Approximate Analytical Solution of an Extended COVID-19 Model Incorporating Two-Dose Vaccination and Physical Control Measures

 Corresponding author: SC Nwasuka Copyright © 2025 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0. Approximate Analytical Solution of an Extended COVID-19 Model Incorporating TwoDose Vaccination and Physical Control Measures Stanley Chinwekele Nwasuka 1, *, Ugochukwu Anuliobi Osisiogu 2, Efor Theresa Ebele 2 and Christie Ishola 3 1 Department of Mathematics and Computer Science, faculty of science, Clifford University, Owerrinta, Abia State.Nigeria. 2 Department of Industrial Mathematics, faculty of Physical Science. Ebonyi State University Abakaliki, Ebonyi State Nigeria 3 Department of Mathematics, faculty of Science National Open university of Nigeria, Nigeria. World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 Publication history: Received on 10 May 2025; revised on 16 June 2025; accepted on 19 June 2025 Article DOI: https://doi.org/10.30574/wjarr.2025.26.3.2344 Abstract The COVID-19 pandemic, which severely disrupted the global economy, remains a vital area of study for effective preparedness against future epidemics. The emergence of different variants has led to successive waves of the disease, prompting the development of numerous mathematical models. This study investigates an extended COVID-19 model that incorporates both first and second doses of vaccination as control strategies, alongside previously established physical preventive measures. The model is demonstrated to be mathematically and epidemiologically well-posed, with the existence and uniqueness of solutions to the state system established prior to analysis. Using the Next-Generation Matrix method, the control reproduction number was derived. Analytical results indicate that the disease-free equilibrium is locally and globally asymptotically stable when the control reproduction number, RcR_cRc, is less than one, and unstable when it exceeds one. Sensitivity analysis was conducted to determine the influence of key parameters on Rc. Findings highlight that improving compliance with hand sanitizing, social distancing, mask usage, testing, isolation, and vaccination significantly aids disease control. Conversely, reducing the rate of contact with exposed individuals, infectiousness development, and transmission probabilities also contributes to containment. Numerical simulations further illustrate the impact of these control measures, emphasizing the effectiveness of vaccination and adherence to physical protocols. The study recommends promoting vaccination and reinforcing compliance with physical preventive measures to mitigate the spread of COVID-19. Keywords: Dual Dose Vaccination; Mathematical modeling; Physical Control Measures; Disease Dynamics; Covid-19; Approximate Analytical Solution 1. Introduction The 2019–20 coronavirus pandemic was caused by the infectious disease corona virus (COVID-19), which was initially discovered in Wuhan, the Chinese capital, in 2019 [48, 38]. On December 31, 2019, it was first reported to the World Health Organization [48].Fever, a dry cough, and breathing problems are the most typical symptoms of Covid-19, while muscle pain, sputum production, diarrhea, and sore