EFFECT OF PARTIALLY AND FULLY VACCINATED INDIVIDUALS IN SOME REGIONS OF INDIA: A MATHEMATICAL STUDY ON COVID-19 OUTBREAK

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INTRODUCTION
A Mathematical model can be defined as the formulation of set of equations that express the essential features of a physical system or a process in mathematical terms. In other words, it is actually a real-world problem that is converted into a mathematical term. There are four steps to construct and solve mathematical modelling. First, we understand the problem and describe it in mathematical terms, secondly, we will construct a basic model in which parameters and variables are defined logically. Thirdly, we will find different ways to solve the model. Finally, we will associate the solution or result of the proposed model to the real-world problem [1]. The main objective of mathematical modelling is to test the effect of changes in a system [2], [3]. In the modern global scenario, infectious disease investigation and control have received a lot of attention [4]. With increasing complications in human everyday life, as well as climatic change due to global warming, pathogens of numerous infectious diseases have become more active than in the past. As a result, in addition to medical diagnosis, a full theoretical examination of the related epidemic system is required. Mathematical models are very beneficial for describing an epidemic system in this theoretical inquiry [5], [6]. Kermack and Mckendric [7] were likely the first to accurately characterise an infectious disease system using a mathematical model. Also, many other researchers worked on mathematical modelling on the transmission of disease such as TB, dengue, HIV-AIDS, Nipah virus etc, which was described in [8], [9], [10], [11], [12], [13], [14]. Here we take COVID-19 as our real world problem and we try to frame a model logically which was described in section 2.
COVID-19 is a deadly infectious disease which was caused by a special type of coronavirus known as severe acute respiratory syndrome coronavirus 2, that was first emerged in December 2019 in the Large town of Wuhan [15]. The viral load is carried from person to person and has spread throughout China, followed by the rest of the world. The COVID-19 virus causes weak to moderate respiratory illness in the majority of persons who are infected with it, but they recover without needing any special treatment [16]. But older people and with low immunity are more likely to develop serious illness. Various safety measures such as social distancing, usage of medical masks and hand hygiene practices are recommended to prevent from the spread of the COVID-19, which was implemented immediately across the globe by the World Health organization. On March 12, 2020 World Health Organization declared the outbreak as pandemic [17].
There were only 1,18,000 cases in 114 countries at the time, with 4,291 individuals deceased.
However, after that the number of cases increases rapidly and in a discontinuous manner. These viruses attack the human population by different waves. Also, the rate of increase, both in the number of cases of infection, recovery and death, varies from region to region. Particularly in India, the first wave occurs between July to November 2020 followed by second wave between Recently, Fanelli and Piazza constructed a SIRD system by classifying the human population into four compartment as susceptible, infected, recovered and deceased classes. They modelled the continents of China, Italy and France using real data and predicted the transmission of COVID-19 in these countries [20]. J. Jiang et al. developed SEIR model to forecast the COVID-19 epidemic trends particularly in USA, New York and Italy [21]. The authors Quian li et al. [22] has proposed the impact of mass influenza vaccination model on COVID-19 epidemic.
Here in our proposed work, we constructed SEIR Model by incorporating the fact of partially vaccinated and fully vaccinated individuals to the population. Then we analyse the behaviour of our model and also we studied their effects in some regions of India. Mainly we have focussed on four regions of India. We consider Tamil Nadu, Maharashtra, West Bengal and Delhi and studied their real data scenario of partially vaccinated and fully vaccinated population and tried to fit those data to our proposed model.
The following is an overview of our paper structure. The forthcoming section presents the four dimensional nonlinear mathematical model, followed by the boundedness and positivity.
Section 3 provides an overview to equilibrium analysis and basic reproduction number. Section 4 deals with both local and global stability analysis of our proposed model. In section 5 we have presented our analytical solution in terms of numerical simulation and finally we have focussed on real data validation in section 6. The paper ends with brief conclusion in section 7.

