Dynamics of cholera epidemics with impulsive vaccination and disinfection

Accepted Manuscript

Dynamics of Cholera Epidemics with Impulsive Vaccination and Disinfection Omprakash Singh Sisodiya, O.P. Misra, Joydip Dhar PII: DOI: Reference:

S0025-5564(17)30426-1 10.1016/j.mbs.2018.02.001 MBS 8026

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Mathematical Biosciences

Received date: Revised date: Accepted date:

6 August 2017 12 November 2017 5 February 2018

Please cite this article as: Omprakash Singh Sisodiya, O.P. Misra, Joydip Dhar, Dynamics of Cholera Epidemics with Impulsive Vaccination and Disinfection, Mathematical Biosciences (2018), doi: 10.1016/j.mbs.2018.02.001

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Highlights • Waterborne diseases have a tremendous influence on human life

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• A mathematical model for the cholera disease dynamics is proposed

• A delayed SIRB epidemic model with impulsive vaccination and disinfection • Moreover sanitation to control the cholera disease

• Obtained a sufficient condition for the permanence of the epidemic

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• Stability of the endemic periodic solution are investigated analytically and numerically

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Dynamics of Cholera Epidemics with Impulsive Vaccination and Disinfection

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Omprakash Singh Sisodiya1∗ , O.P. Misra2 , Joydip Dhar3 1

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School of Mathematics and Allied Sciences, Jiwaji University, Gwalior-474011, M.P., India (e-mail:[email protected]) 2 School of Mathematics and Allied Sciences, Jiwaji University, Gwalior-474011, M.P., India (e-mail: [email protected]) 3 Department of Applied Sciences, ABV-Indian Institute of Information Technology and Management, Gwalior-474015, M.P., India (e-mail: [email protected])

Abstract

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Waterborne diseases have a tremendous influence on human life. The contaminated drinking water causes water-borne disease like cholera. Pulse vaccination is an important and effective strategy for the elimination of infectious diseases. A waterborne disease like cholera can also be controlled by using impulse technique. In this paper, we have proposed a delayed SEIRB epidemic model with impulsive vaccination and disinfection. We have studied the pulse vaccination strategy and sanitation to control the cholera disease. The existence and stability of the disease-free and endemic periodic solution are investigated both analytically and numerically. It is shown that there exists an infection-free periodic solution, using the impulsive dynamical system defined by the stroboscopic map. It is observed that the infection-free periodic solution is globally attractive when the impulse period is less than some critical value. From the analysis of the model, we have obtained a sufficient condition for the permanence of the epidemic with pulse vaccination. The main highlight of this paper is to introduce impulse technique along with latent period into the SEIRB epidemic model to investigate the role of pulse vaccination and disinfection on the dynamics of the cholera epidemics. Keywords: Water-borne disease; Cholera; Pulse vaccination; Pathogens; Impulsive inoculation of water; Sanitation effect

Preprint submitted to Mathematical Biosciences

February 6, 2018

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1. Introduction

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In most of the countries, cholera disease spreads widely due to lack of water sanitation and inadequate water including unhygienic conditions.The cholera disease takes the form of an epidemic in many developing countries. To control the spread of cholera disease, it is necessary to inhibit the growth of V. cholerae pathogen in drinking water and which can be attained by using disinfectants. Many researchers have studied the dynamics of spread of cholera disease using mathematical models [1–12]. For example, Capasso and Paveri-Fontana [13] developed and studied a dynamical model to describe the cholera outbreaks of Italy in 1973. Later on, C.T. Codeco [14] proposed a model to explore the role of unhygienic aquatic reservoirs on the spread of cholera disease. In this work, the author has shown that the reproduction rate of cholera is a function of social and environmental factors. Pascual et al. [15] studied the role of climate variability on the dynamical behavior of cholera disease. Jensen et al. [16] investigated the effect of bacteriophage on the spread and control of cholera outbreaks. Tien and Earn [17] proposed a mathematical model to examine the spread of water borne diseases by assuming that the infection is transmitted through direct contact as well as through drinking water contaminated with pathogens. Rosangela P. Sanches et al. [18] proposed a mathematical model for cholera epidemics which included seasonality, loss of host immunity and control mechanisms to reduce cholera transmission. They analyzed the application of different control mechanisms and emphasized the importance of sanitary conditions, better water treatment and hand/food hygiene in the disease dynamics. Neilan et al.[19] formulated a mathematical model to include essential components such as a hyper-infectious, short-lived bacterial state, a different class for mild human infections and waning disease immunity. Recently, Tian and Wang [20] developed and analyzed a model for cholera considering a separate growth rate function for pathogens. However, A. Mwasa and J.M. Tchuenche [21] formulated and analyzed a deterministic model that captures some basic dynamics of cholera transmission to study the impact of vaccination, treatment, and public health educational campaigns, as control strategies in curtailing the disease. Pulse vaccination is a useful approach for the removal of many communicable diseases such viral infections as yellow fever, poliovirus, hepatitis B, parotitis, rabies and encephalitis B. It depends on how large a fraction of the population do we have to keep vaccinated to prevent the particular disease. 3

