key: cord-0907050-6pjg08if
authors: Habenom, Haile; Suthar, D.L.; Aychluh, Mulualem
title: Effect of vaccination on the transmission dynamics of COVID-19 in Ethiopia
date: 2021-11-27
journal: Results Phys
DOI: 10.1016/j.rinp.2021.105022
sha: 09e6bf10d9c96eb75a8114317e0d465edbf073bb
doc_id: 907050
cord_uid: 6pjg08if

Governments and health officials are eager to gain a thorough understanding of the dynamics of COVID-19 transmission in order to devise strategies to mitigate the pandemic’s negative effects. As a result, we created a new fractional order mathematical model to investigate the dynamics of Covid-19 vaccine transmission in Ethiopia. The nonlinear system of differential equations for the model is represented using Atangana-Baleanu fractional derivative in Caputo sense and the Jacobi spectral collocation method is used to convert this system into an algabraic system of equations, which is then solved using inexact Newton’s method. The fundamental reproduction number, [Formula: see text] for the proposed model is determined using the next generation matrix approach.

Coronavirus disease 2019 (COVID-19) is a major public health concern worldwide that has harmed and killed millions of people around the world. It contributes to several political, economic, and social issues. In December 2019, Wuhan, China, reported an unknown-cause viral pneumonia, marking the beginning of a new Coronavirus outbreak [1, 2] . The World Health Organization (WHO) declared the disease a pandemic [3] on March 11, 2020 . Because of its similarities to SARS-CoV, the virus was given the name Severe Acute Respiratory Syndrome Coronavirus 2. The disease caused by this virus is known as COVID-19. The virus is primarily spread from person to person through contact with the droplets of an infected individual. Droplets are released into the air when an infected person coughs, sneezes, or exhales, and can land in another person's nose or mouth, where they can be inhaled into the lungs. Someone who is infected yet has no symptoms of disease can spread a virus (asymptomatically infected). This is why keeping a distance of at least 2 meters (6 feet) is necessary. Infected droplets could fall to the ground or settle on objects. A person can contract the virus by touching his or her lips, nose, or eyes. Fever, dry cough, and shortness of breath or difficulty breathing are the most common symptoms. Muscle and joint pain, headaches, sore throats, and a loss of taste or smell are all symptoms. would be prioritized for vaccination. By the end of 2021, Ethiopia hopes to have vaccinated 20% of its population [8] .

Compartmental dynamic models in epidemiology are viewed as a group of ordinary differential equations and parameters that track the temporal evolution of the number of people in each of the system's phases. Throughout the last decade, the fractional-order derivative (FOD), which is defined as an extension of the integer derivative to a non-integer order (arbitrary order) operator, has been used to model a range of memory and latency phenomena, including epidemic behavior [9] . In this paper, we investigated mathematical model for COVID-19 that might be used to investigate Coronavirus transmission dynamics with vaccination in Ethiopia. The dynamical transmission mechanism of new Coronavirus is studied using Atangana-Baleanu fractional derivative in the Caputo sense.

Atangana-Baleanu fractional derivative. Definition 1.1. Let a function y(t) ∈ H 1 (a, b) , b > a. The Atangana-Baleanu fractional derivative in Caputo sense of the order δ of y(t) is given by [10] as:

The function E δ (ψ) = ∞ n=0 ψ n Γ (δn + 1)

, δ > 0, is the one parameter Mittag-Leffler function and H 1 (a, b) , b > a is called the Sobolev space of order one is defined as

Definition 1.2. The AB fractional integral of the function y ∈ H 1 (a, b) , b > a is given by [10] as:

ABC fractional derivative has a linearity property:

The Laplace transform of ABC fractional derivative is [11] 

(1.6) Jacobi Polynomials: The Jacobi polynomials J (ε, ζ) n (x) of degree n defined on [−1, 1] can be generated with the recurrence relation [12] :

with beginning values J (ε, ζ) 0 (x) = 1 and J (ε, ζ) 1

where u n = (2n + ε + ζ + 1) (2n + ε + ζ + 2) 2 (n + 1) (n + ε + ζ + 1) v n = (2n + ε + ζ + 1) ζ 2 − ε 2 2 (n + 1) (n + ε + ζ + 1) (2n + ε + ζ) w n = (2n + ε + ζ + 2) (n + ε) (n + ζ) (n + 1) (n + ε + ζ + 1) (2n + ε + ζ) Introducing the variable x = 2t − 1 to define the so-called shifted Jacobi polynomial (SJP) on the interval x ∈ [0, 1]. Let P (ε, ζ) n (t) denote the shifted Jacobi polynomials J (ε, ζ) n (2t − 1), and it may be generated as follows: P (ε, ζ)

where r n = (2n + ε + ζ + 1) (2n + ε + ζ + 2) (n + 1) (n + ε + ζ + 1)