throat are less typical [38].Since the discovery of this virus, numerous investigations and studies, including mathematical models to comprehend the origin and transmission of the virus, have been conducted. Nigeria is one of the 210 countries affected globally. The first case was confirmed in Lagos State on 27 February 2020. This index case was a 44-year old man, an Italian citizen who returned from Milan, Italy, on 24 February and presented at a health facility on 26 February 2020 [38]. As of 3 May 2020, 2,558 cases have been reported in the country across 35 states and the Federal Capital Territory (FCT). Of these numbers, 1,767 (69 %) are male, the age-group 21 – 30 years World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1872 were the most affected (23%), 210 (8%) had international travel history; 400 (15.6%) cases have been discharged, and 87 deaths were recorded, bringing the case fatality rate (CFR) of confirmed cases to 3.4%, with a range from 015.2% by region. Prior to report of the COVID-19 outbreak in Africa, the WHO identified a strong link between the continent and China and has sent out guidelines on preparedness for the outbreak. Nigeria is one of the thirteen top countries identified as high risk for COVID-19 importation based on either direct link or high travel volume to and from China. The WHO also advised that countries develop capacity to promptly detect cases that will enable them to contain the outbreak early so that the health system is not overwhelmed [38, 48].It is worth noting that as of April 13, 2024, the corona virus tracker is no longer being updated due to the unfeasibility of providing statistically valid global totals, as some countries have now stopped reporting. However, historical data remain accessible. As at last updated in April 2024 Nigeria has the following records: Reported cases: 267,188. Deaths: 3155. Recovered: 259,953[35. 38,48]. Covid-19 is still present in Nigeria. As of May 29, 2025, Nigerian recorded 1,565 new covid-19 cases, bringing the total to 95,934confirmed cases. Lagos State has been significantly affected, with 807 new cases reported on the same date [35. 38,48]. The [38] continues to monitor the situation, reporting six Covid-19 deaths in the last 24 hours. 2. Material and methods In this work, we present a deterministic mathematical model with ten (10) human compartments made up of Vaccination V1, Vaccination V2, Susceptible humans, 𝑆, Exposed humans, 𝐸, Quarantined Humans, 𝑄, Undetected Asymptomatic Infectious Humans, Undetected symptomatic infectious humans, 𝐼, Undetected symptomatic infectious humans under self-medication, 𝑀, Detected and hospitalized infectious humans (via testing), 𝐼𝐻 and Recovered humans, 𝑅. The model is set up to show the effects of two doses of vaccination on the health burden of the disease. The following assumptions are made: • Demographic features such as natural birth and death rates are incorporated into the model as can be seen in most recent works such as [34, 36, 40, 41,45]. • A proportion of the susceptible humans recruited into the system is taken to be vaccinated and denoted by 𝑎. • The susceptible humans take the first dose of covid-19 at the rate 𝜅1 while those with the initial vaccination receives the second dose at the rate 𝜅2. • Those with second dose vaccination that receives covid-19 booster are represented by the proportion, 𝜈. They are migrated to the recovered class since they can not get infected for a long while [41, 49]. Table 1 Parameters and interpretation Parameters Values Λ Rate of recruitment of humans 𝑎1 Proportion of recruited humans that receive first dose of vaccination 𝑎2 Proportion of recruited humans that receive second dose of vaccination 𝛿𝑁 Natural mortality rate 𝛿𝐼 Disease-induced death rate of undetected symptomatic infectious humans 𝛿𝐻 Disease-induced death rate of hospitalized detected infectious humans 𝛿𝑀 Disease-induced death rate of self-medicated humans 𝜇 Quarantined humans who do not develop symptoms and are not infected that progressed to susceptible class again 𝛼 Exposed humans that are quarantined (via contact tracing) 𝛽 Effective contact rate 𝜀𝐴 Recovery rate of undetected asymptomatic infectious humans due to strong immune system 𝜀𝐼 Recovery rate of undetected symptomatic infectious humans due to strong immune system 𝜀𝑀 Recovery rate of humans under self-medication World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1873 𝜈 Rate at which humans receive vaccine boosters 𝜅1 Rate at which susceptible humans receive first vaccination 𝜅2 Rate at which humans with first vaccination receive second vaccination 𝜓 Modification parameter that accounts for a reduced transmission from 𝑉1 𝜎 Progression rate from exposed state to infectious state 𝛾 Recovery rate of detected and hospitalized infectious humans due to treatment 𝑘 Fraction of new infectious humans that is asymptomatic 𝜂 Progression rate from quarantined class to hospitalize detected infectious humans 𝜔 Detection rate (via testing) for the undetected asymptomatic infectious class 𝑞 Transition rate from undetected symptomatic infectious class to 𝐼𝐻 𝜏1 Rate of compliance to social distancing 𝜏2 Fraction of undetected symptomatic infectious humans that adhered strictly to COVID-19 safety protocols and avoided self-medication 𝜙 Sensitization rate on the danger of self-medication 𝜃 Progression rate from 𝑀 class to 𝐼𝐻 class due to severity of COVID-19 in humans under selfmedication 𝜌1 Rate of compliance to wearing of Face mask 𝜌2 Rate of compliance to the use of hand sanitizer 𝑐1 Modification parameter that accounts for a reduced transmission from A class 𝑐2 Modification parameter that accounts for increased transmission from M class Our model (1) is an extension of models that have been formulated towards gaining insights into how the novel coronavirus disease is transmitted from one person to another [10] our work presents a major preventive strategy. We introduced two vaccination compartments accounting for humans with first and second dose vaccination. These was not in-cooperated in the existing literature [10] , in their work they did not capture these compartments. 2. Demographic features such as natural birth and death rates are incorporated into the model of [10] as can be seen in most recent works such as [40, 41, 45,47] . In [10] these features were neglected considering that covid-19 was still knew and might not have been influenced much by natural and death rate but after five years of the epidemic, it is quite appropriate to include these features as demonstrated by many works in literature. We incorporated it in the present work. We reformulated our model (1) by introducing the preventive strategy to adequately capture the dynamics of the transmission and preventive measures to curb the transmission and help Nigerian health policy makers to put under control its spread represented by parameters, V1 and V2 where Λ(1−𝑎) stands for recruited humans that receive first dose of vaccination. World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1874 Figure 1 Disease transmission flow diagram Thus, the model becomes; 𝑑𝑆 𝑑𝑡=Λ(1−𝑎)−(𝜆+𝛿𝑁+𝜅1)𝑆+𝜇𝑄, 𝑑𝑉1 𝑑𝑡 =𝑎1Λ+𝜅1𝑆−(𝜆𝜓+𝜅2+𝛿𝑁)𝑉1, 𝑑𝑉2 𝑑𝑡 =𝑎2Λ+𝜅2𝑉1−(𝛿𝑁+𝜈)𝑉2, 𝑑𝐸 𝑑𝑡=𝜆(𝑆+𝜓𝑉1)−(𝛼+𝜎+𝛿𝑁)𝐸, 𝑑𝑄 𝑑𝑡=𝛼𝐸−(𝜂+𝜇+𝛿𝑁)𝑄, 𝑑𝐴 𝑑𝑡=𝑘𝜎𝐸−(𝜔+𝜀𝐴+𝛿𝑁)𝐴, …………….(1) 𝑑𝐼 𝑑𝑡=(1−𝑘)𝜎𝐸−(𝑞+𝛿𝑁+𝛿𝐼+𝜀𝐼)𝐼, 𝑑𝑀 𝑑𝑡 =(1−𝜏2𝜙)𝑞𝐼−(𝜃+𝛿𝑁+𝛿𝑀+𝜀𝑀)𝑀, 𝑑𝐼𝐻 𝑑𝑡 =𝜂𝑄+𝜔𝐴+𝜏2𝜙𝑞𝐼+𝜃𝑀−(𝛾+𝛿𝑁+𝛿𝐻)𝐼𝐻, 𝑑𝑅 𝑑𝑡=𝜀𝐴𝐴+𝜀𝐼𝐼+𝜀𝑀𝑀+𝛾𝐼𝐻+𝜈𝑉2−𝛿𝑁𝑅 Where 𝑎=𝑎1+𝑎2 and 𝜆=𝛽(1−𝜌1)(1−𝜌2)(1−𝜏1)(𝑐1𝐴+𝐼+𝑐2𝑀) 𝑁. World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1875 The total human population is given by: 𝑁=𝑆+𝑉1+𝑉2+𝐸+𝑄+𝐴+𝐼+𝑀+𝐼𝐻+𝑅. 3. Results and discussion Let ||.|| be the maximum norm in Γ∈ ℛ+ 10 taking to be the banach domain for continuous functions where ||𝑌(𝑡)||=∑||𝑌||∞. Let ||𝑆||≤𝑘1,||𝑉1||≤𝑘2,||𝑉2||≤𝑘3,||𝐸||≤𝑘4,||𝑄||≤𝑘5,||𝐴||≤𝑘6,||𝐼||≤𝑘7,||𝑀||≤𝑘8,||𝐼𝐻||≤𝑘9,||𝑅||≤𝑘10 𝑎𝑛𝑑 0 ≤ 𝑤{𝑖=1,...,10}<1. From the system (1), we will have that for any 𝑆1 and 𝑆2 ∈Γ, then ||𝑓(𝑡,𝑆1)−𝑓(𝑡,𝑆2)||=||Λ(1−𝑎)−(𝜆+𝛿𝑁+𝜅1)𝑆1+𝜇𝑄+𝜋𝑅−(Λ(1−𝑐)−(𝜆+𝛿𝑁+𝜅1)𝑆2+𝜇𝑄+𝜋𝑅)|| =||(𝜆+𝛿𝑁+𝜅1)(𝑆1−𝑆2)||≤ (𝜆+𝛿𝑁+𝜅1)||𝑆1−𝑆2|| ≤ 𝑤1||𝑆1−𝑆2|| The Lipschitz continuity in 𝑆 is established with 𝑤1 as the Lipschitz constant. Similarly, we can establish the Lipschitz continuity in other state variables as follows; ||𝑓(𝑡,𝑉11)−𝑓(𝑡,𝑉12)||=‖𝑎1Λ+𝜅1𝑆−(𝜆𝜓+𝜅2+𝛿𝑁)𝑉11−(𝑎1Λ+𝜅1𝑆−(𝜆𝜓+𝜅2+𝛿𝑁)𝑉12)‖ =||(𝜆𝜓+𝜅2+𝛿𝑁)(𝑉11−𝑉12)||≤ (𝜆𝜓+𝜅2+𝛿𝑁)||𝑉11−𝑉12||≤𝑤2‖𝑉11−𝑉12‖ . ||𝑓(𝑡,𝑉21)−𝑓(𝑡,𝑉22)||=‖𝑎2Λ+𝜅2𝑉1−(𝛿𝑁+𝜈)𝑉21−(𝑎2Λ+𝜅2𝑉1−(𝛿𝑁+𝜈)𝑉21)‖=||(𝛿𝑁+𝜈)(𝑉21−𝑉22)|| ≤ (𝛿𝑁+𝜈)||𝑉21−𝑉22||≤𝑤3‖𝑉21−𝑉22‖ . ||𝑓(𝑡,𝐸1)−𝑓(𝑡,𝐸2)||=||𝜆(𝑆+𝜓𝑉1)−(𝛼+𝜎+𝛿𝑁)𝐸1−(𝜆(𝑆+𝜓𝑉1)− (𝛼+𝜎+𝛿𝑁)𝐸2)||=‖(𝛼+𝜎+𝛿𝑁)(𝐸1−𝐸2)‖≤(𝛼+𝜎+𝛿𝑁)‖𝐸1−𝐸2‖ ≤𝑤4‖𝐸1−𝐸2‖. ||𝑓(𝑡,𝑄1)−𝑓(𝑡,𝑄2)||=‖𝛼𝐸−(𝜂+𝜇+𝛿𝑁)𝑄1−(𝛼𝐸−(𝜂+𝜇+𝛿𝑁)𝑄2)‖ =‖(𝜂+𝜇+𝛿𝑁)(𝑄1−𝑄2)‖≤(𝜂+𝜇+𝛿𝑁)‖(𝑄1−𝑄2)‖≤𝑤5‖(𝑄1−𝑄2)‖. ||𝑓(𝑡,𝐴1)−𝑓(𝑡,𝐴2)||=‖𝑘𝜎𝐸−(𝜔+𝜀𝐴+𝛿𝑁)𝐴1−(𝑘𝜎𝐸−(𝜔+𝜀𝐴+𝛿𝑁)𝐴1)‖ =‖(𝜔+𝜀𝐴+𝛿𝑁)(𝐴1−𝐴2)‖≤(𝜔+𝜀𝐴+𝛿𝑁)‖(𝐴1−𝐴2)‖≤𝑤6‖(𝐴1−𝐴2)‖. ||𝑓(𝑡,𝐼1)−𝑓(𝑡,𝐼2)||=‖(1−𝑘)𝜎𝐸−(𝑞+𝛿𝑁+𝛿𝐼+𝜀𝐼)𝐼1−((1−𝑘)𝜎𝐸−(𝑞+𝛿𝑁+𝛿𝐼+𝜀𝐼)𝐼2)‖ =‖(𝑞+𝛿𝑁+𝛿𝐼+𝜀𝐼)(𝐼1−𝐼2)‖≤(𝑞+𝛿𝑁+𝛿𝐼+𝜀𝐼)‖(𝐼1−𝐼2)‖≤𝑤7‖(𝐼1−𝐼2)‖. ||𝑓(𝑡,𝑀1)−𝑓(𝑡,𝑀2)||=||(1−𝜏2𝜙)𝑞𝐼−(𝜃+𝛿𝑁+𝛿𝑀+𝜀𝑀)𝑀1−((1−𝜏2𝜙)𝑞𝐼 −(𝜃+𝛿𝑁+𝛿𝑀+𝜀𝑀)𝑀2||=||(𝜃+𝛿𝑁+𝛿𝑀+𝜀𝑀)(𝑀1−𝑀2)|| ≤(𝜃+𝛿𝑁+𝛿𝑀+𝜀𝑀)‖𝑀1−𝑀2‖≤𝑤8‖𝑀1−𝑀2‖. ||𝑓(𝑡,𝐼𝐻1)−𝑓(𝑡,𝐼𝐻2)||=||𝜂𝑄+𝜔𝐴+𝜏2𝜙𝑞𝐼+𝜃𝑀−(𝛾+𝛿𝑁+𝛿𝐻)𝐼𝐻1− ( 𝜂𝑄+𝜔𝐴 World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1876 +𝜏2𝜙𝑞𝐼+𝜃𝑀−(𝛾+𝛿𝑁+𝛿𝐻)𝐼𝐻2)||=‖(𝛾+𝛿𝑁+𝛿𝐻)(𝐼𝐻1−𝐼𝐻2)‖ ≤(𝛾+𝛿𝑁+𝛿𝐻)‖𝐼𝐻1−𝐼𝐻2‖≤𝑤9‖𝐼𝐻1−𝐼𝐻2‖. ||𝑓(𝑡,𝑅1)−𝑓(𝑡,𝑅2)||=||𝜀𝐴𝐴+𝜀𝐼𝐼+𝜀𝑀𝑀+𝛾𝐼𝐻+𝜈𝑉2−(𝛿𝑁+𝜋)𝑅1−(𝜀𝐴𝐴+𝜀𝐼𝐼 +𝜀𝑀𝑀+𝛾𝐼𝐻+𝜈𝑉2−(𝛿𝑁+𝜋)𝑅2)||=‖(𝛿𝑁+𝜋)(𝑅1−𝑅2)‖≤(𝛿𝑁+𝜋)‖𝑅1−𝑅2‖ 𝑤10‖𝑅1−𝑅2‖. Where 𝑤1=𝜆+𝛿𝑁+𝜅1, 𝑤2=𝜆𝜓+𝜅2+𝛿𝑁, 𝑤3=𝛿𝑁+𝜈, 𝑤4=𝛼+𝜎+𝛿𝑁, 𝑤5=𝜂+𝜇+𝛿𝑁, 𝑤6=𝜔+𝜀𝐴+𝛿𝑁, 𝑤7=𝑞+ 𝛿𝑁+𝛿𝐼+𝜀𝐼, 𝑤8=𝜃+𝛿𝑁+𝛿𝑀+𝜀𝑀, 𝑤9=𝛾+𝛿𝑁+𝛿𝐻, 𝑤10=𝛿𝑁+𝜋. Lipschitz continuity has been established for all the state solutions. Also, all the 𝑤𝑖′𝑠 are guaranteed to be less than one since they represent fractional outflow from the compartments and their sum must be less than one [1, 3,24]. Hence, by Banach fixed point theorem, the solution to the system exists and is unique. 3.1. Disease-Free Equilibrium of The System The disease-free equilibrium of the model system is the steady state solution to (1) when there is no covid-19 infection in the population. That is, it is the solution to (1) when 𝐸=𝑄=𝐴=𝐼=𝑀=𝐼𝐻=0 [4, 5, 6, 8, 9, 32,40]. Let the diseasefree equilibrium be denoted ℇ0, then ℇ0 has the form ℇ0=(𝑆0,𝑉10,𝑉20,𝐸0,𝑄0,𝐴0,𝐼0,𝑀0,𝐼𝐻 0,𝑅0). The nonzero components 𝑆0,𝑉10,𝑉20 and 𝑅0 of ℇ0 are calculated as follows: Λ(1−𝑎)−(𝜆+𝑘1+𝛿𝑁)𝑆0+𝜇𝑄0=0 ⇒ 𝑆0=Λ(1−𝑎) 𝑘1+𝛿𝑁. 𝑎1Λ+𝑘1𝑆0−(𝜆𝜓+𝑘2+𝛿𝑁)𝑉10=0 ⇒ 𝑉10=𝑎1Λ+k1𝑆0 𝑘2+𝛿𝑁=𝑎1Λ(𝑘1+𝛿𝑁)+𝑘1∧(1−𝑎) (𝑘2+𝛿𝑁)(𝑘1+𝛿𝑁). 𝛼𝐸0−(𝜂+𝜇+𝛿𝑁)𝑄0=0 ⇒ 𝑄0=𝛼𝐸0 (𝜂+𝜇+𝛿𝑁)=0 since 𝐸=0. 𝜖𝐴𝐴0+𝜖𝐼𝐼0+𝜖𝑀𝑀0+𝛾𝐼𝐻 0+𝜈𝑉20−𝛿𝑁𝑅0=0 ⇒ 𝑅0=𝜈𝑉20 𝛿𝑁 ⇒ 𝑅0=𝜈𝑎2Λ(𝑘2+𝛿𝑁)(𝑘1+𝛿𝑁)+𝜈𝑘2𝑎1∧(𝑘1+𝛿𝑁)+𝜈𝑘1𝑘2∧(1−𝑎) 𝛿𝑁(𝑘1+𝛿𝑁)(𝑘2+𝛿𝑁)(𝜈+𝛿𝑁). We have ℇ0=(𝑆0,𝑉10,𝑉20,𝐸0,𝑄0,𝐴0,𝐼0,𝑀0,𝐼𝐻 0,𝑅0) =(𝑆0,𝑉10,𝑉20,0,0,0,0,0,0,𝑅0) where, 𝑆0=Λ(1−𝑎) 𝑘1+𝛿𝑁 𝑉10=𝑉20=𝑎2Λ(𝑘2+𝛿𝑁)(𝑘1+𝛿𝑁)+𝑘2𝑎1∧(𝑘1+𝛿𝑁)+𝑘1𝑘2∧(1−𝑎) (𝑘1+𝛿𝑁)(𝑘2+𝛿𝑁)(𝜈+𝛿𝑁) 𝑅0=𝜈𝑎2Λ(𝑘2+𝛿𝑁)(𝑘1+𝛿𝑁)+𝜈𝑘2𝑎1∧(𝑘1+𝛿𝑁)+𝜈𝑘1𝑘2∧(1−𝑎) 𝛿𝑁(𝑘1+𝛿𝑁)(𝑘2+𝛿𝑁)(𝜈+𝛿𝑁) ………..(3) The disease-free equilibrium is useful in determining the basic reproduction number of the disease. 3.2. The Basic Reproduction Number The relevant equations in the calculation of the NGM are the subsystem for the compartments where there can be disease infections. Therefore, the subsystem for the calculation of the basic reproduction number is therefore 𝑑𝐸 𝑑𝑡=𝜆(𝑆+𝜓𝑉1)−(𝛼+𝜎+𝛿𝑁)𝐸, World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1877 𝑑𝑄 𝑑𝑡 =𝛼𝐸−(𝜂+𝜇+𝛿𝑁)𝑄, 𝑑𝐴 𝑑𝑡=𝑘𝜎𝐸−(𝜔+𝜖𝐴+𝛿𝑁)𝐴, …….(4) 𝑑𝐼 𝑑𝑡=(1−𝑘)𝜎𝐸−(𝑞+𝛿𝐼+𝜖𝐼+𝛿𝑁)𝐼, 𝑑𝑀 𝑑𝑡 =(1−𝜏2𝜙)𝑞𝐼−(𝜃+𝜖𝑀+𝛿𝑀+𝛿𝑁)𝑀, 𝑑𝐼𝐻 𝑑𝑡 =𝜂𝑄+𝜔𝐴+𝜏2𝜙𝑞𝐼+𝜃𝑀−(𝛾+𝛿𝑁+𝛿𝐻)𝐼𝐻. Therefore, from the subsystem, we have that ℱ𝑖(𝑥)= ( 𝛽(1−𝜌1)(1−𝜌2)(1−𝜏1)(𝑐1𝐴+𝐼+𝑐2𝑀)(𝑆+𝜓𝑉1) 𝑁 0 0 0 0 0 ) and 𝒱𝑖(𝑥)= ( (𝛼+𝜎+𝛿𝑁)𝐸 −𝛼𝐸+(𝜂+𝜇+𝛿𝑁)𝑄 −𝑘𝜎𝐸+(𝜔+𝜖𝐴+𝛿𝑁)𝐴 − (1−𝑘)𝜎𝐸+(𝑞+𝛿𝐼+𝜖𝐼+𝛿𝑁)𝐼 −(1−𝜏2𝜙)𝑞𝐼+(𝜃+𝜖𝑀+𝛿𝑀+𝛿𝑁)𝑀 −𝜂𝑄−𝜔𝐴−𝜏2𝜙𝑞𝐼−𝜃𝑀+(𝛾+𝛿𝑁+𝛿𝐻)𝐼𝐻 ) Hence, the matrices 𝐹 and 𝑉 are given by 𝐹= ( 00𝐹1𝐹2𝐹30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) Where 𝐹1=𝑐1𝛽(1−𝜌1)(1−𝜌2)(1−𝜏1) 𝑁0(𝑆0+𝜓𝑉0), 𝐹2=𝛽(1−𝜌1)(1−𝜌2)(1−𝜏1) 𝑁0(𝑆0+𝜓𝑉0), 𝐹3=𝑐2𝛽(1−𝜌1)(1−𝜌2)(1−𝜏1) 𝑁0(𝑆0+𝜓𝑉0), with 𝑁0=Λ 𝛿𝐻 and 𝑉= ( 𝛼+𝜎+𝛿𝑁0 0 0 0 0 −𝛼 𝜂+𝜇+𝛿𝑁0 0 0 0 −𝑘𝜎 0 𝜔+𝜖𝐴+𝛿𝑁0 0 0 − (1−𝑘)𝜎 0 0 𝑞+𝛿𝐼+𝜖𝐼+𝛿𝑁0 0 0 0 0 −(1−𝜏2𝜙)𝑞 𝜃+𝜖𝑀+𝛿𝑀+𝛿𝑁0 0 −𝜂 −𝜔 −𝜏2𝜙𝑞 −𝜃 𝛾+𝛿𝑁+𝛿𝐻 ) . The inverse of the matrix, 𝑉 is nonnegative and is given by World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1878 𝑉−1= ( 1 𝐵10 0 0 0 0 𝜎 𝐵1𝐵51 𝐵50 0 0 0 𝑘𝜎 𝐵1𝐵201 𝐵20 0 0 (1−𝑘)𝜎 𝐵1𝐵30 0 1 𝐵30 0 (1−𝑘)(1−𝜏2𝜙)𝜎𝑞 𝐵1𝐵3𝐵40 0 (1−𝜏2𝜙)𝑞 𝐵3𝐵41 𝐵40 𝐵7𝜂 𝐵5𝐵6𝜔 𝐵2𝐵6𝜏2𝜙𝑞𝐵4+𝜃(1−𝜏2𝜙)𝑞 𝐵3𝐵4𝐵6𝜃 𝐵4𝐵61 𝐵6 ) 𝑤ℎ𝑒𝑟𝑒 𝐵1= 𝛼+𝜎+𝛿𝑁,𝐵2=𝜔+𝜖𝐴+𝛿𝑁,𝐵3=𝑞+𝛿𝐼+𝜖𝐼+𝛿𝑁,𝐵4=𝜃+𝜖𝑀+𝛿𝑀+𝛿𝑁, 𝐵5=𝜂+𝜇+𝛿𝑁, 𝐵6=𝛾+ 𝛿𝑁+𝛿𝐻 and 𝐵7=𝜂𝛼𝐵2𝐵3𝐵4+𝜔𝑘𝜎𝐵3𝐵4𝐵5+𝐵2𝐵4𝐵5(1−𝑘)𝜎𝜏2𝜙𝑞+𝜃𝐵2𝐵5(1−𝑘)(1−𝜏2𝜙)𝜎𝑞 𝐵1𝐵2𝐵3𝐵4𝐵5𝐵6. Then, the next-generation matrix (NGM) becomes 𝐹𝑉−1= ( 𝐹1𝑘𝜎 𝐵1𝐵2+𝐹2(1−𝑘)𝜎 𝐵1𝐵3+𝐹3(1−𝑘)(1−𝜏2𝜙)𝜎𝑞 𝐵1𝐵3𝐵40𝐹1 𝐵2𝐹2 𝐵3+𝐹3(1−𝜏2𝜙)𝑞 𝐵3𝐵4𝐹3 𝐵40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) …….(6) The eigenvalues of the matrix 𝐹𝑉−1 are (0,0,0,0,0,𝐹1𝑘𝜎 𝐵1𝐵2+𝐹2(1−𝑘)𝜎 𝐵1𝐵3+𝐹3(1−𝑘)(1−𝜏2𝜙)𝜎𝑞 𝐵1𝐵3𝐵4). The only non-zero eigenvalue of the next generation matrix, 𝐹𝑉−1 is 𝐹1𝑘𝜎 𝐵1𝐵2+𝐹2(1−𝑘)𝜎 𝐵1𝐵3+𝐹3(1−𝑘)(1−𝜏2𝜙)𝜎𝑞 𝐵1𝐵3𝐵4. Therefore, the control reproduction number of COVID-19 in this model is therefore given by ℛ𝑐=𝐹1𝑘𝜎 𝐵1𝐵2+𝐹2(1−𝑘)𝜎 𝐵1𝐵3+𝐹3(1−𝑘)(1−𝜏2𝜙)𝜎𝑞 𝐵1𝐵3𝐵4 ……….(7) This can be rewritten as ℛ𝑐=ℛ𝑐𝐴+ℛ𝑐𝐼+ℛ𝑐𝑀 ………. (8) where ℛ𝑐𝐴=𝑐1𝛽(1−𝜌1)(1−𝜌2)(1−𝜏1)(𝑆0+𝜓𝑉10)𝑘𝜎 𝑁0(𝛼+𝜎+𝛿𝑁)(𝜔+𝜖𝐴+𝛿𝑁), ℛ𝑐𝐼=𝛽(1−𝜌1)(1−𝜌2)(1−𝜏1)(𝑆0+𝜓𝑉10)(1−𝑘)𝜎 𝑁0(𝛼+𝜎+𝛿𝑁)(𝑞+𝛿𝐼+𝜖𝐼+𝛿𝑁), ℛ𝑐𝑀=𝑐2𝛽(1−𝜌1)(1−𝜌2)(1−𝜏1)(𝑆0+𝜓𝑉10)(1−𝑘)(1−𝜏2𝜙)𝜎𝑞 𝑁0(𝛼+𝜎+𝛿𝑁)(𝑞+𝛿𝐼+𝜖𝐼+𝛿𝑁)(𝜃+𝜖𝑀+𝛿𝑀+𝛿𝑁),….(9) World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1879 ℛ𝑐 as calculated, is the average number of persons that can be infected with COVID-19 by index case of the disease throughout his infectious lifetime when placed in a purely susceptible population with vaccination. The reproduction number ℛ𝑐 must be reduced below one in order to ensure that the disease dies out. In this case, we must have 1>ℛ𝑐 < ℛ0, where ℛ0 is the basic reproduction number of COVID-19. 3.3. Global Stability of the Disease-free equilibrium for the covid-19 model We shall use the method described in [16] , to investigate the global asymptotic stability of the disease-free equilibrium point. Hence, we have the system written as 𝑑𝑋1 𝑑𝑡 =𝐹(𝑋1,𝑋2) …… (10) 𝑑𝑋2 𝑑𝑡 =𝐺(𝑋1,𝑋2),𝐺(𝑋1,0) The two conditions (C1) and (C2) listed below must be satisfied to guarantee global asymptotic stability of ℇ0 when ℛ𝑐<1. [11, 18, 19,20, 21] (C1): For 𝑑𝑋1 𝑑𝑡 =𝐹(𝑋1,0), ℇ1 0 is globally asymptotically stable (C2): 𝐺(𝑋1,𝑋2) can be written as 𝐺(𝑋1,𝑋2)=𝐵𝑋2−𝐺(𝑋1,𝑋2),𝐺(𝑋1,𝑋2)≥0 where 𝐵 is the Jacobian matrix of 𝐺(𝑋1,𝑋2), evaluated at ℇ0. If the model system satisfies the above two conditions, then the following theorem holds: [16]. The disease-free equilibrium point ℇ0=(ℇ1 0,𝟎 ) is globally asymptotically stable provided ℛ𝑐<1 and the conditions (C1) and (C2) are satisfied. Following the method described above, the system 𝑑𝑋1 𝑑𝑡 =𝐹(𝑋1,0) at the disease-free equilibrium is given by 𝑑𝑆 𝑑𝑡=Λ(1−𝑎)−(𝑘1+𝛿𝑁)𝑆, 𝑑𝑉1 𝑑𝑡 =𝑎1Λ+𝑘1𝑆−(𝑘2+𝛿𝑁)𝑉1, 𝑑𝑉2 𝑑𝑡 =𝑎2Λ+𝑘2𝑉1−(𝜈+𝛿𝑁)𝑉2, …….. (11) 𝑑𝑅 𝑑𝑡=𝜈𝑉2−𝛿𝑁𝑅, with the disease-free equilibrium ℇ1 0=(𝑆0,𝑉10,𝑉20,𝑅0). On the other hand, the subsystem for the infected individuals become 𝑑𝐸 𝑑𝑡=𝜆(𝑆+𝜓𝑉1)−(𝛼+𝜎+𝛿𝑁)𝐸, 𝑑𝑄 𝑑𝑡 =𝛼𝐸−(𝜂+𝜇+𝛿𝑁)𝑄, 𝑑𝐴 𝑑𝑡=𝑘𝜎𝐸−(𝜔+𝜖𝐴+𝛿𝑁)𝐴, ……..(12) 𝑑𝐼 𝑑𝑡=(1−𝑘)𝜎𝐸−(𝑞+𝛿𝐼+𝜖𝐼+𝛿𝑁)𝐼, 𝑑𝑀 𝑑𝑡 =(1−𝜏2𝜙)𝑞𝐼−(𝜃+𝜖𝑀+𝛿𝑀+𝛿𝑁)𝑀, 𝑑𝐼𝐻 𝑑𝑡 =𝜂𝑄+𝜔𝐴+𝜏2𝜙𝑞𝐼+𝜃𝑀−(𝛾+𝛿𝐻+𝛿𝑁)𝐼𝐻, The system (11) can be rearranged to get World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1886 Thus, the solution of the system given the initial values of the state variables and the parameters is; 𝑆(𝑡)= 206,000,000−32,639,492.353𝑡+49,257,668.6891𝑡2−2,811,207.2036𝑡3 𝑉1(𝑡)=15,000+32,588,461.746𝑡−2,202,454.7803𝑡2+38,303.0752𝑡3 𝑉2(𝑡)=5000+732.3𝑡+814,713.4483𝑡2−38,073.7983𝑡3 𝐸(𝑡)=200,000−24,740.493𝑡+44,997,260.7246𝑡2−4,814,131.851𝑡3 𝑄(𝑡)=7,000+24,806.72𝑡+8,453.6154𝑡2+2,141,850.5735𝑡3 𝐴(𝑡)=30,000+14,941.80𝑡−2,257.2897𝑡2+1,442,269.7586𝑡3 𝐼(𝑡)=150,000−10461𝑡−154.0741𝑡2+1,442,922.3264𝑡3 𝑀(𝑡)=50,000−19,847.81𝑡−100.4083𝑡2+15.2476𝑡3 𝐼𝐻(𝑡)=1719+11,658.2989𝑡+4,271.3037𝑡2+1,326.5180𝑡3 𝑅(𝑡)=164,415+33,000.0807𝑡−1,421.9987𝑡2+411.7099𝑡3 3.4.2. Comparison of the DTM solutions with RK-4 solutions The solutions obtained by the differential transform method (DTM) is compared with the solutions obtained by RungeKutta method of order 4 (RK-4) to check if the DTM solutions is consistent and convergent. The comparison is shown in Figures 2 – 9. The plots showed that both solutions are relatively comparable within some time interval. Though both the differential transform method and Runge-Kutta methods are approximation methods, the DTM is a semi-analytical method while the Runge-Kutta methods are numerical techniques Figure 2 Susceptible humans World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1887 Figure 3 Humans with first vaccination Figure 4 Asymptomatic humans Figure 5 Symptomatic humans World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1888 Figure 6 Hospitalized humans Figure 7 Humans with second vaccination Figure 8 Quarantine Humans World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1889 Figure 9 Recovered humans 3.5. Sensitivity Analysis of the Parameters in the Control Reproduction Number We use forward normalized sensitivity index method to measure the relative change in ℛ𝑐, to the relative change in the model parameter, 𝑥[14, 23,29]. This is defined as 𝒮𝑥ℛ𝑐=𝝏ℛ𝑐 𝝏𝒙 ×𝑥 ℛ𝑐 …….. (17) Table 4 Sensitivity indices of parameters contained in the reproduction number Parameter Sensitivity Index Parameter Sensitivity Index 𝑐1 +0.1445 𝜔 -2.2955×10−11 𝑐2 +0.0209 𝜖𝐴 -2.2955×10−11 𝛽 +1 𝑞 -0.7111 𝑘 +0.7111 𝛿𝐼 -0.0648 𝜎 +0.4264 𝜖𝐼 -0.6176 𝑎1 +0.1058 𝜃 -0.0069 𝜓 +0.1198 𝛿𝑀 -0.0088 𝜙 0 𝜖𝑀 -0.0052 𝜌1 -0.1111 𝜏2 -0.0083 𝜌2 -0.25 𝑘1 -0.8780 𝜏1 -0.25 𝑘2 -0.1197 𝛼 -0.4263 The result of the sensitivity analysis shows that the parameters with positive sensitivity indices are 𝛽,𝑐1,𝑐2,𝑘,𝜎,𝜓 and 𝑎1 These parameters with positive sensitivity indices are the parameters whose values must be reduced in order to stop the spread of corona virus disease. The parameter with the highest sensitivity index is 𝛽, the effective contact rate between the susceptible class and the infectious classes. The sensitivity index shows that a reduction in the contact rate will lower the spread of the disease. This aligns with the use of the control measures represented by the parameters, 𝜌1,𝜌2 and 𝜏1. These parameters aim to reduce the rate of contact between the susceptible class and the infectious classes namely, 𝐴,𝐼,𝑎𝑛𝑑 𝑀. The negative sign of the sensitivity indices of these parameters is an indication that the spread of the disease can be reduced by increasing the value of these parameters. The importance of first dose of vaccination is seen in the sensitivity index with respect to the parameter, 𝑎1, which represents the proportion of those recruited into the susceptible class that have received World Journal of Advanced Research and Reviews, 2025, 26(03), 1871-1893 1890 first dose of the vaccination. The sensitivity index with respect to this parameter advises that more people recruited into the population should receive first dose of the vaccine to help reduce the rate of infection. The parameter, 𝜎 represents the rate at which people infected with the disease becomes infectious. The positive sign of the sensitivity index with respect to this parameter indicates the need to reduce the rate of infectiousness of those infected with the disease. The main aim of receiving doses of corona virus disease vaccine is to reduce the probability of being infected with the virus. The parameter which represents this probability is 𝜓, with sensitivity index, +0.1198. This shows that reducing the probability of infection by increasing the efficacy of the vaccine will help reduce the rate at which people contract the disease. The need to increase the proportion, 𝑘1 of the susceptible class that receive first dose of the vaccine, and the proportion, 𝑘2 of those in 𝑉1, that receive the second dose is seen in the sensitivity indices with respect to these parameters. Quarantining of those that are exposed to corona virus is vital in the management of the disease outbreak. This is observed in the sensitivity index with respect to 𝛼, which indicates the need to increase the rate at which the exposed persons are quarantined. The rates of hospitalization 𝜂,𝜔,𝑞 and 𝜃 for those in the compartments 𝑄,𝐴,𝐼 and 𝑀, respectively show negative sensitivity indices, which indicates that the rates at which infected persons get hospitalized to receive appropriate treatment should be increased for effective management of the disease outbreak 4. Conclusion This paper discusses the ongoing public health risk that the COVID-19 pandemic poses and assesses how well different control measures with a special emphasis on vaccination, hospitalization and quarantine policies address its spread. The study emphasizes the significance of scientifically based public health treatments, given that COVID-19 is characterized by high transmissibility and considerable mortality, particularly among vulnerable populations. We use a deterministic mathematical model to Obtain the disease-free equilibrium state, Compute the control reproduction number, Obtain conditions for the local and global stability of the disease-free equilibrium state and carryout sensitivity analysis on the control reproduction number. The study does thorough mathematical analysis using programs like MATLAB and Maple. Compliance with ethical standards Disclosure of conflict of interest We the authors declare no affiliations with or involvement in any organization or entity with any financial interest such as honoraria, educational grants, employment stock ownership, or other equity interest or non-financial interest such as personal or professional relationships, affiliations, knowledge or belief in the subject matter or materials discussed in this manuscript. Funding This work is self-sponsored and has not attracted any funding from the school authority or Tetfund. Statement of ethical approval This study is based entirely on mathematical modeling and computational analysis of publicly available data. It does not involve any experiments on human participants or animals, nor does it collect or process any personal, sensitive, or identifiable information. Therefore, ethical approval was not required. All procedures performed in this study were in accordance with relevant institutional, national, and international ethical guidelines and regulations for computational research. References [1] Amira, R., and Delfirm F.M.T., (2015): Mathematical Modeling, Simulation and Optimal Control of the 2014 Ebola Outbreak in West Africa. Discrete dynamics in Nature and Society, vol. 2015, Article ID842792, 9 pages, 2015. [2] Anwar, Z., Ebraheem A.,Vedat, S., (2020): “Mathematical Model for Coronavirus Disease 2019 (COVID-19) Containing Isolation Class”. Biomed Research International, vol 5. 2020, Article ID 3452402, p4. 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