Stability analysis of an SIR model with alert class modified saturated incidence rate and Holling functional type-II treatment

Shivram Sharma; Praveen Kumar Sharma

doi:10.1515/cmb-2022-0145

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Stability analysis of an SIR model with alert class modified saturated incidence rate and Holling functional type-II treatment

Shivram Sharma und Praveen Kumar Sharma

Veröffentlicht/Copyright: 8. Mai 2023

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Abstract

This study discusses an SIR epidemic model with modified saturated incidence rates and Holling functional type-II therapy. In this study, we take the new alert compartment (A) in the SIR compartment model. Consider the modified non-linear incidence rate from the susceptible to the infected class and the second non-linear incidence rate from the alert to the infected class. Further, we investigate the elementary reproduction number, the equilibrium points of the model, and their stability. We apply manifold theory to discuss bifurcations of the disease-free equilibrium point. This study shows that the infected population decreases with the Holling functional type II treatment rate. It also shows that the number of infected people decreases when the psychological rate increases and the contact rate decreases.

Keywords: alert class; SIR model with alert class; Holling functional type-II treatment rate; basic reproduction number

MSC 2010: 34D23; 93A30; 93D20

1 Introduction

Many diseases are spreading rapidly at this time, and mathematical modeling helps a lot in studying them and preventing their spread. Many authors [1,2,6,7,9,13,17] have presented various epidemic models. Naresh et al. [10] developed an SIR compartmental model with a time delay for epidemiology. Yi et al. [17] considered the transmission rate in the SEIR epidemic model, and Nicho [11] revealed herd immunity to many diseases in proportion to those immunized. Kar and Batabyal [7] proposed an SIR epidemic model and described how afflicted populations were treated. Van den Driessche and Watmough [14] examined the reproduction numbers and equilibrium points of the epidemic model. Wang and Ruan [15] introduced the piecewise treatment function, which is known as

G ( I ) = c , when I > 0 0 , when I = 0 ,

where c denotes the infective’s constant clearance rate.

Zhang and Suo [18] introduced the Holling functional type-II therapy rate, which is given by

H ( I ) = aI 1 + bI , I ≥ 0 ,

where a is a non-negative constant and b is a saturation constant effect rate, which is always positive. Cooper et al. [3] developed an SIR model for COVID-19 in the context of four countries and explored the various implications of COVID-19’s spread. Erdem et al. [4] suggested an SIQR model in which restricted persons have a lower primary reproduction number and transmission dynamics. Kumar and Kishor [8] introduced the alert (A) class in the SIR model with the various occurrence and treatment rates.

We will formulate an SIR epidemic model with an alert class, two modified explicit saturated incidence rates, and Holling functional type-II treatment rates and determine the equilibrium points, primary reproduction number, disease-free stability (DF), and endemic equilibrium point of the system. In the last section, we do the simulation, which validates our hypothesis and the principal results. At last, we give a detailed conclusion about the presented study.

2 Formulation of model

This section will discuss the model’s formulation, basic reproduction numbers, and DF and EE points. Here total Population N is divided into four compartments, namely, susceptible (S), alert (A), infected (I), and recovered individuals (R), respectively. Susceptible (S) people are healthy people who can be infected under the right circumstances. Alert (A) individuals are those who are aware of the effects of the disease, infection, and symptoms and take the required steps to control the infection, as well as those who can contact the infected under the right circumstances. Infected people are those who have already been infected with the virus and can pass it on to others who are susceptible and alert.

Infected people lose their infectivity over time, either through auto-recovery (because of their body’s immune system) or treatment, and then move to the recovered class.

Let π be the constant rate of susceptible recruitment, either the birth rate or immigration rate. Assume that susceptible individuals move from susceptible to alert at a rate of δ ( δ denotes the rate of alertness of susceptible individuals), and β is the transmission rate of susceptible individuals to infected individuals.

When an infected person comes in contact with a susceptible person, the susceptible can get infected at one rate, and when the same infected person comes in contact with an aware person, the aware can go to the infected compartment at another rate, i.e., an infected person will infect a susceptible and aware person at different rates. Therefore, we will take two different incidence rates when an infected person comes in contact with a susceptible and alert person. In this study, we take an incidence rate β SI 1 + α I 2 from susceptible to infected individuals, where α is a psychological effect rate in infected individuals. Let γ AI 1 + α I be the second incidence rate from alert to infected individuals, where γ denotes the alerts transmission rate to infected individuals. Suppose μ , d , and θ represent the natural fatality rate, disease-related death rate, and infected individuals' recovery rate. Let H ( I ) = aI 1 + bI be the treatment rate of Holling functional type-II, a representing the cure rate and b the rate with resource limitations.

As a result, we get the following system of nonlinear ordinary differential equations:

d S d t = π − ( μ + δ ) S − β SI 1 + α I 2 ,

(1) d A d t = δ S − μ A − γ AI 1 + α I ,

d I d t = β SI 1 + α I 2 + γ AI 1 + α I − ( μ + d + θ ) I − aI 1 + bI ,

d R d t = aI 1 + bI + θ I − μ R ,

where S ( 0 ) ≥ 0 , A ( 0 ) ≥ 0 , I ( 0 ) ≥ 0 , and R ( 0 ) ≥ 0 .

Now, we shall find the feasible region of the model (1), since the sum of all compartments is equal to the total population, i.e., N ( t ) = S ( t ) + A ( t ) + I ( t ) + R ( t ) . From system (1), we get N ( t ) → π μ as t → ∞ .

Therefore, the bounded and positively invariant feasible region for the system (1) is W = ( S , A , I , R ) ∈ R + 4 : 0 < S + A + I + R < π μ .

As a result, we have the following lemma.

Lemma 1

The positively invariant region of the system (1) is a set W = ( S , A , I , R ) ∈ R + 4 : 0 < S + A + I + R < π μ .

Since the recovered compartment is not present in the first three equations of the model, hence, as a result, we have the following reduced system of equations:

d S d t = π − ( μ + δ ) S − β SI 1 + α I 2 ,

(2) d A d t = δ S − μ A − γ AI 1 + α I ,

d I d t = β SI 1 + α I 2 + γ AI 1 + α I − ( μ + d + θ ) I − aI 1 + bI

The basic reproduction number of the model is ℜ 0 = ( β μ + γ δ ) μ ( μ + δ ) ( μ + d + θ + a ) , which can be determined by the next generation method given by Diekmann et al. [5].

2.1 DF and EE points

By equating all the equations of system (2) to zero and solving them, we get the disease-free equilibrium (DFE), and EE exists when the primary reproduction number is greater than 1. Thus, the DFE and EE are given by E 0 = ( S 0 , A 0 , 0 ) and E 1 = ( S * , A * , I * ) , respectively,

where S 0 = π μ + δ , A 0 = δ π μ ( μ + δ ) , S * = π ( 1 + α I * 2 ) ( μ + δ ) ( 1 + α I * 2 ) + β I * , A * = δ S * ( 1 + α I * ) μ ( 1 + α I * ) + γ I * , and I * is given by

(3) c 5 I * 4 + c 4 I * 3 + c 3 I * 2 + c 2 I * + c 1 = 0 ,

where c 5 = b α ( δ + μ ) ( μ α + γ ) ( μ + d + θ ) > 0 ,

c 4 = ( μ α + γ ) [ α ( δ + μ ) ( μ + d + θ + a ) + b β ( μ + d + θ ) ] + μ α b ( δ + μ ) ( μ + d + θ ) − b α γ δ ,

c 3 = ( μ α + γ ) [ β ( μ + d + θ + a ) + b ( δ + μ ) ( μ + d + θ ) ] + μ [ α ( δ + μ ) ( μ + d + θ + a ) + b β ( μ + d + θ ) ] − b β ( μ α + γ ) − π α γ δ ,

c 2 = ( δ + μ ) ( μ α + γ ) ( μ + d + θ + a ) + μ [ β ( μ + d + θ + a ) + b ( δ + μ ) ( μ + d + θ ) ] − π β ( μ α + γ ) − b ( β μ + γ δ ) ,

c 1 = μ ( δ + μ ) ( μ α + γ ) ( μ + d + θ + a ) − π ( β μ + γ δ ) .

Equation (3) will have a unique positive real root if

c 5 > 0 , c 4 > 0 , c 3 > 0 , c 2 > 0 , c 1 < 0 .
c 5 > 0 , c 4 > 0 , c 3 > 0 , c 2 < 0 , c 1 < 0 .
c 5 > 0 , c 4 > 0 , c 3 < 0 , c 2 < 0 , c 1 < 0 .
c 5 > 0 , c 4 < 0 , c 3 < 0 , c 2 < 0 , c 1 < 0 .

Here c 5 > 0 , and c 1 < 0 , for ℜ 0 > 1 , whereas the coefficients c 2 , c 3 , and c 4 are positive or negative under the conditions given below:

Let us take m = μ + d + θ , n = δ + μ ,

c 2 > 0 , when ( π β ( μ α + γ ) + b ( β μ + γ δ ) ) < ( ( δ + μ ) ( μ α + γ ) ( m + a ) + μ β ( m + a ) + μ bmn ) . < 0 , when ( π β ( μ α + γ ) + b ( β μ + γ δ ) ) > ( ( δ + μ ) ( μ α + γ ) ( m + a ) + μ β ( m + a ) + μ bmn .

c 3 > 0 , when ( b β ( μ α + γ ) + π α γ δ ) < ( μ α + γ ) [ β ( m + a ) + bmn ] + μ α n ( m + a ) + μ b β m . < 0 , when ( b β ( μ α + γ ) + π α γ δ ) > ( μ α + γ ) [ β ( m + a ) + bmn ] + μ α n ( m + a ) + μ b β m .

c 4 > 0 , when b α γ δ < ( μ α + γ ) [ α n ( m + a ) + b β m ] + μ α bmn . < 0 , when b α γ δ > ( μ α + γ ) [ α n ( m + a ) + b β m ] + μ α bmn .

After getting the value of I * the values of S * and A * can be determined. A unique positive equilibrium E 1 = ( S * , A * , I * ) exists if any preceding requirements are satisfied.

Hence, as a result, we have the following lemma:

Lemma 2

DFE E 0 = ( S 0 , A 0 , 0 ) where S 0 = π μ + δ , A 0 = δ π μ ( μ + δ ) of the system (2) exists, when ℜ 0 ≤ 1 , and EE E 1 = ( S * , A * , I * ) exists when ℜ 0 > 1 and any one of the above mentioned four conditions is satisfied.

2.2 Basic reproduction number

In this section, we find the primary reproduction number by Diekmann et al. [5].

Let Y = ( I , A , S ) T . By System (2), we have

d Y d t = F ( Y ) − V ( Y )

where F ( Y ) = β SI 1 + α I 2 + γ AI 1 + α I 0 0 , and V ( Y ) = ( μ + d + θ ) I + aI 1 + bI − δ S + μ A + γ AI 1 + α I − π + ( μ + δ ) S + β SI 1 + α I 2 .

At DFE E 0 = ( S 0 , A 0 , 0 ) , the Jacobian matrices of F ( Y ) and V ( Y ) are DF ( E 0 ) = F 1 0 0 0 , and DV ( E 0 ) = V 1 0 0 0 , respectively.

where

F 1 = β π μ + δ + γ δ π μ ( μ + δ ) 0 0 0 0 0 0 0 0 , and V 1 = μ + d + θ + a 0 0 γ δ π μ ( μ + δ ) μ − δ β π μ + δ 0 μ + δ .

System’s next generation matrix is given by

F 1 V 1 − 1 = π ( β μ + γ δ ) μ ( μ + δ ) ( μ + d + θ + a ) 0 0 0 0 0 0 0 0 .

The spectral radius of F 1 V 1 − 1 = ℜ 0 = π ( β μ + γ δ ) μ ( μ + δ ) ( μ + d + θ + a ) .

To examine the stability of the system (2) at the equilibrium points, the linearized matrix of system (2) is given as follows:

J = − ( μ + δ ) − β I 1 + α I 2 0 − β S ( 1 − α I 2 ) ( 1 + α I 2 ) 2 δ − μ − γ I 1 + α I − γ A ( 1 + α I ) 2 β I 1 + α I 2 γ I 1 + α I β S ( 1 − α I 2 ) ( 1 + α I 2 ) 2 + γ A ( 1 + α I ) 2 − ( μ + d + θ ) − a ( 1 + b I ) 2 .

The Jacobian matrix at DFE E 0 = ( S 0 , A 0 , 0 ) is given by as follows:

J [ E 0 ] = − ( μ + δ ) 0 − β π μ + δ δ − μ − γ δ π μ ( μ + δ ) 0 0 β π μ + δ + γ δ π μ ( μ + δ ) − ( μ + d + θ + a ) .

The characteristic values of the above Jacobian matrix J [ E 0 ] are − μ , − ( μ + δ ) , ( μ + d + θ + a ) ( ℜ 0 − 1 ) . The first two characteristic values are negative, while the third characteristic value is negative when ℜ o < 1 . The DFE is asymptotically stable (by the Routh–Hurwitz criterion).

From the above discussion, as a result, we have the following theorem:

Theorem 1

The DFE E 0 = ( S 0 , A 0 , 0 ) is stable when ℜ 0 < 1 and unstable when ℜ 0 > 1 .

Stability analysis of the DFE at ℜ 0 = 1 :

Using the center manifold theory [2,12], we will study the stability of DFE at ℜ 0 = 1 .

Let φ = β = β * = μ ( μ + δ ) ( μ + d + θ + a ) − π γ δ π μ be the transmission parameter.

Let S = x 1 , A = x 2 , I = x 3 , then the system (2) can be written as follows:

d x 1 d t = π − ( μ + δ ) x 1 − β x 1 x 3 1 + α x 3 2 ≡ f 1 ,

(4) d x 2 d t = δ x 1 − μ x 2 − γ x 2 x 3 1 + α x 3 ≡ f 2 ,

d x 3 d t = β x 1 x 3 1 + α I x 3 2 + γ x 2 x 3 1 + α x 3 − ( μ + d + θ ) x 3 − a x 3 1 + b x 3 ≡ f 3 .

The Jacobian matrix J ∗ at ℜ 0 = 1 , i.e., at β = β * = μ ( μ + δ ) ( μ + d + θ + a ) − π γ δ π μ is

J [ E 0 ] = − ( μ + δ ) 0 − β * π μ + δ δ − μ − γ δ π μ ( μ + δ ) 0 0 0 .

Here zero is the simple characteristic value of a matrix J * . Let u = [ u 1 , u 2 , u 3 ] and w = [ w 1 , w 2 , w 3 ] T be the left and right characteristic vectors of the matrix J * corresponding to the zero-characteristic value, respectively. We get u 1 = 0 , u 2 = 0 , and u 3 = 1 , and w 1 = − β * π μ + δ , w 2 = − δ π ( γ δ + γ μ + μ β * ) μ 2 ( μ + δ ) 2 , and w 3 = 1 .

Non-zero partial derivatives of f 1 , f 2 , f 3 of the system (3) at ℜ 0 = 1 , i.e., at β = β * = μ ( μ + δ ) ( μ + d + θ + a ) − π γ δ π μ are

∂ 2 f 3 ∂ x 1 x 3 E 0 = β * , ∂ 2 f 3 ∂ x 2 x 3 E 0 = γ , ∂ 2 f 3 ∂ x 3 β * E 0 = π μ + δ , ∂ 2 f 3 ∂ x 3 2 E 0 = 2 γ α δ π μ ( μ + δ ) + 2 ab ,

The bifurcation constants a 1 and b 1 are determined by using bifurcation theory as follows:

a 1 = ∑ k , i , j = 1 3 u k w i w j ∂ 2 f 3 ∂ x i x j E 0 = − 2 β * 2 π ( μ + δ ) 2 − 2 δ γ π ( γ δ + γ μ + μ β * ) μ 2 ( μ + δ ) 2 − 2 α δ γ π μ ( μ + δ ) + 2 ab < 0 ,

iff ab < β * 2 π ( μ + δ ) 2 + δ γ π ( γ δ + γ μ + μ β * ) μ 2 ( μ + δ ) 2 + α δ γ π μ ( μ + δ ) .

b 1 = ∑ k , i , = 1 3 u k w i ∂ 2 f 3 ∂ x i β * E 0 = π μ + δ > 0 .