The analytical form of the SJP P (ε, ζ) n (t) of degree n:

A square integrable function y(t) on [0, 1] can be expressed as follows in terms of the SJP:

The coefficients a i are given by

The first m + 1-terms of shifted Jacobi polynomials are considered in practice, then we have

is an approximate function in terms of the SJP given by in eq. (1.12). Suppose that δ ∈ (0, 1) then, we obtain:

In this paper, we divided Ethiopia's total population into nine groups: susceptible-S (individuals who are free to the disease can become infected by coming into touch with infected people), vaccinated-V (individuals who have received the COVID-19 vaccine), exposed-E (individuals who have been exposed to the infection but have not yet become infected and cannot transmit the infection to susceptible individuals), asymptomatic infectious-A (having Covid-19 but showing no symptoms at all), symptomatic infectious-I (individuals who are infected with the disease are capable of transmitting the infection to uninfected individuals.), quarantined-Q (to limit Coronavirus transmission, infected individuals are quarantined in a certain place and are isolated and cared for at home), hospitalized-H (infected people are being treated in hospitals), recovered-R (those who have recovered from COVID-19), and death-D (COVID-19 caused individuals to die). The whole population in this model defined as:

The model is based on the following assumptions:

(1) All individuals in the population are initially considered as susceptible. i.e. S(0) = T (0).

(2) People who have been vaccinated may become infected. (6) Individuals become infectious after they have been exposed. (7) Vertical transmission (mother to her unborn baby) is not considered. (8) All cases of infection are considered to be among humans.

All new recruited individuals are assumed to be susceptible and are recruited at a rate . Susceptible people are vaccinated at a rate τ . Vaccinated people may become infected, but at lower rate than unvaccinated people, because some vaccine does not provide immunity to everyone who receive it http://www.who.ch/.

The effective contact rate β is multiplied by a factor ρ(0 ≤ ρ ≤ 1), where 1−ρ describes the vaccine efficiency, ρ = 0 represent the vaccine that provide perfect effective protection against covid-19 infection, while ρ = 1 the vaccine provide no protection at all. The parameter π is the rate at which the exposed individuals become infectious with symptoms of the coronavirus, and σ(1 − π) is the rate that the exposed individuals become asymptomatic infectious. The parameter η and ψ are the rate constants that the symptomatic and asymptomatic individuals enter to the hospital for more treatment. The home-based isolation program is one of the controlling strategies for the spread of the pandemic. The parameter ϕ(1 − η) is the rate at which symptomatic infected individuals transfer to the home-based isolation and individuals transfer from HBIC to the treatment center at the rate δ for better care. Individuals may transfer from treatment centers to HBIC care after improvement at rate γ. Atangana-Baleanu fractional order derivative in Caputo sense is used to described the dynamics of these population as follow:

with positive starting conditions

The parameter ς is the recovery rate of asymptomatically infected, is the recovery rate of hospitalized individuals, ξ is the recovery rate of quarantined infected individuals, they were in home-based isolation and care HBIC. β is transmission rate and ν is modification parameter for the reduction in transmissiblity for symptomatic cases.

The proposed model eq. (2.2) in the following form:

,

Applying the Atangana-Baleanu fractional integral to model eq. (2.6) which is equivalent to model eq. (2.2), we obtain

denote the Banach space of all continuous functions from [0, s] to R endowed with the norm defined by [14, 15 ]

x represent the vector presented as {S, V, E, A, I, Q, H, R, D} and are contractions for 0 ≤ δ i < 1, i = 1, 2, . . . , 9

Proof: For the state variable S, we have

So, Υ 1 (t, S) satisfies the Lipschitz condition with Lipschitz constant δ 1 . Moreover, if 0 ≤ δ 1 < 1, then Υ 1 (t, S) is a contraction. In the similar way, we can show the existence of Lipschitz constants δ i , i = 2, 3, . . . , 9 and a contraction principle for

