key: cord-0077303-m9stflkx
authors: Gupta, Shefali; Bhatia, Sumit Kaur; Arya, Naina
title: Effect of incubation delay and pollution on the transmission dynamics of infectious disease
date: 2022-04-20
journal: Ann Univ Ferrara
DOI: 10.1007/s11565-022-00399-5
sha: f1cb182a8d5443fe92589e5b0b837b7596f5c966
doc_id: 77303
cord_uid: m9stflkx

Emerging infectious diseases pose serious threat to human population. Studies suggest that there is correlation between population’s pollution status and emerging infectious diseases. We propose a delayed SIS model to examine the effects of environmental contamination on human health, which can lead to the spread of numerous diseases. A threshold parameter called basic reproduction number has been obtained for the system. Within the sight of time delay, stability analysis for equilibrium points has been obtained. The existence of Hopf bifurcation around endemic equilibrium point pertaining to time delay as a critical parameter is observed. Our study suggests that pollution can have detrimental effects on the spread of disease. Analytical results are supported by numerical simulations.

in populace [1] [2] [3] [4] [5] [6] [7] . Different factors such as economic, geographic conditions, media coverage and so on, have a signinificant role in the transmission of these ailments [8] [9] [10] [11] . Every country on the planet is experiencing the uncontrollable spread and effects of certain infections caused by agents such as bacteria, virus, fungi, and others, transmitted through direct or indirect contact with an infected person infected with diseases such as tuberculosis, AIDS, Dengue fever, and Covid-19, among others. The effects of these sicknesses bring about the higher rate of demise. Various strategies like vaccination, isolation, drugs and so forth, have been in regular practice by the doctors and the organizations to manage the unfold of different ailments. In spite of all the methods followed in practice, there are few diseases that are an irreplaceable part of the society, such as lifestyle diseases. Numerous models and demonstrating methods are accessible in writing [12] [13] [14] [15] which helps us to analyze the impacts of epidemic diseases on populace. Environmental pollution continues to be a major cause of health danger around the world. Environmental pollution, which encompasses pollution of the air, water, and soil, inuences the wellbeing of the populace [11, [16] [17] [18] . In [18] , authors studied epidemic model to study the impact of environmental pollution on spread of infectious diseases. Ecological pressure additionally harms the individuals and they tend to grow more vulnerable to certain ailments. Toxin exposure during pregnancy has an effect on newborns, making them more susceptible to illness. Poisonous synthetic substances existing in the environment as a result of increased contamination cause uncontrollable illnesses [17, 19, 20] , which adds to the quick spread of epidemics. Thus, epidemic control becomes challenging for authorities, physicians, and health organizations in the modern society. We investigate a mathematical model that incorporates pollution since environmental contamination is an important aspect that cannot be overlooked while investigating disease dynamics.

Time delays are incorporated in a disease transmission model for a wide range of biological reasons. The delay model is used to describe infectious disease dynamics in an attempt to gain a better understanding of increasingly complex models [2, 21, 22] . The incubation period is defined as the time between host infection and the onset of the symptoms. It is known that disease penetrates and transmits invisibly much before the visible disease symptoms appear, it is difficult to assess health risks and avoid, identify, and control the epidemic growth. As a result, in order to examine disease dynamics, the incubation period must be included in the model's architecture in order to suggest appropriate disease control techniques [6, 23] . Dengue fever, Chikungunya virus, Lyme disease, and malaria are examples of vector-borne diseases that have a seven-day incubation period. Cooke [5] also presented a model for infectious disease transmission involving a susceptible and infected population. To better understand the dynamics, the author in [5] included an incubation period in a disease transmission model. In [18] , authors provided detailed dynamical analysis of the disease model. They demonstrated the impact of environmental pollution on the dynamics of disease. But they did not include incubation period in their study. Based on the above literature, we study an SIS model under the effect of pollution incorporating incubation time delay. We have considered the time period in which the infectious pathogen develops in the vector before the infected vector infects susceptible persons. The main objective of our work is to establish the role of incubation delay and provide the implication of the results on disease dynamics of the model under the effect of pollution. The paper has been organized as follows: A mathematical model with time delay has been proposed along with its boundedness in Sect. 2. In Sect. 3, the equilibrium points and threshold parameter (R 0 ) has been obtained. In Sect. 4, we have obtained local stability of equilibria followed by the direction and stability of Hopf-bifurcation. Numerical simulations of the model are presented in Sect. 5. The model system is concluded in Sect. 6.