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Pulse vaccination seems more reasonable in term of both economic and periodic than traditional continuous constant vaccination in the real world. In particular, Agur et al. [22] first proposed a pulse vaccination strategy, which has been established as an important and effective strategy for reducing the disease burden in particular environment. At each vaccination cycle, a fixed portion of susceptible people is vaccinated. This kind of vaccination is called impulsive since all the vaccine doses are applied within a short duration, for a target disease. Epidemiological models with pulse vaccination have been set up and studied for many years, and most of the research literature related to these type of models are formulated using impulsive ODE by the researchers [23–29]. Bacaer and Ouifki have formally defined basic reproduction number in periodic models with and without delay [30].They have shown that the epidemic threshold depends not only on the mean contact rate but also on the amplitude of fluctuations. However, impulsive equations with time delay have been studied [31, 32]. More recently Browne et al. studied pulse vaccination in a metapopulation model with environmental transmission [33]. Controlling waterborne diseases has been an increasingly complex issue in recent years. Therefore, in this paper, we have proposed a nonlinear SIRB mathematical model to examine the dynamics of water-borne diseases like cholera by incorporating time delay with impulsive effects to study the roles of latent period, pulse vaccination and impulsive use of disinfectants. It is assumed in the model that infection spreads through the ingestion of contaminated water by susceptibles. Since the duration of the spreads of cholera disease is for a limited period in a particular season. Therefore as an approximation, we have considered the same impulse period for vaccination and inoculation of water. Also, it is assumed that the disinfectants are introduced to kill pathogens with a rate proportional to the density of pathogens in the aquatic environment. As the pathogens discharged by infectives move to the aquatic habitat, hence the growth of pathogens is proportional to the number of infectives. Using stability theory of impulsive differential equations, the model is analyzed. It is noted that the system exhibits two equilibria, namely, the disease-free and the endemic equilibrium. The analysis shows that under a certain set of conditions, the cholera disease may be controlled by using disinfectants impulsively, but a long delay in their use may destabilize the system. The organization of this paper is as follows: In the second section, we investigate the SEIRB epidemic model with pulse vaccination strategy and impulsively using disinfectants with delay. To establish our main results, 4

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we also state two lemmas which will be essential for proofs. In Section 3, using the discrete dynamical system determined by the stroboscopic map, we establish sufficient condition for the global attractivity of the infectionfree periodic solution. The sufficient conditions for the permanence of the model are obtained in Section 4. A numerical simulation is also conducted to confirm the analytical results in Section 5. In the final section, the conclusion is presented. 2. Mathematical Model

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In the following model, we study a Human population N (t) that is partitioned into four classes, the Susceptibles S(t), Exposed E(t), Infectives I(t) and Recovered or removed class R(t) at time t. The class B(t) denotes the concentration of toxigenic Vibrio cholerae in water. We mainly consider the pulse vaccination strategy and an impulsive introduction of disinfectants in water to control the cholera disease. The model is proposed under following assumptions:

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(1) The susceptible individuals are born at the rate Λ. The constant natural death rate is µ and taken same for all hosts S(t), E(t), I(t) and R(t).

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(2) Considering p as the natural growth rate of Vibrio cholerae in water through environmental factors like (floods, temperature, and seasonal changes). Vibrios have a natural washout rate c in the environment. We assuming c > p throughout this paper ( c < p biologically and epidemiologically meaningless).

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(3) Susceptible individuals acquire cholera infection by ingesting environB ). mental vibrios from contaminated aquatic reservoirs, at the rate θ1 β1 ( K+B Here, β1 is the rate of contact with contaminated water per unit time, K is the pathogen concentration that yields 50 % chance of infection cholera. The coefficient θ1 = (1 − ρ), where ρ is the efficiency of sanitation to reduce the contact with bacteria. (4) Suppose that τ is the latent period of the disease and consider the death of the exposed individuals during latent period of disease, that is the B(t−τ ) β( K+B(t−τ )S(t − τ ) term, where β = β1 e−µτ . )

(5) Infection is only spread through pathogen, and there is no human to human transmission. 5

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(6) Infected individuals are recovered or removed at a rate α, and cholera induced mortality rate of infected individuals is assumed d. Infected individuals contribute to V. cholerae in the aquatic environment at a rate η.

