DYNAMICS OF AN SIR PANDEMIC MODEL USING CONSTRAINED MEDICAL RESOURCES WITH TIME DELAY

. The dynamics of the SIR epidemic model are examined in this paper with ﬁnite medical resources and variable supply efﬁciency are examined along with the implications of time delay. This work demonstrates the stability of endemic equilibrium as well as the incidence of backward bifurcation can be signiﬁcantly impacted by the inclusion of time delay. The theoretical results are supported and supplemented with numerical simulations


INTRODUCTION
In research on epidemiology, mathematical modeling has become increasingly important in transmission of a communicable illness. Several distinct models for epidemics were proposed and thoroughly examined in the research on disease prevention and management [1]. The classic epidemic model typically presupposed that the eradication of infections is inversely correlated with the infective population. This suggests that there are extremely abundant medical resources for infectious diseases, such as medications, antibiotics, hospital wards, and isolation facilities [2], [3], [4], [5] and [6]. The maximum hospital resource availability per unit of time and shows the negative impact of infected patients delaying treatment, which has a substantial effect on the spread of infectious illnesses [7], [8], [9] and [10]. The Chronic illness transmission is significantly impacted by both the lack of medical resources, which is influenced by a variety of factors such as control tactics, drug or vaccine development [11], [12], [13]. In 2004, Wang and Ruan [15], [16] added a constant to SIR model by isolating infectious agents to eliminate the sickness. A constant was used in a SIR model that mimicked a restricted capacity for treatment in order to examine the impact of this limitation on the spread of infectious disease [15] adjusted the ongoing approach to (1) g(I) =      rI * , 0 ≤ I * ≤ I * 0 rI * 0, I * > I * 0 In addition, Wang [16] changed the constant treatment so that the maximum value, rI * 0, was taken when the maximum amount of patients could be treated and the rate of treatment matched the number of infections. The dynamics of various epidemic models with average and standard incidence rates [17], [18] and [19] were investigated by several authors who embraced the staged treatment function. Linhua Zhou and Meng Fan [14] most recently develop a new continuously distinct treatment function in response to the saturation phenomenon of the limited medical resources and meticulously examined the dynamics of the subsequent SIR model.
Here, β SI 1+KI is saturated and indicates either the inhibitory effect or the psychological influence. Where α ≥ 0 denotes the total number of medical assets made available in one time unit and ω > 0 is a constant for part-saturation, which evaluates the efficiency of the availability of medical assets in that it is more effective if is lower. The dynamics of the SIR model (2) are widely explored in [14] have been thoroughly researched and some intriguing findings, like backward bifurcation and local equilibrium stability, the simulation's dynamics are still far from being finished. We continue to examine the SIR epidemic model's dynamics (2) in [14]. From the aforementioned inspiration study, we have structured our work with delay. We further investigate the Dynamics of a SIR Pandemic Model with Limited Medical Resources based on the earlier arguments.
The proposed time delay model is, where, τ represent the time delay. The percentages S(t), I(t), and R(t) correspond to the number of susceptible, infected, and recovered individuals at time t. In section 2, we examine the existence of backward bifurcation and endemic equilibrium as well as the impact of scarce medical resources and the effectiveness of their supply on the backward bifurcation with time delay.
We examine the local stability of the equilibria by examining the eigenvalues of the Jacobian matrix. Also, provide a model-wide analysis look at the endemic equilibrium's and the diseasefree equilibrium's local and global asymptotic stability. In section 3, The intervention analytical and numerical bifurcation analyses are provided to further the theoretical findings.

BACKWARD BIFURCATION
The anticipated amount of new sick people that a single diseased person introduces into a population free of illness is known as the basic reproduction number, which is typically denoted by R 0 , is one of the essential ideas in dealing with endemic models and is crucial to epidemiology. One frequently notices the threshold property, which states that the disease will not spread if R 0 < 1 and will do so if R 0 > 1. Forward bifurcation in this context refers to the transition from an endemic equilibrium to equilibrium free of illness. The endemic equilibrium and the disease-free equilibrium coexist when R 0 < 1 and the essential criterion for the complete eradication of the disease cannot be the basic reproduction number, as is the case with according to an increasing number of studies. To get certain essential thresholds for disease control, it is crucial to locate the backward bifurcations. By examining the occurrence of the disease-free and the endemic equilibrium, we explore the feasibility of backward bifurcation in the α-ω plane in this section. According to [20], next generation approach, indicates that the fundamental reproduction number of (3) is, Similar to the non-delayed model, the delayed model's disease-free equilibrium is described by The algebraic equation below can be solved to produce the endemic equilibrium: which has the following quadratic equation solution: Where, The roots of the quadratic equation can be obtained using the quadratic formula: The corresponding values of susceptible can be obtained using the equation: These values of susceptible represent the candidates for the endemic equilibrium of the delayed model. Where, where I 2 is the quadratic equation's bigger root.
Here, Ω 1 , Ω 2 ,and Ω 3 are defined as in the original theorem, and α 0 (ω) is given by: Theorem 1. Assuming R * > 1, we can make the following statements about the equilibrium of this DDE: (i) If R 0 > 1, then the endemic equilibrium E 2 of DDE is distinct.
Proof. By the model (5) we have: disease transmission, and the recovery of infected individuals, respectively. However, the calculations become much more complicated due to the presence of time delays.
In general, the existence of temporal delays can significantly affect the system's dynamics and cause complicated behaviors to emerge in emergency situations.
Let τ be the time delay, then the modified equation becomes, In this equation, N (t) stands for the nation's size at time t, r for intrinsic growth, K for carrying capacity, C for per capita mortality rate, and τ for time delay. Where N(t − τ) denotes the total number of infected people at time t − τ.
The condition for the existence of an endemic equilibrium E 2 is given by the positive root of equation (6).
The following prerequisites must be met for endemic equilibrium E 2 to exist in the presence of temporal delay: (1) If R 0 < 1, then it has no endemic equilibrium.
(2) If R 0 > 1, then (6) has a unique positive root when C < 0 and the condition for the existence of E 2 is given by P 1 < 1, where and, Therefore, in the presence of time delay, the condition for the existence of endemic equilibrium E 2 depends on the value of which in turn depends on the parameters of the system and the time delay τ. We introduce time delay in the model and consider the following delay differential equations: where τ is time delay, and N = S 1 + S 2 + I 1 + I 2 is the total population. The conclusion reached in the absence of time delay still hold in the presence of time delay. We consider the same cases as before: As the phenomenon, we have ∆ = 0, and I 1 (t) = I 2 (t) for every t. Also, we have C > 0 and B < 0. So a special endemic equilibrium E 1 = E 2 exists in (5), which persists in the presence of time delay.
In this case, we have B ≥ 0 and C ≥ 0, and no positive root exists for (6). Hence, there is no endemic equilibrium in (5). This conclusion reached in the absence of time delay still hold in the presence of time delay, the evidence is conclusive.
Theorem (1) provides a comprehensive understanding of the endemic equilibrium exists in the presence of time delay. The parameters ω and α play an important part in determining the dynamics of the delayed differential equation (5). Specifically, if (ω, α) ∈ Ω 1 ∪ Ω 2 , then no endemic equilibrium exists in (5). For 0 < R 0 < 1 and a distinctive endemic E 2 when R 0 < 1.
From (5), a forward bifurcation is visible at R 0 = 1, where the healthy equilibrium to one single endemic equilibrium E 0 changes.
Proof. In particular applications, an endemic backward bifurcation with equilibrium when R 0 < 1 is very significant. The actual critical threshold for curing a disease is P * . The reversible bifurcation from the disease-free equilibrium E 0 at R 0 = 1 result in the establishment of both endemic balances E 1 and E 2 for P * < R 0 < 1, as shown in previous theorem with time delay.
The phase portrait of (5) is shown in Fig. 1b for (ω, α) ∈ Ω 3 and R 0 with time delay, where E 0 is an equilibrium devoid of disease and E 1 and E 2 are two endemic equilibrium states. According to the premise that E 1 refers to a saddle and the fact that E 2 and E 0 are asymptotically stable at the local level proves that a backward bifurcation has taken place.
2.1. Dynamics on a global scale and the stability of equilibria. Here, we discuss the analysis of the globalized dynamics dynamics of (5) by investigating the regional consistency of equilibrium, which will involve examining the Jacobian matrix eigenvalues for each equilibrium position. Specifically, we will start by considering the (5), Jacobian matrix evaluated at the optimal equilibrium E 0 after a certain time delay. and Theorem 3. If R * ≤ 1 and (ω, α) ∈ Ω 1 ∪ Ω 2 , then the endemic balance E 1 of (5) is locally unstable if D > 0 and locally symmetrically stable if D < 0. If R * > 1 and (ω, α) ∈ Ω 3 , then E 1 is locally symmetrically stable, if R 0 < P 1 and erratic if R 0 > P 1 . If R 0 > 1 and R * > 1, then E 1 is erratic. If E 1 exists, it is saddle.
Proof. To prove the theorem, we first consider the system (5), Jacobian matrix evaluated at endemic equilibrium E 1 : where I(E 1 ) is the value of I at the endemic equilibriumE 1 , and we have used the notation I(E 1 ) to indicate their dependence. To analysis the stability of E 1 , we can use the characteristic equation of J(E 1 ), which is a third-order polynomial of the form: At τ = 0, the endemic equilibrium, E 1 , is locally asymptotically stable. This criteria indicates that the intrinsic equilibrium E 1 is locally asymptotically stable in the absence of a temporal delay, which may be confirmed by looking at the Jacobian matrix J(E 1 ) assessed at τ = 0.
Hence, E 1 is a saddle point.
Proof. To explore we introduce the presence of time delay τ in the equation for the infected individuals: which leads to the following delayed system.
It is trivial to demonstrate that the Jacobian matrix E 2 exists. The Jacobian matrix J(E 2 ) of this delayed system evaluated at the endemic equilibrium E 2 , which is defined below: Next, We determine J(E 2 ) 's characteristic equation, which is given by: where, Conditions that must be met in order for E 2 to be locally asymptotically stable are that m 1 > 0, m 2 > 0, and m 3 > 0. However, the conditions for m 1 and m 2 are complicated expressions.
We begin the linearizing the system about E 2 : Then, using time delay, we write the system as follows: Therefore, To obtain the characteristic equation, we assume that the solution of the form, e rt , r + µ + Λ 11 e −rτ 1 Λ 1 + Λ 12 e −rτ 2 Λ 2 + Λ 13 e −rτ 3 = 0, By solving r, the determinant of the coefficients of the system is given by where, Since, E 2 is a positive constant and the other terms are also positive.
Proof. We will assume that the conditions for the existence of E 2 are satisfied.
First, we linearize the system around E 2 and obtain the Jacobian matrix: where δ is the time delay. The characteristic equation of J(E 2 ) is given by According to this criterion, E 2 is asymptotically stable if all the coefficient of the characteristic equation is positive, and there are no sign changes in the sequence of the coefficients. Let's define the following expressions: . Then, the condition for stability is P 1 , P 2 , P 3 , q 1 , q 2 > 0. Now, If all the eigenvalues are negative, then the endemic equilibrium E 2 is locally asymptotically stable. If any real components of eigenvalue are positive then E 2 is unstable. Since, E 2 is locally asymptotically stable at µ > 0, and E 2 is unstable at µ < 0.
Proof. Assume by contradiction that there exists a closed orbit in the phase space of system (5), this indicates that a periodic solution exists with period T > 0. Without loss of generality, we consider the orbit lies in the first quadrant of the phase space.
Let us construct a suitable Dulac function that satisfies the conditions of the Bendixson-Dulac theorem. Let: Then, We have, Therefore, for all (x, y), ∂ ∂ x (D( f 1 )) + ∂ ∂ y (D( f 2 )) lies in the first quadrant. Since, ∂ D ∂ x and ∂ D ∂ y are both continuous on the first quadrant, h(x, y) is a continuously differentiable function that occurs in such a way that, Thus, by the Bendixson-Dulac theorem [21], there are no closed orbits in the phase space of system (5).
Proof. To prove the global asymptotic stability of E 2 , we use the Lyapunov function approach. Consider, Taking the time derivative of V, Using the expression for I * from Theorem 7, we can simplify the above expression as where, µ = 2A(m 2 ω 2 +Bm 2 +m 1 ) ω . Note that µ > 0 for the given conditions on R 0 and α. Therefore, dV dt is negative definite and (I(t) − I * )(I(t), S(t)) is a Lyapunov function for the system. This suggests that the system's entire set of solutions is covered to the set I = I * , S = 0 as t → ∞.
Since, I * is the only positive solution making it unique to the equation ψ(I) = 0, the only equilibrium in this set is E 2 . Hence the proof.
Theorem 8. The disease-free equilibrium E 0 is globally asymptotically stable if one of the following conditions are satisfied.
Proof. From the system (5), In this case, the ideal state of equilibrium E 0 is the solitary equilibrium point of the system. It is simple to demonstrate that E 0 is locally stable using the Routh-Hurwitz criteria. Furthermore, the region D = ((S, I), I ≥ 0, S + I ≥ 0) is positively invariant with regard to the integrity of the system (5). This means that every solution trajectory, a closed orbit or equilibrium that is located in D. Since, condition (i) is holds; No endemic equilibrium exists in the system (5), and hence, there is no closed orbit in D. Since, every solution trajectory beginning in D will therefore go closer to the disease-free equilibrium E 0 . By the Poincare-Bendixson theory, E 0 is asymptotically stable everywhere. The evidence for case (i) is now complete.
In this case, there exists at least one endemic equilibrium E * of the system (5). It is simple to demonstrate that E * is locally stable using the Routh-Hurwitz criteria. Furthermore, with regard to the system, the region known as D is still positively invariant (5). Since, condition (ii) is holds; the system (5) has no closed orbits in D. Therefore, every solution trajectory starting in D will approach either the equilibrium without disease E 0 or the epidemic equilibrium E * . By the Lasalle invariance principle [22], every solution trajectory starting in D will approach the largest invariant set contained in the union of the unaffected equilibrium E 0 and the infectious equilibrium E * . If the initial condition (S(0), I(0)) in D, then S(t) ≥ 0 for all t ≥ 0. Since, R 0 < P * , the endemic equilibrium E * is unstable. Therefore, each solution trajectory will move closer to the disease-free equilibrium E 0 as t → ∞ starting from D. By the Poincare-Bendixson theory, E 0 is asymptotically stable. The evidence for case (ii) is now complete. Therefore, we have shown that if either condition (i) or condition (ii) is satisfied. Consequently, the disease free equilibrium E 0 is globally asymptotically stable.

NUMERICAL SIMULATION
A numerical simulation is used to examine how infectious illnesses spread throughout a pop-

CONCLUSION
We investigated the dynamics of SIR epidemic model of supply variability in time delay and resource constraints on availability of medical care. We demonstrated the increase in temporal delay can have a considerable impact on both the occurrence of backward bifurcation and the beginning and stability of endemic equilibrium. The SIR compartmental model is also obtained from the value of balanced infectious sizes versus R 0 . Finally, our theoretical conclusions were strengthened and verified by numerical simulations for the system with a time delay.