To examine the effects of contamination and disease we propose an epidemic SIS model. We start with some assumptions, such as total population is categorized into subgroups: susceptible and infected. The population under variable N is sub categorized as S, _P and I . Here, S is the susceptible population who aren't suffering from pollutants, _P is the population affected by pollution and I is the class of infected people. We have considered the following assumptions for our model:

(i) When infectives come into contact with susceptibles, the susceptibles becomes infected at a rate λ rate. (ii) Because prenatal exposure to pollution has a variety of negative consequences, it is anticipated that a fraction m of all infants will enter to the S class, while the remaining fraction (1 − m) will enter into _P class. (iii) At a consistent rate θ populace will enter from S into _P. (iv) Because pollution has a variety of negative effects on stressed people, including a weakened immune system, it is hypothesised that the transition rate to the infected class is higher for pollution-affected persons (_P) than for those who are not (S). Let τ > 0 represent the incubation period of the disease, defined as a fixed time during which the infectious agents develop in the vector, and it is only after that time that the infected vector can infect a susceptible individual. Consequently, proposed mathematical model is as follows :

All parameters of the model are assumed to be positive. The initial conditions of the model (1) are as follows: 

, C is the Banach space of continuous functions.

This section focuses on the study of positivity and boundedness of solutions, for which we have the following lemma Proof Assuming that one solution of the system (1) is atleast not positive, then we've the subsequent cases: Case I: there exists t 1 such that S(0) > 0, S(t 1 ) = 0, S (t 1 ) < 0, _P(t) > 0, I (t) > 0, 0 ≤ t < t 1 . Case II: there exists t 2 such that _P(0) > 0, _P(t 2 ) = 0, _P (t 2 ) < 0, S(t) > 0,

Case III: there exists t 3 such that I (0) > 0,

If Case I holds, then we get S (t 1 ) = M > 0 which contradicts S (t 1 ) < 0. If Case II holds, then we get _P (t 2 ) = 0 which contradicts _P (t 2 ) < 0. If Case III holds, then we get I (t 3 ) = 0 which contradicts I (t 3 ) < 0. Therefore, due to arbitrariness of S, _P and I , all solutions of the system remain positive for all t ≥ 0.

is positively invariant for system (1) and attracts all the solutions starting in the interior of positive orthant.

is a positively invariant set of system (1) . Hence, lemma is proved.

Therefore, in this paper, we consider the dynamics of the model system on the set K .

We find the equilibrium points of (1) in this section. There are only two types of equilibrium points namely:

(i) E 0 : Disease-Free Equilibrium Point (ii) E * : Endemic-Equilibrium Point (i) Disease-free equilibrium point: The solution of the following algebraic equations gives disease-free equilibrium point E 0 = (S 0 , _P 0 , I 0 ) for model system (1) .

Substituting the value of I 0 = 0 in Eq. (2) gives

Now, substitute the value of I 0 = 0 in equation (3), we get (1−a)M +θ S 0 −μ_P 0 = 0

Hence, the disease-free equilibrium point is

Before finding endemic equilibrium point E * , we firstly find the threshold parameter called basic reproduction number. The parameter R 0 helps us to find the number of secondary infections of the infected and characterize the spread of infectious disease. The next generation matrix method [4, 24] is used in calculating R 0 . The matrix F and V for (1) are given below as:

Let F be the jacobian of F and V be the jacobian of V at disease free equilibrium point.

R 0 is the spectral radius of F V −1 i.e.,

Now, we evaluate the endemic equilibrium point E * = (S * , _P * , I * ). From Eq. (2), we get

Substitute the values of S * and _P * in Eq. (4), we get the quadratic equation of I as:

It is evident from the above expressions that A is always negative. Now it is clear, by Descartes's rule of sign, that Eq. (5) always has a unique positive root whenever R 0 > 1. Moreover, Eq. (5) may have more than one positive roots if R 0 < 1 and B > 0 but in the next section we will show that it is not possible.

The objective of this section is to study stability analysis of disease-free equilibrium point as well as of endemic equilibrium point.

For the local stability of disease free equilibrium point E 0 , we prove the following theorem:

Clearly, two of the negative eigenvalues of the above equation are −(θ + μ) and −μ and other root is the solution of f ( ) = 0, where f ( ) is as follows:

Assume that R 0 < 1. To obtain a contradiction, suppose that f has a root δ 2 C such that

which contradicts our assumption. Therefore, in this case, the eigenvalue has negative real part if

implies that f has at least one positive root. Hence, E 0 is unstable in this case. Hence, the theorem.

Analysis for R 0 = 1 We will employ centre manifold theory [3] to investigate the equilibrium point's stability behaviour. It is obvious from the value of basic reproduction number that it is proportional to λ, so let the bifurcation parameter be λ.

, and jacobian martix around disease free equilibrium point of the system (1) has one of the characteristic values as 0 and the other characteristic values are negative. Now, Jacobian matrix for the disease-free point for model system (1) is :

For λ = λ * , Jacobian has characteristic value 0 while remaining characteristic values are negative. Right eigenvector,

Similarly, left eigenvector (q 1 , q 2 , q 3 ) can be evaluated to be (0, 0, 1).

Using theorem 4.1 as given in [3] , the coecients c and d can be calculated as:

For model system (1), the values of c and d are as follows:

Thus, we can state the below theorem from [3] .

The disease-free equilibrium changes its stability from stable to unstable at R 0 = 1 and there exists a positive equilibrium as R 0 crosses one. Hence, the system exhibits transcritical bifurcation with bifurcation parameter λ * at R 0 = 1.

As c < 0 and d > 0, the existence of backward bifurcation is unfeasible and hence Eq. (5) will not have positive root for R 0 < 1.

The system (1) has a positive endemic equilibrium point E * = (S * , _P * , I * ), as shown in the previous section. Here, we investigate the local stability of E * . The Jacobian matrix corresponding to (S * , _P * , I * ) is:

The characteristic equation corresponding to the endemic equilibrium point is:

where

Case 1 When τ = 0 Put τ = 0 in Eq. (6), we get

All roots of Eq. (7) must have negative real parts according to the Routh-Hurwitz criteria if c 2 > 0, c 1 > 0, c 0 > 0 and c 2 c 1 > c 0 holds.

By the above analysis we have the following theorem:

The endemic point E * is locally stable for τ = 0 if following inequalities hold:

Case 2 When τ > 0 We obtain conditions under which the roots of the Eq. (6) will have negative real parts. Let = ϕ(τ ) + i (τ ) be the eigen value of characteristic Eq. (6),where ϕ(τ ) and (τ ) depends on delay τ . We have already shown that the endemic equilibrium point E * is stable when τ = 0, which implies that ϕ(τ ) < 0 for sufficiently small τ . By increasing τ , the real part of the root of the Eq. (6) reaches the value zero at τ = τ * , i.e., ϕ(τ * ) = 0, which implies = i (τ * ) for a specific value of τ * > 0. So, we get = i (τ * ) which is purely imaginary root which means that the Eq. (6) will have root with positive real part and then E * becomes unstable. If such τ * does not exist then it would mean that Eq. (6) will not have a purely imaginary root for all delay and E * will always be stable. Assume = i to be root of the Eq. (6) with > 0. Put = i in (6), separating real and imaginary parts, we get:

Now, squaring and adding Eqs. (8) and (9), we get

We can rewrite the above equation as:

Putting k = 2 , in Eq. (10), reduces it to the following:

The discussion of the roots of Eq. (11) is similar to the discussion in [21] , and we put forward the following lemma: Furthermore, let us assume that Eq. (11) has positive roots. Without the loss of generality, we suppose that the Eq. (11) has three positive solutions, say k 1 , k 2 and k 3 respectively. Then Eq. (10) has three positive solution n = √ k n , where n = 1, 2, 3. Each positive solution (τ ) of (10) is also defined as the solution of Eqs. (12) and (13) which are given below. We obtain Eqs. (12) and (13) from Eq. (6) as follows:

From (13), we get

for n = 1, 2, 3; i = 0, 1, 2, 3, . . . where b j 's and a j 's for j = 0, 1, 2 are considered to be bounded functions of τ , then ±i are imaginary roots of Eq. (6) for τ = τ n Let us take τ * = min{τ n , 0 = n , n = 1, 2, 3} at τ = τ * . Let λ = ϕ(τ ) + i (τ ) be solution of Eq. (6), satisfying ϕ(τ * ) = 0, (τ * ) = 0 . Now, we have subsequent transversality conditions.

Proof Differentiate Eq. (6) w.r.t. τ we get,

Put λ = i in the above expression and simplifying, we get,

Now, we rationalise each term in numerator and denominator and using A( , τ ) = |E(i , τ )| 2 − |F(i , τ )| 2 = 0, we get,

where U , V , R and Z are mentioned above in lemma 5 by Eqs. (14) , (15) , (16) and (17) . Thus, we get,

Let us assume that if d Re(λ) dτ < 0, then characteristic equation has positive real parts when τ < τ * which contradicts the local stability of the endemic equilibrium point. Thus, d Re(λ) dτ > 0 is satisfied. Because of continuity, real part of (τ ) becomes positive for τ > τ * and accordingly stable state converts to unstable state. Since the loss of stability relates to the root =i of the characteristic equation therefore there will be periodic solutions. Using the previous analysis, we get the subsequent theorem. 

According to the analysis done in the previous section, we obtained certain conditions under which a given system undergoes Hopf-bifurcation, with time lag (τ ) being a critical parameter. In this section, we will use the normal form and the center manifold theory established in [21, 22, 25] to check the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions. Throughout this section, we will assume that either (D3) or (D4) holds and U R − V Z = 0. If we take v 1 = S − S * , v 2 = _P − _P * and v 3 = I − I * , then the Taylor expansion up to second order terms for system at E * becomes

where

Also, Eqs. (18)- (20) can be rewritten as

where

Obviously, O ζ is a one-parameter family of bounded linear operators on C. According to the Riesz representation theorem, for ω ε [−τ, 0] there is a matrix function of 3 × 3 and ρ(ω, ζ ) of bounded variation s.t.

for all γ C. Now, we can take ρ(ω, ζ ) = Q 1 κ(ω) − Q 2 κ(ω + τ ) where Dirac delta function is κ.

Then Eq. (22) can be written as

For χ C 1 , the adjoint operator Z * can be defined as

and for χ , φ C 1 . Define the form as

where ρ(ω) = ρ(ω, 0). It can be verified that Z * and Z(0) are adjoint operators with respect to this bilinear form" and ±iσ 0 are eigenvalues of Z(0), also they are eigenvalues value of Z * . Let p C 1 be an eigenvector of Z(0) associated with iσ 0 such that p(ω) = ( p 1 , 1, p 2 ) T e iσ 0 ω . At that point Z(0)p(ω) = iσ 0 p(ω) for all ω [−τ, 0]. When ω = 0, we can use (21), (23), (24) and (25) to get

Similarly, assume that the eigenvector p * of Z * comparing to −iσ 0 takes the form

Solving the above equation gives p * 2 = −a 6 (iσ 0 + a 1 ) (a 8 + a 13 iσ 0 τ * + iσ 0 )(iσ 0 + a 1 ) − a 4 a 7 a 3 − a 3 a 11 e iσ 0 τ * Now, we will find the value of D to certify that ( p * , p) = 1. By (28), we have

, we get ( p * , p) = 1. Now, we will calculate the center manifold coordinates C 0 at ζ = 0. So let us take v t to be the solution of Eq. (26) for ζ = 0. Let us define

For center manifold C 0 ,

where

and z,z are the center manifold coordinates C 0 in the ways of p * andp * respectively where X is real if v t is real. From (29) and (30), we get For a solution v t C 0 of Eqs. (26), (24) , (25) and (27) with ζ = 0 gives

where f 0 (z,z) = F (0, X (z,z, ω) ). Equation (33) is also written aṡ

Put (26) and (33) intoẊ =u t −ż p −zp, we havė

Again, we haveẊ = x zż + xzz on C 0 . Put Eqs. (29), (30) and (34) into (37) and comparing coefficients with Eq. (36), we get

As

Now, we compare the coefficients with Eq. (35), we obtain

Now, we follow the same procedure as in [22] and we get

where O 1 and O 2 are both two dimensional vectors such that

2 )

and Y n denotes the n × n identity matrix. Hence,

and the specific articulations for O 1 and O 2 are acquired by solving the above linear equation.

Based on the above analysis, we can write each h i j in terms of the parameters of the system, and we can thus compute the following quantities:

,

From [21] , we can form the subsequent conclusions. 

To observe and understand the dynamical conduct, we perform numerical simulations for the system (1) with the help of MATLAB. We investigate to explore the effect that natural contamination may have on the spread of illness. Consequently, we conduct a mathematical investigation to check the effect of pollution associated parameters on the disease dynamics of system (1). Figure 1 presents the disease-free equilibrium point for the system (1). The parametric value set considered in the simulation is: M = 10, a = 0.9832, θ = 0.00009, μ = 0.1, λ = 0.0001, η = 0.0001, δ = 0.001, λ = 0.1785, ξ = 0.9875, φ = 0.9541. It can be seen from Fig. 1 that the trajectories approach E 0 (S 0 = 98.23, _P 0 = 1.768, I 0 = 0). It can easily be calculated that R 0 = 0.0049 < 1, for the set of parameters, which also satisfies the condition of Theorem 3. Let θ = 0.7, η = 0.9, δ = 0.1, ξ = 0.1 and μ = 0.0001. For these set of parametric values, we obtain R 0 = 9.6552 > 1 and the trajectories approach the endemic equilibrium point, which is, (S * = 58.83, _P * = 15.34, I * = 10.45), as represented in Fig.  2 . In order to check the effect of pollution on the disease dynamics of the system (1), we increase δ from 0.1 to 0.4. It can be seen that for δ = 0.4, R 0 = 10.1631, which is more in comparison with the case when δ = 0.1. Also, the endemic equilibrium point for δ = 0.4 is S * = 14.71, _P * = 15.36, I * = 10.47, as shown in Fig. 3 . So it can be concluded that as pollution increases, the number of stressed and infectives also increases. Therefore, it is important to control pollution in the system. We will now take the parameters to be, M = 250, θ = 0.059865, μ = 0.01. Figure  5a , b shows that the endemic equilibrium point looses its stability and undergoes Hopfbifurcation at τ = τ * = 28. Since, we know that the Hopf-bifurcation is supercritical, there exists periodic solutions for τ > 28 and endemic equilibrium point is unstable. Now, for the same set of parameters, if we take τ < 28, we observe that the system is stable around endemic equilibrium point, as shown in Fig. 4a , b. It can be easily checked that parameters satisfy the conditions of Theorem 9. Again to check the effect of pollution on the disease dynamics of the model (1), we increase δ. Increase in transmission rate due to natural contamination (δ) stabilizes the system around endemic equilibrium point, as represented in Fig. 6a , b but the number of infected and stressed individuals increases and overall population size decreases. Also, we notice that by decreasing λ (disease transmission rate), the system stabilizes around endemic equilibrium point, as shown in Fig. 7a, b . Thus, it is very important to keep check on 

An SIS model is developed to contemplate the impact of environmental contamination on the spread of epidemic illness. We categorized total populace into three divisions, which are susceptible populace, stressed populace, and infected populace. Analysis of the model indicates that there are two types of equilibrium points: (a) disease-free equilibrium point, and (b) endemic equilibrium point. The disease-free equilibrium point's local stability has been proved in Theorem 3, which shows that the system is stable around disease free equilibrium point if R 0 < 1. The existence of endemicequilibrium point has been established. We discussed stability analysis for endemic equilibrium point (E * ) for τ ≥ 0. We have shown that the time delay plays a crucial role in shaping the dynamics of the system. When τ < τ * the system is stable around Endemic equilibrium point of the system is stable for τ = 28, δ = 0.001, λ = 0.00005 endemic equilibrium point but it loses its stability and a Hopf-bifurcation occurs at a threshold value τ * as shown in Theorem 9. The endemic equilibrium point become unstable for τ > τ * . Our model predicts that with the increase in the transmission rate due to natural contamination (δ), the system becomes stable around endemic equilibrium point. However, this leads in a population drop and an increase in diseased persons, which is not beneficial for our society. As a result, our model suggests that a healthy community requires a check on the transmission rate produced by natural contamination (δ) and disease transmission rate (λ).

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