THE MODEL FORMULATION
We consider the whole human population N(t) and we have divided into four compartments namely Susceptible Individuals S(t), Exposed Individuals E(t), Infected Individuals I(t) and Recovered Individuals R(t). To build our system, the following assumptions has been made: Individuals are recruited in the region at a fixed rate A and joins the susceptible compartment.
The susceptible population become exposed to the infection, when infected individual become contact at the rate α. Also we assume that, the exposed population become infected at a constant rate δ > 0. Here we have incorporated partially vaccinated population V p and fully vaccinated population V f into the model to make it more realistic. So those who are partially vaccinated may get exposed and the fully vaccinated individuals directly moves to recovery compartment.
Other than vaccinated individuals, the remaining infected individuals are recovered due to treatment at the rate γ.
So, by considering the above divisions and assumptions a simple nonlinear mathematical model is formulated.
Further the model (1) is simplified as below: where, Parameters description of the model (1) is given in the Table 1. Also the transfer diagram of our model is described in Figure 1. α Rate of interaction between susceptible and infected individuals.
µ Natural death rate.

V p
Partially vaccinated population.

V f
Fully vaccinated population.
δ Exposed population becomes infected population at a constant rate δ > 0.
γ Rate of medication to recovery.

Positivity and Boundedness of the model
In this subsection, we proved the positivity and boundedness of the system. From (1), we Clearly, all the population and parameters are non-negative in the region. Further from the model (1), the total population is N = S +E +I +R and the rate of change of the total population is given by, In absence of disease we have I = 0, therefore dN dt = A − µN. This gives that lim sup N ≤ A µ . Therefore all the solutions of S, E, I and R are bounded by A µ . Hence the biologically feasible region of the model (1) is provided by positive invariant set:

EQUILIBRIUM ANALYSIS OF THE MODEL
In this, we analyse the existence of equilibrium points for the system (2). Here we obtained two equilibrium points namely E 0 = (S 0 , E 0 , I 0 , R 0 ) and E 1 = (S * , E * , I * , R * ).

Disease Free Equilibrium Points (E 0 )
For the proposed system, we obtained disease free equilibrium points The estimated number of secondary infections developed in susceptible individuals by a typical infected individual during his or her whole infectious period is denoted as the basic reproduction number. The reproduction number R 0 for the model is found out using the next generation matrix method which was mentioned in [23]. For this we consider, F (t), V + (t) and V − (t) . Where F (t) is the rate at which new infections appear in the compartment, V + (t) is the rate at which individuals are transferred into the compartment and V − (t) is the rate at which individuals are transferred out of the compartment. In this way, F (t), V + (t) and V − (t) are associated with the system (2). Which are obtained as, At last, the matrix FV −1 is given by, Thus the basic reproduction number of the system (2) is obtained by the dominant eigen value of FV −1 . Here, we obtain R 0 = Aαδ k 1 k 2 k 3 .

Endemic Equilibrium Points (E 1 )
For the proposed system, we obtained the endemic equilibrium points where, , I * is the root of the following quadratic equation. where, Clearly, M 1 > 0, M 3 < 0 and M 2 < 0 when R 0 > 1. By using Descartes rule of signs, we get Therefore, unique positive endemic equilibrium root exists.
So obviously the other root is negative. So we neglect the negative root.

STABILITY ANALYSIS
In this section we find both local and global stability of disease-free equilibrium and endemic equilibrium points. The Jacobi matrix for the model (2) is given by,

Local Stability Analysis
Disease Free Equilibrium (DFE)

Theorem 1
If R 0 < 1, E 0 is locally asymptotically stable and otherwise it is unstable.

Proof:
From model (2), the Jacobi matrix at disease free equilibrium Clearly, −µ is one of the eigen value of J 0 matrix. The other three eigen value are computed from the below matrix. The characteristic equation corresponding to matrix A 0 is obtained from, where, a 2 = a 22 a 33 − a 32 a 23 + a 11 a 33 − a 31 a 13 + a 11 a 22 − a 21 a 12 , According to Routh Hurwitz Criteria disease free equilibrium, E 0 is locally asymptotically stable if it satisfies a 1 > 0, a 3 > 0 and a 1 a 2 > a 3 . Here we get a 1 = k 1 + k 2 + k 3 , So from above clearly we can say, a 1 > 0, a 2 > 0 when R 0 < 1 if and only if k 2 k 3 − Aαδ k 1 > 0 and a 3 > 0 when R 0 < 1 if and only if k 1 k 2 k 3 − Aαδ > 0.

Theorem 2
If R 0 > 1, E 1 is locally asymptotically stable and otherwise it is unstable.

Proof:
From model (2), the Jacobi matrix at endemic equilibrium E 1 = (S * , E * , I * , R * ) is obtained as follows, Clearly, −µ is one of the eigen value of J 1 matrix. The other three eigen value are computed from the below matrix.
The characteristic equation corresponding to matrix B 1 is obtained from, Now, according to Routh Hurwitz Criteria endemic equilibrium, E 1 is locally asymptotically greater than one, if and only if k 2 k 3 − S * αδ < 0 and b 3 > 0 when R 0 is greater than one, if and Hence endemic equilibrium E 1 = (S * , E * , I * , R * ) is locally asymptotically stable whenever R 0 is greater than one. This completes the proof.

Global Stability Analysis
Disease Free Equilibrium (DFE)

Theorem 3
If R 0 < 1, Disease free equilibrium of system (2) is globally asymptotically stable and unstable otherwise.

Proof:
The Lyapunov function for our proof is constructed as follows, Now, differentiating with respect to time we get, Since we know, S ≤ 0, E ≤ 0 and R ≤ V f A µk 1 . So, we replace and proceed as, Clearly, for I = 0, we getV 1 = 0 andV 1 ≤ 0 if and only if R 0 < 1. It follows from Lasalle's Invariance Principle, that is every solution to the system (2) with initial condition in ξ approaches disease free equilibrium as t → ∞. Since we already know region ξ is positively invariant, which was described in (4).
Therefore E 0 is globally asymptotically stable in the region ξ whenever R 0 is less than one.
Hence the proof.

Endemic Equilibrium (EE)
In this subsection, the global stability of endemic equilibrium is derived with the help of Poincare Bendixson technique. Let g(x) : H ⊂ R n → R n be a map, where H is an open set and if E = g(x) be a differential equation, then by using the initial condition x(t)= x 0 , the solution can be determined.
• H is simply connected.
• There exists a compact absorbing subset K of H.
• In H,x is the globally stable equilibrium point wheneveṙ ∂ x is the second additive compound Jacobian matrix of order two and Ψ denotes the Lozinskii measure [24].

Theorem 4
The system is globally asymptotically stable in the region ξ , whenever R 0 is greater than one.

Proof:
Leaving the recovered population, the Jacobi matrix can be written as, Also we know that, for A 1 = (a i j ), To find the matrix W , we define a diagonal matrix P = diag {1, E I , E I }. Let f be the vector field of the model, then Hence W matrix is given by, Therefore, block matrix can be written as, Now, the Lozinskii measure of W is obtained from Ψ(W ) ≤ Sup {g 1 , g 2 }, where g 1 = Ψ(w 11 ) + |w 12 | and g 2 = |w 21 | + Ψ(w 22 ).

Now from model
By substituting in g 1 and g 2 we get, I(0), R(0)) in the absorbing set. Thereforeq 2 is strictly less than zero, which satisfies the condition. Therefore, the endemic equilibrium is globally stable in the region ξ . Hence the proof.

NUMERICAL SIMULATION
In this section, we simulate the proposed model (1) by assuming different set of parameters.
We consider the following set. A = 17000, α = 0.0006, δ = 0.0079, γ = 0.0929, µ = 0.989, .0001 and V f = 0.03. For the above set we obtained R 0 as 0.0159, which is less than one. Therefore, disease free equilibrium is stable. This fact is illustrated in Figure 2 (a). Next, we make some changes in parameter which corresponds to endemic equilibrium. For this case we obtained R 0 as greater than one. Therefore, endemic equilibrium is stable. Which is also illustrated in Figure 2 (b). is obtained as 0.0959 which is less than one. Also from the Figure 3 (a), we can clearly note that the infection will vanish out in a short period of time. Figure 3 (b) demonstrates the effect of partially vaccinated individuals on the exposed population. This clearly indicates that the exposed population increases gradually whenever we increase the parameter V p as 0.1, 0.2, 0.3 and 0.4 respectively.   consider 5% of population are fully vaccinated and 15% are partially vaccinated from the total assumed population. From 5 (a) we interpret that 5% of V f and 15% of V p to the population, gradually increases the recovered population and gradually decreases the exposed population.
Also, the value of R 0 is obtained as 4.765. Whereas in Figure 5 (b), we consider 60% are fully vaccinated and 30% are partially vaccinated individuals. From this we interpret that 60% of V f and 30% of V p to the population, drastically increases the recovered population and drastically decreases the exposed population. Also, the value of R 0 is obtained as 1.059, which is closer to one. In another way, we can say that increase in parameter V f decreases the basic reproduction number R 0 . Effect of both V p and V f , when V p = 30% and V f = 60%.  From the above data, we can see the more number of doses administrated in West Bengal and less number administrated in Delhi. Also, we make some reasonable assumptions to the parameter, which was described in Table 4 and Table 5. By using python software, we tried to fit the simulated and observed data from all the considered regions during the time period from January 2021 to December 2021. Figure 6 demonstrates the real scenario of partially vaccinated and fully vaccinated individuals of the regions of Tamil Nadu, Maharashtra, West Bengal and Delhi for the time period of twelve months. In Figure 7, we tried to fit the monthly cumulative partially and fully vaccinated data of four regions. In that blue dots represents the observed vaccinated data and the red curve represents the corresponding fitted curve from our proposed model (1). Figure 7   Overall, from the graphs, we observe that in the regions of Tamil Nadu, Maharashtra and Delhi, the fully vaccinated population real data increases drastically. That is administration of V f to the population increases, whereas in West Bengal the fully vaccinated data increases from the month of November. So, at the beginning the vaccination drive in all the considered region was slow but at the second half of the year, especially from the month of August it increases gradually. Mainly administration of V f to the population increases drastically.

REAL DATA VALIDATION
So, from the proposed model we can comment that, increase in V f increases the recovered population. So that completely vaccinated individuals will have less impact to the infection and moves directly to the recovered class. Due to sudden jump in vaccination drive, we predict that the infected people from virus decreases. So that the infection due to virus may start decreasing in the forthcoming waves.

CONCLUSION
In this paper, a four-dimensional SEIR model of COVID-19 is developed and analysed with the effects of partially and fully vaccinated individuals in some regions of India. From our proposed model we find some of the mathematical findings, which was discussed here. Our model exhibits both disease free equilibrium and endemic equilibrium. Both are locally asymptotically stable whenever R 0 < 1 and R 0 > 1 respectively. Furthermore, Lyapunov function is constructed to show that, our disease-free equilibrium is globally asymptotically stable whenever R 0 < 1.
Next, by using Poincare Bendixson technique we also proved that our endemic equilibrium is globally asymptotically stable in the region ξ , provided R 0 > 1. Then numerical simulation is performed for different set of parameters and from that we observed that, the exposed population is directly proportional to the partially vaccinated individuals and the recovered population is directly proportional to the fully vaccinated individuals. Finally, we validated our proposed model with the real-world data, which indicates that the infection due to virus may start decreasing in the forthcoming waves, since in all the studied regions, the fully vaccination drive has a sudden jump at the second half of the year 2021. So that from our model we comment that, infected people due to virus may decrease due to the increase in fully vaccination drive to the individuals particularly in the regions of Tamil Nadu, Maharashtra, West Bengal and Delhi.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.