EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2450-2460 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Model of COVID-19 Epidemic Based on Vaccination and Treatment Fahdah Alshammari1, Alaa Mustafa1,∗, Ehssan Omer2, Fatima Alnoor3 1 Department of Biology, Faculty of Science and Arts-RAFHA, Northrn Border University, Arar 73213 Kingdom of Saudi Arabia 2 Department of Mathematics, Faculty of Science -Arar, Northrn Border University, Arar 75211 Kingdom of Saudi Arabia 3 Department of Mathematics, Faculty of Science and Arts-TURIAF, Northrn Border University, Arar 75211 Kingdom of Saudi Arabia Abstract. This article presents a mathematical model of the COVID-19 transmission mechanism, considering therapeutic interventions like immunization and recovery or treatment. The model shows that the disease-free and endemic equilibriums are globally asymptotically stable when effective reproduction numbers are less than or larger than unity. The critical vaccination threshold depends on the vaccine’s ability to prevent or cure the illness. The model predicts the effectiveness of vaccination based on factors like vaccination efficiency, scheduling, and relaxation of social measures. The subsiding of the epidemic as vaccination is implemented depends on the scale of relaxation of social measures. 2020 Mathematics Subject Classifications: 37N25 Key Words and Phrases: Covid-19 Epidemic, Mathematical Modeling, Vaccination, Social measures, Transmission 1. Introduction Years of study and testing are frequently needed to create vaccinations that are safe and effective against infectious agents. In comparison, COVID-19 vaccine development and distribution took less than a year [18]. The effectiveness of the epidemiological models suggested for the COVID-19 outbreak may be impacted by the answers to the many unanswered issues created by this rapid development. For instance, vaccine efficiency rates vary depending on the level and length of immunization for various groups [23] , [24] , [13] , [17]. According to the epidemic status of the vaccinated individuals—vulnerable, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4805 Email addresses: bfahdah1402@hotmail.com (F. Alshammari), alal200949@yahoo.com (A.Mustafa), ehssanomerphdd@gmail.com (E.Omer), fattoshaa1988@gmail.com (F. Alnoor) https://www.ejpam.com 2450 © 2023 EJPAM All rights reserved. A. Mustafa et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2450-2460 2451 infected, or having received vaccine doses—we will examine the efficacy of vaccination [22]. Since the start of the COVID-19 pandemic [[11], [14], [12], [5], [21], [4][19]], mathematical modeling has been utilized to examine the reliability of parameters, forecast its progression, and contrast various. Despite these limitations, a first simplified approach can lead to a model, making it possible to predict effective vaccination coverage at the population level that will prevent the appearance of successive epidemic waves. [15] Daily vaccination data, even if they are global and unrefined (for example, [14] by age group or social classification), make it possible to better understand the effect of vaccination policy and test the consequences of changes in this policy to improve its effectiveness[19]. We use an epidemic model to understand the complex interactions between epidemic control and epidemic data. [[18], [9]] Our model considers the changes in public health policy [[1], [20], [12]], such as confinement, social distancing measures, etc., through the time-dependent transmission rate in the model. The data consists of the daily number of reported cases and the daily number of second doses of vaccine. We refer to [[13], [17], [9], [5] [15]] for more results on the subject. In the study, we propose a new model for vaccination implementation. We can connect the model with vaccination to a model without vaccination. We will find a simple transformation for the epidemic data to combine the daily reported case data and the cumulative number of vaccinated individuals[12]. We can use the model to explore controlling the dynamics of virus propagation, for example, by rapidly slowing down an epidemic wave. In this new model, we will explicitly take into account the variable corresponding to the size of the vaccinated population, and we will simulate the increasing efficacy of several vaccination scenarios. Then, we will apply our model to the COVID-19 epidemic. By applying the maximum principle, optimal control of the model can mitigate the spread of COVID-19[7]. 2. Mathematical COVID-19 Epidemic models. S.