. Now for t = t n , n = 1, 2, . . . , defined the following recursive form

Taking the norm on both sides of the resulting equation and the difference between consecutive terms in the above equation, we receive:

Furthermore, by taking w = S, eq. (2.12) can be reduced to the following expressions:

As a result, we have 13) and is contraction for 0 ≤ δ 1 < 1. The remaining expressions can be reduced to the following expressions:

(2.14) y = {V, E, A, I, Q, H, R, D} and are contractions for 0 ≤ δ i < 1, i = 2, 3, . . . , 9.

Theorem 2.2. The AB fractional model given in eq. (2.2) has a solution if we can find γ satisfying the inequality

Proof: From eq. (2.13) and eq. (2.14) we have

The existence of the solution (the existence of a fixed point) is confirmed by Theorem 2.1, and we have to show that the functions S(t), V (t), E(t), A(t), I(t), Q(t), H(t), R(t), D(t) are solutions of model eq. (2.2) Let us assume that the following is satisfied:

(2.18)

Repeating the process recursively leads to

Taking t = γ yields

Applying the limit to both sides of eq. (2.20) as n → ∞ we see that q in (t) → 0 for 

Taking the norm both sides, we get

Similarly, we can show that V (t) = V 1 (t), L(t) = L 1 (t), A(t) = A 1 (t), I(t) = I 1 (t), Q(t) = Q 1 (t), Applying the Laplace transform on eq. (2.24) and then taking the inverse Laplace transform, we arrived at

Following the work in [13] , we observe that T (t) ≤ υ as t → ∞. Hence Φ is the biologically feasible region of the model eq. (2.2). 

of the system eq. (2.2) are positive for all t ≥ 0 .

ABC 0

Applying the Laplace transform in the previous inequality (2.27), we arrive at 

υ, 0, 0, 0, 0, 0, 0, 0 . 

The required effective reproduction number is the dominant eigenvalue of the next-generation matrix ST −1 .

That is

Theorem 2.6. The dfe point (2.29) is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.

Proof: We linearize the system of equations (2.2) as follows:

In this matrix, three of the eigenvalues are negative. That is s 1 = s 2 = −υ, s 3 = −τ − υ. The remaining eigenvalues can be obtained from the following characteristic equation:

where

The coefficients d 2 , d 3 , d 4 and d 5 are positive when R 0 < 1, and the coefficient d 1 is positive. Further, the Routh-Hurwitz criteria for fifth-order polynomials are d k > 0, k = 1, 2, 3, 4, 5,

can be easily satisfied by using the above coefficients. So, Ξ 0 is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.

To determine the endemic equilibrium point by adjusting the model equations in (2.2) to zero and solve simultaneously, we get the endemic equilibrium point X = (S * , V * , E * , A * , I * , Q * , H * , R * ): where

H * υ and we have the following quadratic equation for I * :

since b 2 > 0 and b 0 < 0 when R 0 > 1. Therefore, there exists a unique positive value for I * , and as a result, a unique endemic equilibrium point X when R 0 > 1.

Theorem 2.7. The dfe point in eq. (2.29) is globally asymptotically stable for R 0 ≤ 1.

Proof: Assume the Lyapunov function:

Differentiate H(t) using Atangana-Baleanu fractional derivative, we get ABC 0

Substituting the appropriate expressions from the model equations

Since, S ≤ S 0 and at ρ = 1,

We used positive parameters in our model equations; it follows that D α 0 H(t) ≤ 0 for R 0 ≤ 1 and D α 0 H(t) = 0 if and only if, E = 0. Thus, by LaSalle's invariance principle Ξ 0 is globally asymptotically stable in a positively invariant region if R 0 ≤ 1.

Approximate S(t), V (t), E(t), A(t), I(t), Q(t), H(t), R(t) and D(t) of the model equation (2.2) using shifted Jacobi polynomials with m = 3 and fractional order 0 < α ≤ 1 as:

Now, using the idea of Lemma 1.3 and equation (3.1) on the model equation (2.2) we get:

Then, we have 36 equations that can be solved using inexact Newton's method, for the unknowns a i , b i , c i , d i , e i , f i , g i , h i , and u i for i = 0, 1, 2, 3.