Hence a 1 < 0 iff ab < β * 2 π ( μ + δ ) 2 + δ γ π ( γ δ + γ μ + μ β * ) μ 2 ( μ + δ ) 2 + α δ γ π μ ( μ + δ ) and b 1 > 0 so the DFE is not stable. The Endemic Equilibrium (EE) points of the system exists when ℜ 0 crosses one.

As per the above discussion, we have the following theorem (at ℜ 0 = 1 ):

Theorem 2

When ℜ 0 = 1 , DFE is unstable, and when ℜ 0 crosses 1, we get the EE.

Now we will discuss the stability of the EE E 1 = ( S * , A * , I * ) .

The Jacobian matrix J ( E 1 ) is given by

J ( E 1 ) = − ( μ + δ ) − β I * 1 + α I * 2 0 − β S * ( 1 − α I * 2 ) ( 1 + α I * 2 ) 2 δ − μ − γ I * 1 + α I * − γ A * ( 1 + α I * ) 2 β I * 1 + α I * 2 γ I * 1 + α I * β S * ( 1 − α I * 2 ) ( 1 + α I * 2 ) 2 + γ A * ( 1 + α I * ) 2 − ( μ + d + θ ) − a ( 1 + b I * ) 2 .

The characteristic equation of J ( E 1 ) is

(5) λ 3 + p 0 λ 2 + p 1 λ + p 2 = 0 ,

where

p 0 = 2 μ + δ + β I * 1 + α I * 2 + γ I * 1 + α I * + α β S * I * 2 ( 1 + α I * 2 ) 2 + a ( 1 + b I * ) 2 + ( μ + d + θ ) − β S * ( 1 + α I * 2 ) 2 − γ A * ( 1 + α I * ) 2 ,

p 1 = μ + δ + β I * 1 + α I * 2 μ + γ I * 1 + α I * + 2 μ + δ + β I * 1 + α I * 2 + γ I * 1 + α I * a ( 1 + b I * ) 2 + ( μ + d + θ ) − β S * ( 1 + α I * 2 ) 2 − γ A * ( 1 + α I * ) 2 + 2 μ + δ + γ I * 1 + α I * α β S * I * 2 ( 1 + α I * 2 ) 2 + γ 2 A * I * ( 1 + α I * ) 3 + β 2 S * I * ( 1 + α I * 2 ) 3 ,

p 2 = μ + δ + β I * 1 + α I * 2 μ + γ I * 1 + α I * a ( 1 + b I * ) 2 + ( μ + d + θ ) − β S * ( 1 + α I * 2 ) 2 − γ A * ( 1 + α I * ) 2 + ( μ α I * 2 + δ ) β γ S * I * ( 1 + α I * 2 ) 2 ( 1 + α I * ) + ( μ + δ ) μ α β S * I * 2 ( 1 + α I * 2 ) 2 + γ 2 A * I * ( 1 + α I * ) 3 μ + δ + β I * 1 + α I * 2 + μ + γ I * 1 + α I * β 2 S * I * ( 1 + α I * 2 ) 3 .

If β S * ( 1 + α I * 2 ) 2 + γ A * ( 1 + α I * ) 2 ≤ a ( 1 + b I * ) 2 + ( μ + d + θ ) , then p 0 , p 1 , and p 2 will be positive. By Descartes's rule of the signs [16], it is evident that all the characteristic values of equation (5) are negative. The EE E 1 = ( S * , A * , I * ) is stable, when the condition β S * ( 1 + α I * 2 ) 2 + γ A * ( 1 + α I * ) 2 ≤ a ( 1 + b I * ) 2 + ( μ + d + θ ) satisfies (by the Routh–Hurwitz criterion).

Hence, as a result, we have the following theorem:

Theorem 3

If β S * ( 1 + α I * 2 ) 2 + γ A * ( 1 + α I * ) 2 ≤ a ( 1 + b I * ) 2 + ( μ + d + θ ) , and ℜ 0 > 1 , then EE E 1 = ( S * , A * , I * ) is stable.

3 Simulation

First, we take the initial values of the susceptible, alert, infected, and recovered compartments. S ( 0 ) = 50 , A ( 0 ) = 20 , I ( 0 ) = 9 , and R ( 0 ) = 10 . Now, we consider the following data to test the validity of our result: π = 2.2 , α = 0.6 , β = 0.002 , μ = 0.006 , d = 0.06 , γ = 0.002 , δ = 0.003 , a = 0.3 , b = 0.3 , and θ = 0.5 . With these values, we get the primary reproduction number ℜ 0 = 0.8467 < 1 .

Figure 1 shows that when ℜ 0 < 1 , the susceptible and alert population increases rapidly to S 0 = 244.45 , and A 0 = 122.22 , respectively, and the infected population suddenly drops to zero, the recovered population grows rapidly and after that it decreases rapidly and after time 450, it increases and becomes stable. Hence, we conclude that when ℜ 0 = 0.8467 < 1 , the susceptible, alert, infected, and recovered population approaches DFE point. Thus proving that theorem 1 is verified.

$Figure 1 When ℜ 0 = 0.8467 < 1 {{\rm{\Re }}}_{0}=0.8467\lt 1 .$

Figure 1

When ℜ 0 = 0.8467 < 1 .

Let us consider the following data

π = 2 . 2 , α = 0 . 6 , β = 0 . 4 , μ = 0 . 006 , d = 0 . 06 , γ = 0 . 002 , δ = 0 . 003 , a = 0 . 3 , b = 0 . 3 , and θ = 0 . 003

On using these values, we have β S * ( 1 + α I * 2 ) 2 + γ A * ( 1 + α I * ) 2 ≤ a ( 1 + b I * ) 2 + ( μ + d + θ ) , ℜ 0 = 1.90 > 1 , c 1 < 0 , c 2 > 0 , c 3 > 0 , and c 4 > 0 .

Figure 2 shows that the susceptible, infected, alert, and recovered population approaches the EE point, which implies that Theorem 3 is verified.

$Figure 2 When ℜ 0 = 1.90 > 1 {{\rm{\Re }}}_{0}=1.90\gt 1 .$

Figure 2

When ℜ 0 = 1.90 > 1 .

Figure 3 shows that the infected population decreases when the psychological rate increases.

Figure 3

Infected population with psychological rate.

Figure 4 shows that the number of infected individuals increases when the contact rate increases. To control the infected individuals, we must reduce the contact rate via alertness about the disease.

Figure 4

Infected population with contact rate.

Figure 5 shows that the infected population decreases with the treatment of Holling functional type-II rate.

Figure 5

Infected population with and without Holling type II treatment.

4 Conclusion

This study considered an SIR epidemic model with alert classes and a Holling functional type-II treatment incidence rate. The presented study shows that if the psychological rate increases and the contact rate decreases, the number of infected people decreases. We have shown that the infected population decreases with the Holling functional type-II treatment rate.

These days, the epidemic disease, COVID-19, is a big challenge for all of us. We should create awareness among people (especially those who go to the alert compartment for treatment) to wear a mask, use sanitizer, and follow the government guidelines. Therefore, various epidemic diseases can be controlled by applying the concepts of the presented model.

The proposed study will help in controlling the epidemic diseases in which the susceptible people are made aware about the disease.

dr.shivramsharma@mp.gov.in; praveen_jan1980@rediffmail.com

Funding information: Authors state no funding involved.
Author contributions: All authors designed, analyzed, and prepared the presented work.
Conflict of interest: Authors state no conflict of interest.
Ethical approval: The conducted research is not related to either human or animals use.
Data availability statement: Not applicable.

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Received: 2022-08-17

Revised: 2023-02-05

Accepted: 2023-02-05

Published Online: 2023-05-08

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alert class; SIR model with alert class; Holling functional type-II treatment rate; basic reproduction number

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