(t) = −ρ(t)[I(t) + ωN(t)]S(t)− εV aData(t) S(t) Uv(t) E.(t) = −ρ(t)[I(t) + ωN(t)]S(t)− βE(t)− εV aData(t) e(t) Uv(t) I .(t) = βE(t)− σI(t)− εV aData(t) E(t) Uv(t) (1) N .(t) = σ(1− r)I(t)− γN(t)− εV aData(t) E(t) Uv(t) R.(t) = σrI(t)− γR(t)) where, when t, S(t) is the proportion of susceptible, individual health, E(t) is the proportion of those who were exposed, I(t) is the proportion of infected yet asymptomatic A. Mustafa et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2450-2460 2452 persons, N(t) is the proportion of symptomatic infected people who go undetected, and R(t) is the proportion of symptomatic infected people who have been reported[10].Uv(t) is the proportion of unvaccinated people. The biological interpretations of all parameters are provided in Table 1. The primary information is added to the system (1). S(t1) = S1, E(t1) = E1, I(t1) = I1, N(t1) = N1, R(t1) = R1 (2) The model features, ∅(t) is the transmission rate that varies with time, 1/β is the median amount of the exposure time, 1/σ is the maximum range of the infectious period without symptoms, And for the sake of simplicity, we divide the group of symptomatic infectious persons into two fractions: the fraction 0 ≤ r ≤ 1 showing severe symptoms and the fraction 1−r showing moderate symptoms, which is considered to be undiagnosed. For both unreported and reported symptomatic individuals, the amount 1/γ represents the average length of the symptomatic infectious period. [23] The factor (0 ≤ ω ≤ 1) represents the relative contributions of undetected and unreported clinical infectious persons to the infection of susceptible individuals. The potency of the vaccine is the parameter 0 ≤ ε ≤ 1. This implies that the vaccination is totally effective at ε = 1 and completely ineffective at ε = 0, respectively. The equation is satisfied by the total number of removed persons t → M(t), immunized (recovered or vaccinated), and/or dead[6]. M ′ (t) = γ[R(t) +N(t)] + eVa(t)[ S(t) + E(t) + I(t) +N(t) Uv(t) ] (3) In this instance, Va(t) there is steady stream of properly vaccinated people. It implies that ∫ t t1 VA(α).dα Is the overall vaccination rate between time points a and b[2]. Since no people received vaccinations at the beginning of the disease (i.e., for t = t1), we can assume that the total population P at time t1 is P = S1 + E1 + I1 +R1 +N1 The total number of people who have received vaccinations is provided by LVa(t1) = 0 and LV ′ a(t) = Va(t), it is comparable to LVa(t) = ∫ t t1 VA(α).dα The number of people that aren’t vaccinations is Uv(t) = P − LVa(t) A. Mustafa et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2450-2460 2453 It’s expected that a fraction 0 < r ≤ 1 of infectious persons is reported at the end of the asymptomatic infectious period. Therefore, the following connection links the epidemic model to the total number of reported cases LR(t). , for t ≥ t1 , and LR0(t1) = LR0. LR ′ a(t) = σrI(t), (4) Beginning at time t1, we define the fraction of people who are not fully immunized at time t as G(t) = exp (ε ∫ t t1 P − Va(α) P − Va(α) dα). Then, G(t) = exp (ε(ln [P − Va(t)]− ln [P − Va(t1])) Therefore, G(t) = ( (P − LVa(t)) (P − LVa(t1)) )ε (5) If LVa(t) is the total number of recipients of the second dose vaccination. When G(0) = 1 , the function t,W (t) is non-increasing. Define LRa(t) = ∫ t t1 LR ′ a(α) G(α) dα) (6) 2.1. Measuring the data transmission. The parameters ω, β, σ, γ, r, S1, E1, I1, N1, and the five subsequent equations t ≥ t1 fully indicate the data transmission. ∅(t) = ∅̂(t) G(t) (7) therefore, at t ≥ t1 Î(t) = (I(t) G(t) , N̂(t) = N(t) G(t) , and ∅̂(t) = ∅(t)G(t). ∅̂(t) = 1 ∅̂(t)ωN̂(t) × L̂E(t) + β ˆLE′(t) E1 + S1 − ˆLE′(t) + β ˆLE′(t) and ∅̂(t) = LR ′ a(t( σr , L̂E(t) = 1 βσr (L̂R′ a(t)− σrI1 + σ(LR) ′ a(t)) N̂(t) = e−γ(t−t1)N1 + ∫ t t1 e−γ(t−s0 1− r r L̂R ′ a(s1).ds1 A. Mustafa et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2450-2460 2454 The data used in Formula (6) to define(LRa)(t) are those represented by functions t → LRa(t) the cumulative number of reported cases and t → Va(t) resources spent on second doses of vaccination (t) [12]. We arrive at the following equation to define the rate of transmission as a function of the total number of reported cases t → LRa(t) and the total number of immunized people t → LVa(t)[11]. 2.2. Statistics homogenized by G(t) The everyday number of identified cases normalized by G(t), or the percentage of those who weren’t effectively immunized at time t, L̂Ra(t) ′ = LRa(t) ′ G(t) = LRa(t) ′ ( P P − LVa(t) )ε Case frequency on a regular schedule standardized by W(t) is L̂Ra(t) = ∫ t t1 LR ′ a(α) G(α) dα)( P P − LVa(t) )ε 2.3. Reproduction Numbers Utilizing the Formula, we can get the daily transmission rate t → ρ(t).[16] The issue of the immediate reproduction numbers may then be considered by utilizing model (6) (see [24] for more information). We apply our approach to calculate the transmission rate and take into account the immediate reproduction number with vaccination to explore the role of vaccination in the COVID-19 data[13]. RV (t) = ρ(t)S(t) γσ × γ + ωσ(1− r) (8) The reproduction rate with vaccination R1 V (t) = ρ(t)G(t)S1 γσ × γ + ωσ(1− r) (9) The reproduction rate Without vaccination R1(t) = ρ(t)S1 γσ × (γ + ωσ(1− r), (10) 3. Mathematical Analysis of the Epidemic Model In this section, we explain the COVID-19 model’s boundedness, positiveness, epidemic and endemic stability, and fundamental reproduction frequency[4]. A. Mustafa et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2450-2460 2455 Theorem 1. If S1 ≥ 0, E1 ≥ 0, I1 ≥ 0, N1 ≥ 0, R1 ≥ 0,then the solutions of system (1) remain Positiveness for all t > 0. Proof. Proof. We obtain the following from system (1.1)’s first equation: ds dt ≥ −ωS (11)∫ ds ≥ −ωS.dt S(t) ≥ S1e −ωt (12) Therefore, for any t > 0, S(t) stays positive. We also get E(t) ≥ 0, I(t) ≥ 0, N(t) ≥ 0, andR(t) ≥ 0, using the remaining equations of system (1). Theorem 1 is verified. Theorem 2. All solutions of the proposed model with Positiveness initial conditions are bounded and P (t) ≤ ρ ω for all t > 0. Proof. The overall population’s growth rate may be expressed as the sum of all the equations in the system (1). dP dt = dS dt + dE dt + dI dt + dN dt + dR dt (13) Equation (3) gives the following formula: dP dt = ρ− ωP (14) The solution is obtained by integrating two sides of Equation (14) as follows: P (t) = ρ ω + (P1 − ρ ω )e−ωt. (15) Therefore, for t → ∞ , we have that P (t) ≤ ρ ω (16) Thus, Theorem 1 as well as Equation (16) conclude in 0 ≤ P (t) ≤ ρ ω Therefore, S(t), E(t), I(t), N(t), and R(t) are bounded.Theorem 2 is so verified A. Mustafa et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2450-2460 2456 3.1. Epidemic equilibrium and the model’s essential reproduction number When all derivatives are set to zero with I = 0, the epidemic equilibrium Y 0 a system (1) is attained, which results in: Y 0 = ρ ω + w′ , 0, 0, 0, wρ ω(ω + w) (17) The disease-free equilibrium Y 0 is locally asymptotically stable if R1 < 1 ,and unstable if R1 > 1 [1]. 3.2. Transmission rate The model calculates the time-dependent transmission rate before the last date of daily reported cases using daily reported cases data. A rolling weekly average is used to smooth the data[17]. 3.3. Vaccination rate The study aims to determine the effectiveness of vaccination in removing susceptible individuals from infection risk. The daily number of vaccinated individuals and cumu- lative vaccination rate are graphed from December 2020 to June 2021. The efficiency of vaccination is assumed to be 98%, in TABLE 2 and the removal of susceptible is the fraction of the number vaccinated, excluding previously infected individuals. Future daily vaccination rates are chosen for illustration due to their uncertain values[3]. 4. Result and Discussion The Covid-19 pandemic was predicted using the suggested modeling technique. The Centers for Disease Prevention and Control have data accessible [8]. The number of fatalities as well as those retrieved and verified are among them. Daily assimilation of the overall mortality toll, active cases, and recovered cases took place. 4.1. The Number of Instantaneous Reproduction On the fundamental reproduction number, we see essentially little effect on the vac- cine’s effectiveness. This results from compensating effects between R(t) and S(t), which are assessed to change the fixed number of cumulative reported cases. We hardly notice any change. It indicates that the cumulative infection rate is so low relative to the size of the whole population that the R(t) remains essentially unchanged from R1(t)[22] .This indicates that the total number of infected people is insufficient to significantly affect the rate of basic reproduction. To put it another way, the changes in the population of vul- nerable individuals are not big enough to be seen immediately. The R1 is shown by the blue curve R(t). Given that the vaccine is entirely effective, A. Mustafa et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2450-2460 2457 the instantaneous reproduction number should be understood as the instantaneous re- production number in the absence of immunization. [2] The same explanation applies to ε = 0.75, 0.5, 0.25, and 0. This indicates that vaccination significantly affects the dynamics of the epidemic. The effectiveness of the vaccination is a key factor in this influence. We can see that without vaccination. The final peak of the red curve, which represents R1(t) about Theorem 2, is present for ε = 0.75 [9]. 4.2. Impact of Vaccination Began a three-phase COVID-19 immunization campaign on July 17, 2022, to December 20, 2022, making it the first Arab nation to introduce the Pfizer-BioNTech vaccine. [23] The first phase focuses on those over 65 and those who are most at risk for the disease. People over 50 and those with particular chronic conditions, such as diabetes and asthma, will be included in the second phase. In the third step, the vaccine will be administered to the remaining population. The Ministry has, however, identified people who shouldn’t receive the vaccination. Women who are pregnant or nursing, those who want to get pregnant within the next two months, those who experience severe allergic reactions, and those who have been exposed to the virus within the last 90 days are among them[19]. 4.3. Future Work and Modeling Expansions. . Despite these flaws, the model effectively illustrates how vaccination policies affect epidemic dynamics, in part because precise formulas allow for the computation of crucial variables like transmission rate. In addition, the model permits the incorporation of other components, including age, immunization loss, and cross-immunization, when supported by actual data. The efficiency of the immunization strategy is weighed down by this most recent phenomenon, which merits more study. The vaccination policy may be changed in light of the response via cross-immunity against epitopes shared by several Corona viruses. Vaccination provides new immunity in addition to any potential pre-existing cross-immunity [19]. For instance, when age groups are taken into account, young people are those whose cross-immunity is still active. 5. Conclusions The study presents an enhanced SEIR model for including vaccines in epidemic mod- els, considering seven infection phases and vaccination. The model demonstrates non- negativity, boundedness, epidemic equilibrium, existence, distinctiveness of endemic equi- librium, and fundamental reproduction number. It incorporates COVID-19 dynamics, transmission, reported and unreported cases, and social distancing measures. The model predicts vaccination effectiveness in the US, with varying vaccination rates due to hesi- tancy and opposition. Future research will explore other countries and locations. REFERENCES 2458 Table 1: Descriptions of parameters in system(1). Parameter Description Value ω Unreported symptomatic infectious cases that can spread the disease [25] 1 β Average time expended having exposed Estimated 1 σ The average duration of the infectious period without indications Estimated 1 γ Frequency of the symptomatic infectious phase, on average Estimated r proportion of symptomatic infectious disease Estimated S1 Number of people who are at risk at time t1 Estimated E1 quantity of individuals who were exposed at time t1 Estimated I1 Number of infected people that are asymptomatic at time t1 Estimated N1 Indicative infectious cases that were not identified at time t1 Estimated ε Vaccine impact 0.03 Table 2: Simulations of daily reported incidents using models AR.(t), susceptibles S.(t), and cumulative reported cases MR.(t),MR.(t) since the last data point were calculated using t = 1 July 2022 for entirely vaccinated = 98%, 88%, 78%, and social behaviour scaling factor= 0.03, 0.023, 0.016, 0.010.. Vaccinated ε ε = 0.03 ε = 0.023 ε = 0.016 ε = 0.01 98% AR.(t) = 829 AR.(t) = 461 AR.(t) = 124 AR.(t) = 25 S.(t) = 15513000 S.(t) = 18684000 S.(t) = 20992000 S.(t) = 22561000 MR.(t) − MR.(t0) = 4513000 MR.(t) − MR.(t0) = 3074000 S.(t) = 20992000 S.(t) = 22561000 88% AR.(t) = 3742 AR.(t) = 1867 AR.(t) = 589 AR.(t) = 81 S.(t) = 21891000 S.(t) = 27054000 S.(t) = 29723000 S.(t) = 31641000 MR.(t) − MR.(t0) = 4786000 MR.(t) − MR.(t0) = 3074000 S.(t) = 1356000 S.(t) = 1565000 78% AR.(t) = 11246 AR.(t) = 8379 AR.(t) = 2170 AR.(t) = 378 S.(t) = 26847000 S.(t) = 37241000 S.(t) = 41592000 S.(t) = 38683000 MR.(t) − MR.(t0) = 5893000 MR.(t) − MR.(t0) = 3936000 S.(t) = 21320000 S.(t) = 885000 Acknowledgements The authors gratefully acknowledge the approval and the support of this research study by the grand no: SCAR-2022-11-1743 from the deanship of scientific research at Northern Border University Arar, K.S.A. References [1] Eid Alhmadi. 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