The total population of Ethiopia up to June 07, 2021, is approximately 117028692. The initial number of exposed population infectious but not detected by testing has been assumed to be E The sensitivity of quantity R 0 concerning parameter p in eq. (2.2) is given by The graphical interpretations of Figures 5 and 6 show the influence of the fractional order α on approximate solutions of the exposed, death, asymptomatic and symptomatic cases. Numerical results of the behavior of exposed, asymptomatic, symptomatic and death cases in a relation to the vaccination parameter τ are shown in the Figures 7 and 8 below.

(a) (b) Figure 7 Impacts of the vaccination parameter on exposed and asymptomatic cases (c) (d) Figure 8 : Impacts of the vaccination parameter on symptomatic and death cases

Increasing the vaccine efficiency 1 − ρ leads to lowering the number of infected individuals and it is shown in Figure 9 . (e) (f) Figure 10 : Influence of the effectiveness of vaccine on symptomatic and asymptomatic cases

Ethiopia started a first-phase COVID-19 vaccination campaign on March 13, 2021. The first phase focuses on health workers and those who are most exposed to the disease. This dynamics is studied using Atangana-Baleanu fractional order derivative system of equations and the system is converted into an algebraic system of equations using Jacobi spectral collocation method, which is then solved using inexact Newton's method. The simulated results are in good agreement with real data reported in Ethiopia between February 29, 2021 and June 07, 2021. The real data from reproted cases in Ethiopia for asymptomatic and symptomatic confirmed cases, as well as their numerical results, are shown in Figures 4(a) and 4(b) . The simulated results are in good agreement with data reproted in Ethiopia between February 29, 2021 and June 07, 2021, according to our results. We run the COVID-19 model eq. (2.2) for different choices of fractional order α using the values of the parameters in Table 1 . The behavior of approximate solutions of a model eq. (2.2) for different values of fractional order α is shown in Figures 5 and 6 . The fractional order α has its own impact on a model's numerical solutions. We observed a significant reduction in the number of infected individuals in a fractional order α = 0.45. Based on Table 1 , the basic reproductive number is R 0 = 1.2297 > 1, indicating that the endemic equilibrium point is locally asymptotically stable and Ξ 0 is locally asymptotically unstable. Figures 7(a) , 7(b), 8(c), and 8(d) indicate that increasing the vaccination parameter τ from 0.000001593 to 0.001593 reduces the number of exposed, asymptomatic, symptomatic, and death cases significantly. For τ = 0.00001593, total number of exposed cases decreased by 8.04%, asymptomatic cases were reduced by at least 19.84%, symptomatically infected cases were reduced by at least 23.09%, and the number of deaths decreased by at least 9.07%. Finally, a notable reduction is observed for τ = 0.001593, results show a reduction of up to 42.99% of asymptomatic cases and 14.74% for exposed cases, 46.52% of symptomatic cases and 15.61% of death cases. To minimize the spread of coronavirus disease in Ethiopia, it is advised that responsible bodies, policymakers, or the Ethiopian government improve access to a wide range of vaccines. The COVID-19 vaccine, like other vaccines, does not fully protect everyone who takes it. So, even if we have been vaccinated, we must continue to take other preventative measures to battle the pandemic. In Figure 9 , reducing the contact rate from infected to the susceptible population leads to a decrease in the symptomatic and asymptomatic number of cases. Decreasing the modification parameter ν from 0.8882 to 0.0989, the basic reproduction number R 0 reduced from 2.8218 to 0.7276. That is, the disease free equilibrium point becomes stable for ν ≤ 0.2017, provided that the remaining parameters are kept constant. Also, from Figure  10 , increasing the vaccine efficiency by 1 − ρ causes a decrease in the numbers of confirmed cases, both symptomatic and asymptomatic. This result shows an increase in the numbers of susceptible population without vaccination and with vaccination, while the number of exposed cases shows a reduction. Vaccination is a powerful health tool, and the Ethiopian government should prioritize importing vaccines from licensed vaccine manufacturing institutes.

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We use roots of shifted Jacobi polynomials P (ε, ζ) 3 (t) for suitable collocation points. Also, by substituting equation (3.1) in the starting conditions eq. (2.3) we can obtain nine equations: