Joseph Ackora-Prah¹, Ebenezer Bonyah², [...] Bennedict Barnes¹, Maxwell Akwasi Boateng¹, Ishmael Takyi¹, Samuel Akwasi Adarkwa³

Samuel Okyere ¹, Joseph Ackora-Prah¹, [...] Ebenezer Bonyah², Bennedict Barnes¹, Maxwell Akwasi Boateng¹, Ishmael Takyi¹, Samuel Akwasi Adarkwa³

PUBLISHED 15 May 2023

Author details Author details

¹ Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ashanti Region, Ghana
² Mathematics Education, Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development, Kumasi, Ashanti Region, Ghana
³ Statistical Sciences, Kumasi Technical University, Kumasi, Ashanti Region, Ghana

Samuel Okyere
Roles: Conceptualization, Data Curation, Formal Analysis, Investigation, Methodology, Resources, Software, Validation, Writing – Original Draft Preparation

Joseph Ackora-Prah
Roles: Conceptualization, Resources, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

Ebenezer Bonyah
Roles: Conceptualization, Formal Analysis, Resources, Software, Supervision

Bennedict Barnes
Roles: Conceptualization, Resources, Software, Validation

Maxwell Akwasi Boateng
Roles: Conceptualization, Resources, Software, Validation

Ishmael Takyi
Roles: Conceptualization, Resources, Software, Supervision, Validation, Writing – Review & Editing

Samuel Akwasi Adarkwa
Roles: Conceptualization, Data Curation, Resources, Software, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS

This article is included in the Emerging Diseases and Outbreaks gateway.

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.

Keywords

Hypertension, Fractional Derivative, Basic Reproduction Number, COVID-19, Double Vaccination

Corresponding author: Samuel Okyere

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

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

How to cite: Okyere S, Ackora-Prah J, Bonyah E et al. Hypertension and COVID-19 fractional derivative model with double dose vaccination [version 1; peer review: 2 approved with reservations]. F1000Research 2023, 12:495 (https://doi.org/10.12688/f1000research.133768.1) First published: 15 May 2023, 12:495 (https://doi.org/10.12688/f1000research.133768.1) Latest published: 15 May 2023, 12:495 (https://doi.org/10.12688/f1000research.133768.1)

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 Giangreco³³ 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_V₁, Individuals with second dose vaccination H_V₂, and Recovered individuals H_R. Figure 1 displays the model’s schematic.

Figure 1. Dynamic flow of the Hypentension-COVID-19 model with double dose vaccination.

The system is described below using the fractional-order derivatives.

(1)

\begin{array}{l} _{k}^{ABC} D_{t}^{k} H_{S} = σ^{k} + ξ_{6}^{k} H_{R} - σ_{2}^{k} (\frac{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} (\frac{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} (\frac{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} (\frac{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} (\frac{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} . \end{array}

with initial conditions

H_S(0) = H_S₀, H_h(0) = H_h₀, H_E(0) = H_E₀, H_I(0) = H_I₀, H_C(0) = H_C₀, H_V₁(0) = H_V₁, H_V₂(0) = H_V₂, H_R(0) = H_R₀.

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

(2)

{{}_{k}^{C}D}_{t}^{k} h (t) = \frac{1}{Γ (1 - k)} \int_{0}^{t} {(τ - q)}^{- k} h (p) dp, 0 < k \leq 1 .

where k is the fractional operator.

Definition 2: Let $h \in T^{1} (x_{1}, x_{2}), x_{2} > x_{1} \in [0, 1]$ , then, the new Caputo fractional derivative as stated in Ref. 43 is:

(3)

D_{t}^{k} h (t) = \frac{H (k)}{(1 - k)} \int_{x_{2}}^{t} h^{'} (x) exp (- k (\frac{(t - x)}{1 - k})) dx,

where H(k) denotes a normalized function such that H(0) = H(1) = 1. However, if the function

h \notin T^{1} (x_{1}, x_{2})

then equation (3) has the form

(4)

D_{t}^{k} h (t) = \frac{kH (k)}{1 - k} \int_{x_{2}}^{t} h (t) - h (x) exp (- k (\frac{(t - x)}{1 - k})) dx,

If we let $ζ = \frac{1 - k}{k} \in [0 \infty]$ then $= \frac{1}{1 - ζ} \in [0, 1]$ , the above equation assumes the form

(5)

D_{t}^{ζ} h (t) = \frac{N (ζ)}{ζ} \int_{x_{2}}^{t} h^{'} (x) exp (- (\frac{(t - x)}{ζ})) dx, N (0) = N (\infty) = 1 .

Definition 3: Let $h \in T^{1} (x_{1}, x_{2}), x_{2} > x_{1} \in [0, 1]$ , then, the new fractional derivative as defined by Ref. 43 is given as:

(6)

{{}_{k}^{ABC}D}_{t}^{k} h (t) = \frac{H (k)}{(1 - k)} \int_{x_{2}}^{t} h^{'} (x) E_{k} (- k (\frac{{(t - x)}^{k}}{1 - k})) dx,

Laplace transform of equation (6) is given by

(7)

L (({}_{0}^{ABC}D {}_{t}^{k}{h (t)}) (p)) = \frac{H (k)}{1 - k} \frac{L ((h (t))) - p^{k - 1} h (0)}{p^{k} + \frac{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 $h \in T^{1} (x_{1}, x_{2}), x_{2} > x_{1} \in [0, 1]$ , then, the new fractional derivative as defined by Ref. 43 is given as:

(8)

{{}_{k}^{ABR}D}_{t}^{k} h (t) = \frac{H (k)}{1 - k} \frac{d}{dt} \int_{x_{2}}^{t} h (x) E_{k} (- k (\frac{{(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:

(9)

L (({}_{0}^{ABR}D {}_{t}^{k}{h (t)}) (p)) = \frac{H (k)}{1 - k} \frac{p^{k} L ((h (t))) (p)}{p^{k} + \frac{k}{1 - k}}

Existence and uniqueness of the solutions

Let G(T) denote a banach space, where T = [0, b], and $M = G (T) \times G (T) \times G (T) \times G (T) \times G (T) \times G (T) \times G (T) \times G (T)$ 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 $‖H_{S}‖ = {Sup}_{π \in T} |H_{S}|, ‖H_{h}‖ = {Sup}_{π \in T} |H_{h}|, ‖H_{E}‖ = {Sup}_{π \in T} |H_{E}|, ‖H_{I}‖ = {Sup}_{π \in T} |H_{I}|, ‖H_{C}‖ = {Sup}_{π \in T} |H_{C}|, ‖H_{V_{1}}‖ = {Sup}_{π \in T} |H_{V_{1}}|, ‖H_{V_{2}}‖ = {Sup}_{π \in T} |H_{V_{2}}|, ‖H_{R}‖ = {Sup}_{π \in T} |H_{R}|$ . Using the Atangana-Baleanu-Caputo integral operator on the system (1), we have

(10)

\begin{array}{l} H_{S} (t) - H_{S} (0) =_{k}^{ABC} D_{t}^{k} H_{S} (t) \{σ^{k} + ξ_{6}^{k} H_{R} - σ_{2}^{k} (\frac{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} (\frac{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} (\frac{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} (\frac{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} (\frac{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}\} . \end{array}

Applying Liouville and Caputo (LC) definition of fractional-order derivatives gives,

(11)

\begin{array}{l} H_{S} (t) - H_{S} (0) = \frac{1 - k}{B (k)} π_{1} (k, t, H_{S} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{1} (k, τ, H_{S} (τ)) d τ, \\ H_{h} (t) - H_{h} (0) = \frac{1 - k}{B (k)} π_{2} (k, t, H_{h} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{2} (k, τ, H_{h} (τ)) d τ, \\ H_{E} (t) - H_{E} (0) = \frac{1 - k}{B (k)} π_{3} (k, t, H_{E} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{3} (k, τ, H_{E} (τ)) d τ, \\ H_{I} (t) - H_{I} (0) = \frac{1 - k}{B (k)} π_{4} (k, t, H_{I} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{4} (k, τ, H_{I} (τ)) d τ, \\ H_{C} (t) - H_{C} (0) = \frac{1 - k}{B (k)} π_{5} (k, t, H_{C} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{5} (k, τ, H_{C} (τ)) d τ, \\ H_{V_{1}} (t) - H_{V_{1}} (0) = \frac{1 - k}{B (k)} π_{6} (k, t, H_{V_{1}} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{6} (k, τ, H_{V_{1}} (τ)) d τ, \\ H_{V_{2}} (t) - H_{V_{2}} (0) = \frac{1 - k}{B (k)} π_{7} (k, t, H_{V_{2}} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{7} (k, τ, H_{V_{2}} (τ)) d τ, \\ H_{R} (t) - H_{R} (0) = \frac{1 - k}{B (k)} π_{8} (k, t, H_{R} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{8} (k, τ, H_{R} (τ)) d τ, \end{array}

where

(12)

\{\begin{cases} π_{1} (k, τ, H_{S} (t)) = σ^{k} + ξ_{6}^{k} H_{R} - σ_{2}^{k} (\frac{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} (\frac{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} (\frac{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} (\frac{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} (\frac{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} . \end{cases}

The ABC derivatives only meets the Lipschitz requirement⁴⁴^,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,

(13)

‖π_{1} (k, t, H_{S} (t)) - π_{1} (k, t, H_{S}^{*} (t))‖ = ‖- [σ_{2}^{k} (\frac{H_{I} + H_{C}}{N}) H_{S} + (σ_{1}^{k} + σ_{7}^{k} + ξ_{1}^{k})] (H_{S} (t) - H_{S}^{*} (t))‖

Considering

(14)

T_{1} = ‖- [σ_{2}^{k} (\frac{H_{I} + H_{C}}{N}) H_{S} + (σ_{1}^{k} + σ_{7}^{k} + ξ_{1}^{k})]‖,

Equation (13) simplifies to

(15)

‖π_{1} (k, t, H_{S} (t)) - π_{1} (k, t, H_{S}^{*} (t))‖ \leq T_{1} ‖ (H_{S} (t) - H_{S}^{*} (t)) ‖

Similarly,

(16)

\begin{array}{l} ‖ π_{2} (k, t, H_{h} (t)) - π_{2} (k, t, H_{h}^{*} (t)) ‖ \leq T_{2} ‖ H_{h} (t) - H_{h}^{*} (t) ‖, \\ ‖ π_{3} (k, t, H_{E} (t)) - π_{3} (k, t, H_{E}^{*} (t)) ‖ \leq T_{3} ‖ H_{E} (t) - H_{E}^{*} (t) ‖, \\ ‖ π_{4} (k, t, H_{I} (t)) - π_{4} (k, t, H_{I}^{*} (t)) ‖ \leq T_{4} ‖ H_{I} (t) - H_{I}^{*} (t) ‖, \\ ‖ π_{5} (k, t, H_{C} (t)) - π_{5} (k, t, H_{C}^{*} (t)) ‖ \leq T_{5} ‖ H_{C} (t) - H_{C}^{*} (t) ‖, \\ ‖ π_{6} (k, t, H_{V_{1}} (t)) - π_{6} (k, t, H_{V_{1}}^{*} (t)) ‖ \leq 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)) ‖ \leq T_{7} ‖ H_{V_{2}} (t) - H_{V_{2}}^{*} (t) ‖, \\ ‖ π_{8} (k, t, H_{R} (t)) - π_{8} (k, t, H_{R}^{*} (t)) ‖ \leq T_{8} ‖ H_{R} (t) - H_{R}^{*} (t) ‖ . \end{array}

where

(17)

\begin{array}{l} T_{2} = ‖- σ_{2}^{k} (\frac{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} (\frac{H_{I} + H_{C}}{N}) - (σ_{1}^{k} + ξ_{3}^{k})‖, \\ T_{7} = ‖- σ_{2}^{k} (\frac{H_{I} + H_{C}}{N}) - σ_{1}^{k}‖, \\ T_{8} = ‖- (ξ_{6}^{k} + σ_{1}^{k})‖, \end{array}

Lipschitz condition is now valid. Now repeatedly applying system (10) gives,

(18)

\begin{array}{l} {H_{S}}_{n} (t) - H_{S} (0) = \frac{1 - k}{B (k)} π_{1} (k, t, {H_{S}}_{(n - 1)} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{1} (k, τ, {H_{S}}_{(n - 1)} (τ)) dτ, \\ {H_{h}}_{n} (t) - H_{h} (0) = \frac{1 - k}{B (k)} π_{2} (k, t, {H_{h}}_{(n - 1)} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{2} (k, τ, {H_{h}}_{(n - 1)} (τ)) dτ, \\ {H_{E}}_{n} (t) - H_{E} (0) = \frac{1 - k}{B (k)} π_{3} (k, t, {H_{E}}_{(n - 1)} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{3} (k, τ, {H_{E}}_{n - 1} (τ)) dτ, \\ {H_{I}}_{n} (t) - H_{I} (0) = \frac{1 - k}{B (k)} π_{4} (k, t, {H_{I}}_{(n - 1)} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{4} (k, τ, {H_{I}}_{(n - 1)} (τ)) dτ, \\ {H_{C}}_{n} (t) - H_{C} (0) = \frac{1 - k}{B (k)} π_{5} (k, t, {H_{C}}_{(n - 1)} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{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) = \frac{1 - k}{B (k)} π_{6} (k, t, {H_{V_{1}}}_{(n - 1)} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{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) = \frac{1 - k}{B (k)} π_{7} (k, t, {H_{V_{2}}}_{(n - 1)} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{7} (k, τ, {H_{V_{2}}}_{(n - 1)} (τ)) dτ, \\ {H_{R}}_{n} (t) - H_{R} (0) = \frac{1 - k}{B (k)} π_{8} (k, t, {H_{R}}_{(n - 1)} (t)) + \frac{k}{B (k) Γ (k)} \times \int_{0}^{t} {(t - τ)}^{k - 1} π_{8} (k, τ, {H_{R}}_{(n - 1)} (τ)) dτ, \end{array}

Difference of consecutive terms yields

(19)

\begin{matrix} Ω_{{H_{S}}_{n}} (t) = {H_{S}}_{n} (t) - {H_{S}}_{(n - 1)} (t) = \frac{1 - k}{B (k)} (π_{1} (k, t, {H_{S}}_{(n - 1)} (t)) - π_{1} (k, t, {H_{S}}_{(n - 2)} (t))) \\ + \frac{k}{B (k) Γ (k)} \int_{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) = \frac{1 - k}{B (k)} (π_{2} (k, t, {H_{h}}_{(n - 1)} (t)) - π_{2} (k, t, {H_{h}}_{(n - 2)} (t))) \\ + \frac{k}{B (k) Γ (k)} \int_{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) = \frac{1 - k}{B (k)} (π_{3} (k, t, {H_{E}}_{(n - 1)} (t)) - π_{3} (k, t, {H_{E}}_{(n - 2)} (t))) \\ + \frac{k}{B (k) Γ (k)} \int_{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) = \frac{1 - k}{B (k)} (π_{4} (k, t, {H_{I}}_{(n - 1)} (t)) - π_{4} (k, t, {H_{I}}_{(n - 2)} (t))) \\ + \frac{k}{B (k) Γ (k)} \int_{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) = \frac{1 - k}{B (k)} (π_{5} (k, t, {H_{C}}_{(n - 1)} (t)) - π_{4} (k, t, {H_{C}}_{(n - 2)} (t))) \\ + \frac{k}{B (k) Γ (k)} \int_{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) = \frac{1 - k}{B (k)} (π_{6} (k, t, {H_{V_{1}}}_{(n - 1)} (t)) - π_{6} (k, t, {H_{V_{1}}}_{(n - 2)} (t))) \\ + \frac{k}{B (k) Γ (k)} \int_{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) = \frac{1 - k}{B (k)} (π_{7} (k, t, {H_{V_{2}}}_{(n - 1)} (t)) - π_{7} (k, t, {H_{V_{2}}}_{(n - 2)} (t))) \\ + \frac{k}{B (k) Γ (k)} \int_{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) = \frac{1 - k}{B (k)} (π_{8} (k, t, {H_{R}}_{(n - 1)} (t)) - π_{8} (k, t, {H_{R}}_{(n - 2)} (t))) \\ + \frac{k}{B (k) Γ (k)} \int_{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, \end{matrix}

where

${H_{S}}_{n} (t) = \sum_{i = 0}^{n} Ω_{{H_{S}}_{n}} (t), {H_{h}}_{n} (t) = \sum_{i = 0}^{n} Ω_{{H_{h}}_{n}} (t), {H_{E}}_{n} (t) = \sum_{i = 0}^{n} Ω_{{H_{E}}_{n}} (t), {H_{I}}_{n} (t) = \sum_{i = 0}^{n} Ω_{{H_{I}}_{n}} (t), {H_{C}}_{n} (t) = \sum_{i = 0}^{n} Ω_{{H_{C}}_{n}} (t),$ ${H_{V_{1}}}_{n} (t) = \sum_{i = 0}^{n} Ω_{{H_{V_{1}}}_{n}} (t), {H_{V_{2}}}_{n} (t) = \sum_{i = 0}^{n} Ω_{{H_{V_{2}}}_{n}} (t), {H_{R}}_{n} (t) = \sum_{i = 0}^{n} Ω_{{H_{R}}_{n}} (t)$ . Taking into consideration equations (15)–(16) and considering $Ω_{{H_{S}}_{n - 1}} (t) = {H_{S}}_{(n - 1)} (t) - {H_{S}}_{(n - 2)} (t), Ω_{{H_{h}}_{(n - 1)}} (t) = {H_{h}}_{(n - 1)} (t) - {H_{h}}_{(n - 2)} (t),$ $Ω_{{H_{E}}_{(n - 1)}} (t) = {H_{E}}_{(n - 1)} (t) - {H_{E}}_{(n - 2)} (t), Ω_{{H_{I}}_{(n - 1)}} (t) = {H_{I}}_{(n - 1)} (t) - {H_{I}}_{(n - 2)} (t), Ω_{{H_{C}}_{(n - 1)}} (t) = {H_{C}}_{(n - 1)} (t) - {H_{C}}_{(n - 2)} (t),$ $Ω_{{H_{V_{1}}}_{(n - 1)}} (t) = {H_{V_{1}}}_{(n - 1)} (t) - {H_{V_{1}}}_{(n - 2)} (t), Ω_{{H_{V_{2}}}_{(n - 1)}} (t) = {H_{V_{2}}}_{(n - 1)} (t) - {H_{V_{2}}}_{(n - 2)} (t), Ω_{{H_{R}}_{(n - 1)}} (t) = {H_{R}}_{(n - 1)} (t) - {H_{R}}_{(n - 2)} (t),$

(20)

\begin{array}{l} ‖Ω_{{H_{S}}_{n}} (t)‖ \leq \frac{1 - k}{B (k)} T_{1} ‖Ω_{{H_{S}}_{(n - 1)}} (t)‖ \frac{k}{B (k) Γ (k)} T_{1} \times \int_{0}^{t} {(t - τ)}^{k - 1} ‖Ω_{{H_{S}}_{(n - 1)}} (τ)‖ dτ, \\ ‖Ω_{{H_{h}}_{n}} (t)‖ \leq \frac{1 - k}{B (k)} T_{2} ‖Ω_{{H_{h}}_{(n - 1)}} (t)‖ \frac{k}{B (k) Γ (k)} T_{2} \times \int_{0}^{t} {(t - τ)}^{k - 1} ‖Ω_{{H_{h}}_{(n - 1)}} (τ)‖ dτ, \\ ‖Ω_{{H_{E}}_{n}} (t)‖ \leq \frac{1 - k}{B (k)} T_{3} ‖Ω_{{H_{E}}_{(n - 1)}} (t)‖ \frac{k}{B (k) Γ (k)} T_{3} \times \int_{0}^{t} {(t - τ)}^{k - 1} ‖Ω_{{H_{E}}_{(n - 1)}} (τ)‖ dτ, \\ ‖Ω_{{H_{I}}_{n}} (t)‖ \leq \frac{1 - k}{B (k)} T_{4} ‖Ω_{{H_{I}}_{(n - 1)}} (t)‖ \frac{k}{B (k) Γ (k)} T_{4} \times \int_{0}^{t} {(t - τ)}^{k - 1} ‖Ω_{{H_{I}}_{(n - 1)}} (τ)‖ dτ, \\ ‖Ω_{{H_{C}}_{n}} (t)‖ \leq \frac{1 - k}{B (k)} T_{5} ‖Ω_{{H_{C}}_{(n - 1)}} (t)‖ \frac{k}{B (k) Γ (k)} T_{5} \times \int_{0}^{t} {(t - τ)}^{k - 1} ‖Ω_{{H_{C}}_{(n - 1)}} (τ)‖ dτ, \\ ‖Ω_{{H_{V_{1}}}_{n}} (t)‖ \leq \frac{1 - k}{B (k)} T_{6} ‖Ω_{{H_{V_{1}}}_{(n - 1)}} (t)‖ \frac{k}{B (k) Γ (k)} T_{6} \times \int_{0}^{t} {(t - τ)}^{k - 1} ‖Ω_{{H_{V_{1}}}_{(n - 1)}} (τ)‖ dτ, \\ ‖Ω_{{H_{V_{2}}}_{n}} (t)‖ \leq \frac{1 - k}{B (k)} T_{7} ‖Ω_{{H_{V_{2}}}_{(n - 1)}} (t)‖ \frac{k}{B (k) Γ (k)} T_{7} \times \int_{0}^{t} {(t - τ)}^{k - 1} ‖Ω_{{H_{V_{2}}}_{(n - 1)}} (τ)‖ dτ, \\ ‖Ω_{{H_{R}}_{n}} (t)‖ \leq \frac{1 - k}{B (k)} T_{8} ‖Ω_{{H_{R}}_{(n - 1)}} (t)‖ \frac{k}{B (k) Γ (k)} T_{8} \times \int_{0}^{t} {(t - τ)}^{k - 1} ‖Ω_{{H_{R}}_{n (n - 1)}} (τ)‖ dτ, \end{array}

Theorem: The system (1) has a unique solution for $t \in [0, b]$ subject to the condition $\frac{1 - k}{B (k)} T_{i} + \frac{k}{B (k) Γ (k)} b^{k} n_{i} < 1, i = 1, 2, 3, . \dots \dots, 8$ 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.

(21)

\begin{array}{l} ‖{Ω_{H_{S}}}_{n} (t)‖ \leq ‖{H_{S}}_{o} (t)‖ {(\frac{1 - k}{B (k)} T_{1} + \frac{k b^{k}}{B (k) Γ (k)} T_{1})}^{n}, \\ ‖{Ω_{H_{h}}}_{n} (t)‖ \leq ‖{H_{h}}_{o} (t)‖ {(\frac{1 - k}{B (k)} T_{2} + \frac{k b^{k}}{B (k) Γ (k)} T_{2})}^{n}, \\ ‖{Ω_{H_{E}}}_{n} (t)‖ \leq ‖H_{E_{O}} (t)‖ {(\frac{1 - k}{B (k)} T_{3} + \frac{k b^{k}}{B (k) Γ (k)} T_{3})}^{n}, \\ ‖{Ω_{I}}_{n} (t)‖ \leq ‖{H_{I}}_{o} (t)‖ {(\frac{1 - k}{B (k)} T_{4} + \frac{k b^{k}}{B (k) Γ (k)} T_{4})}^{n}, \\ ‖{Ω_{H_{C}}}_{n} (t)‖ \leq ‖H_{C_{O}} (t)‖ {(\frac{1 - k}{B (k)} T_{5} + \frac{k b^{k}}{B (k) Γ (k)} T_{5})}^{n}, \\ ‖{Ω_{H_{V_{1}}}}_{n} (t)‖ \leq ‖{H_{V_{1}}}_{o} (t)‖ {(\frac{1 - k}{B (k)} T_{6} + \frac{k b^{k}}{B (k) Γ (k)} T_{6})}^{n}, \\ ‖Ω_{H_{V_{2_{n}}}} (t)‖ \leq ‖{H_{V_{2}}}_{o} (t)‖ {(\frac{1 - k}{B (k)} T_{7} + \frac{k b^{k}}{B (k) Γ (k)} T_{7})}^{n}, \\ ‖Ω_{H_{R_{n}}} (t)‖ \leq ‖H_{R_{O}} (t)‖ {(\frac{1 - k}{B (k)} T_{8} + \frac{k b^{k}}{B (k) Γ (k)} T_{8})}^{n}, \end{array}

and

$‖Ω_{H_{S}} (t)‖ \to 0, ‖Ω_{H_{h}} (t)‖ \to 0, ‖Ω_{H_{E}} (t)‖ \to 0, ‖Ω_{H_{I}} (t)‖ \to 0, ‖Ω_{H_{C}} (t)‖ \to 0, ‖Ω_{H_{V_{1}}} (t)‖ \to 0, ‖Ω_{H_{V_{2}}} (t)‖ \to 0$ $||Ω_{H_{R}} (t)|| \to 0$ as $n \to \infty$ . Incorporating the triangular inequality and for any $j$ , system (21) yields

(22)

\begin{array}{l} ‖{H_{S}}_{(n + j)} (t) - {H_{S}}_{n} (t)‖ \leq \sum_{i = n + 1}^{n + j} F_{1}^{j} = \frac{F_{1}^{n + 1} - F_{1}^{n + k + 1}}{1 - F_{1}}, \\ ‖{H_{h}}_{(n + j)} (t) - {H_{h}}_{n} (t)‖ \leq \sum_{i = n + 1}^{n + j} F_{2}^{j} = \frac{F_{2}^{n + 1} - F_{2}^{n + k + 1}}{1 - F_{2}}, \\ ‖H_{E (n + j)} (t) - {H_{E}}_{n} (t)‖ \leq \sum_{i = n + 1}^{n + j} F_{3}^{j} = \frac{F_{3}^{n + 1} - F_{3}^{n + k + 1}}{1 - F_{3}}, \\ ‖H_{I (n + j)} (t) - H_{I_{n}} (t)‖ \leq \sum_{i = n + 1}^{n + j} F_{4}^{j} = \frac{F_{4}^{n + 1} - F_{4}^{n + k + 1}}{1 - F_{4}}, \\ ‖{H_{C}}_{(n + j)} (t) - {H_{C}}_{n} (t)‖ \leq \sum_{i = n + 1}^{n + j} F_{5}^{j} = \frac{F_{5}^{n + 1} - F_{5}^{n + k + 1}}{1 - T_{5}}, \\ ‖{H_{V_{1}}}_{(n + j)} (t) - {H_{V_{1}}}_{n} (t)‖ \leq \sum_{i = n + 1}^{n + j} F_{6}^{j} = \frac{F_{6}^{n + 1} - F_{6}^{n + k + 1}}{1 - F_{6}}, \\ ‖{H_{V_{2}}}_{(n + j)} (t) - {H_{V_{2}}}_{n} (t)‖ \leq \sum_{i = n + 1}^{n + j} F_{7}^{j} = \frac{F_{7}^{n + 1} - F_{7}^{n + k + 1}}{1 - F_{7}} \\ ‖{H_{R}}_{(n + j)} (t) - {H_{R}}_{n} (t)‖ \leq \sum_{i = n + 1}^{n + j} F_{8}^{j} = \frac{F_{8}^{n + 1} - F_{8}^{n + k + 1}}{1 - F_{8}} . \end{array}

where

F_{i} = \frac{1 - k}{B (k)} T_{i} + \frac{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_V₁(0) = 0, H_V₂(0) = 0, H_R(0) = 74047, H_S(0) = N(0) − H_h(0) − H_E(0) − H_I(0) − H_V1 − H_V₂ − 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.

Table 1. Description of the variables [Author's own table].

Parameter	Description	Estimate (per day)	Source
σ	New susceptible recruitment	29.08	⁴¹^,⁴⁶
σ₂	COVID-19 transmission rate	0.9	⁴¹
σ₄	Infectious rate	0.25	¹⁷^,⁴¹
σ₁	natural mortality rate	0.4252912 ×10⁻⁴	⁴¹
σ₅	COVID-19 disease-induced death rate	1.6728 ×10⁻⁵	³²^,⁴¹
σ₆	Recovery rate of hypertension patients from COVID-19	1/14	⁴¹
σ₈	Hypertension disease-induced death rate	0.05	⁴¹^,⁴⁷
σ₇	hypertension prevalence rate	0.01	Assume
σ₃	Recovery rate of COVID-19 patients	1/14	⁴¹
σ₉	COVID-19 disease-induced death of hypertension patients	0.0144	⁴⁸^,⁴⁹
ζ₁	First dose vaccination rate	0.12	²¹
ζ₃	Second dose vaccination rate	0.34	⁴²
ζ₆	Loss of immunity	0.0167	²¹

Figure 2. Dynamics of the susceptible compartment.

Figure 3. Dynamics of exposed compartment.

Figure 4. Behaviour of the infected compartment.

Figure 5. Dynamics of the COVID-19 patients with hypertension compartment.

Figure 6. Dynamics of individuals with hypertension.

Figure 7. Dynamics of the recovered compartment.

Figure 8. Dynamics of first dose compartment.

Figure 9. 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 reference⁵¹ has an old version of this work which was submitted as pre-print.

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