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(7) θ(0 < θ < 1) is the fraction of the susceptible population to whom the vaccine is inoculated at times t = nT , n ∈ N + = 1, 2, 3..., where, T is the impulsive period, i.e. the time between two consecutive pulse vaccinations. Pulse vaccination at multiple times with the fixed period, nT . Vaccination provides temporary immunity, i.e., R(t) contains vaccinated as well as recovered individuals. After immunity period is over, the recovered individuals join the susceptible class at the rate ω. (8) Water sanitation leads to the death of vibrios. So we assume that ξ(0 < ξ < 1) is the death rate of the bacteria population due to pulse inoculation of disinfectant in the aquatic reservoir at time t = nT , to control the toxigenic Vibrio cholerae in water.

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t 6= nT, n ∈ N +

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B(t) S 0 (t) = Λ + ωR(t) − µS(t) − θ1 β1 ( K+B(t) )S(t), B(t) B(t−τ ) 0 E (t) = θ1 β1 ( K+B(t) )S(t) − θ1 β( K+B(t−τ ) )S(t − τ ) − µE(t), B(t−τ ) I 0 (t) = θ1 β( K+B(t−τ )S(t − τ ) − (µ + d + α)I(t), ) 0 R (t) = αI(t) − (µ + ω)R(t), B 0 (t) = (p − c)B(t) + ηI(t),  S(t+ ) = (1 − θ)S(t),       E(t+ ) = E(t),  + I(t ) = I(t), t = nT, n ∈ N +   R(t+ ) = R(t) + θS(t),     B(t+ ) = (1 − ξ)B(t). 

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The model is as follows:

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(1) Where N (t) = S(t)+E(t)+I(t)+R(t) denotes the total human population at time ’t’. All coefficients are positive constants. The model system describes the dynamics of human population and biomass of V. cholerae (pathogen population). The total human population size N (t) can be determined by the differential equation N 0 (t) = Λ − µN (t) − dI(t), 6

(2)

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which is derived by adding the equations in system (1).Thus the total population size may vary in time. From equation (2) we have

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It follows that

Λ − (µ + d)N (t) ≤ N 0 (t) ≤ Λ − µN (t). Λ Λ ≤ lim Inf N (t) ≤ lim SupN (t) ≤ . t→∞ t→∞ µ+d µ

In the same manner, we get from system (1),

B 0 (t) = (p − c)B(t) + ηI(t) ⇒ lim B(t) ≤

Λη . µ(c − p)

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t→∞

Since the equation for the variable E in model (1) is independent of other equations, then the dynamical behavior of (1) is determined by the following system:

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B(t) S 0 (t) = Λ + ωR(t) − µS(t) − θ1 β( K+B(t) )S(t), B(t−τ ) I 0 (t) = θ1 β( K+B(t−τ ) )S(t − τ ) − (µ + d + α)I(t), R0 (t) = αI(t) − (µ + ω)R(t), N 0 (t) = Λ − µN (t) − dI(t), B 0 (t) = (p − c)B(t) + ηI(t),  S(t+ ) = (1 − θ)S(t),       I(t+ ) = I(t),  + R(t ) = R(t) + θS(t), t = nT, n ∈ N +    N (t+ ) = N (t),    + B(t ) = (1 − ξ)B(t). 

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The initial condition of systems (3) is given as:

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t 6= nT, n ∈ N +

(3)

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φ(ζ) = (φ1 (ζ), φ2 (ζ), φ3 (ζ), φ4 (ζ), φ5 (ζ)) ∈ C+ , φi (0) > 0, i = 1, 2, 3, 4, 5, (4) 5 where C+ = C([−τ, 0], R+ ). From biological considerations, we discuss systems (3) in the closed set 5 Ω = {(S, I, R, N, B) ∈ R+ : 0 ≤ S+I+R ≤

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Λ Λ ηΛ , N ≤ , B(t) ≤ , c > p}. µ µ µ(c − p) (5)

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It is easy to show that Ω is positively invariant with respect to (3). To prove our main results, we need the following lemmas. Lemmas 2.1: Let consider the following impulsive differential equations u0 (t) = a − bu(t), u(t+ ) = (1 − θ)u(t),

t 6= nT t = nT, n ∈ N + .

(6)

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where a, b > 0, 0 < θ < 1. Then there exist a unique positive periodic solution of (6) a a uee (t) = + (u∗ − )e−b(t−nT ) , nT < t ≤ (n + 1)T (7) b b Which is globally asymptotically stable, where a(1 − θ)(1 − e−bT ) b(1 − (1 − θ)e−bT )

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u∗ =

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Proof: By calculating, we obtain the solution of first equation of system (6) between two pulses a a + (u(nT + ) − )e−b(t−nT ) , nT < t ≤ (n + 1)T. b b Using the second equation of system (6), we deduce the stroboscopic map such that

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u(t) =

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a a u((n + 1)T + ) = (1 − θ)( + (u(nT + ) − )e−bT ) , f (u(nT + )), b b a a −bT where f (u) = (1 − θ)( b + (u − b )e ). It is easy to show that the map f has −bT ) a unique positive fixed point u∗ = a(1−θ)(1−e . b(1−(1−θ)e−bT ) ∗ ∗ Obviously, u satisfies u < f (u) < u as 0 < u < u∗ and u > f (u) > u∗ as u > u∗ . It implies that the corresponding periodic solution of system (6): a a + (u∗ − )e−b(t−nT ) , nT < t ≤ (n + 1)T b b is globally asymptotically stable. The proof of lemma 2.1 is completed. Lemmas 2.2 (see [34, 35]): Let us consider the following delay differential equation x0 (t) = a1 x(t − τ ) − a2 x(t),

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uee (t) =

where a1 , a2 , τ > 0; x(t) > 0 for −τ ≤ t ≤ 0. We have: (1) if a1 < a2 , then limt→∞ x(t) = 0;

(2) if a1 > a2 , then limt→∞ x(t) = +∞. 8

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3. Global attractivity of infection- free periodic solution

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In this section, we demonstrate the existence of the disease free periodic solution (DFPS) of systems (3), in which infectious individuals are entirely absent from the population permanently i.e., I(t) = 0, for all t ≥ 0, for this, pathogen population assumed to be absent from the environment i.e., B(t) = 0, for all t ≥ 0. Under this condition system (3) will be converted into the following impulsive system: S 0 (t) = Λ + ωR(t) − µS(t)   R0 (t) = −µR(t) − ωR(t), t 6= nT, n ∈ N +   N 0 (t) = Λ − µN (t)  + S(t ) = (1 − θ)S(t),   R(t+ ) = R(t) + θS(t), t = nT, n ∈ N +   + N (t ) = N (t),

(9)

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where N = S + R. From the third and sixth equations of system (9), we can easily obtain Λ lim N (t) = . t→∞ µ

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Now, we consider the following limit system of (9) by taking R(t) = in equation for S 0 (t), then we get S 0 (t) = (µ + ω)(

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Finally, we have

S 0 (t) = (µ + ω)( Λµ − S(t)), S(t+ ) = (1 − θ)S(t),

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(

Λ µ

− S(t)

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t 6= nT, t = nT

n ∈ N+

(10)

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By lemma 2.1, we know that periodic solution of system (10), See (t) =

Λ Λ + (S ∗ − )e−(µ+ω)(t−nT ) , nT < t ≤ (n + 1)T, µ µ

(11)

which is globally asymptotically stable, where S∗ =

Λ(1 − θ)(1 − e−(µ+ω)T ) . µ(1 − (1 − θ)e−(µ+ω)T ) 9

(12)

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R1 =

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Next we determine the global attractivity of the infection free periodic solution (See (t), 0, Λµ − See (t), 0) of system (3). Theorem: 3.1 If R1 < 1, then the disease free periodic solution (See (t), 0, Λµ − See (t), 0) of system (3) is globally attractive, where Λθ1 βη(1 − e−(µ+ω)T ) . µK(c − p)(µ + d + α)(1 − (1 − θ)e−(µ+ω)T )

Proof: Since R1 < 1, we can choose ε0 > 0 small enough such that

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θ1 βη Λ(1 − e−(µ+ω)T ) ( + ε0 ) < (µ + d + α) K(c − p) µ(1 − (1 − θ)e−(µ+ω)T )

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It follows from the first equation of system (3) such that S 0 (t) ≤ (µ + ω)( Λµ − S(t)). So we consider the following comparison impulsive differential system t 6= nT, x0 (t) = (µ + ω)( Λµ − x(t)), + x(t ) = (1 − θ)x(t), t = nT

n ∈ N+

(14)

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In view of lemma 2.1, we see that periodic solution of system (14)is Λ Λ + (s∗ − )e−(µ+ω)(t−nT ) , nT < t ≤ (n + 1)T, µ µ

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xe (t) = See (t) =

(15)

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which is globally asymptotically stable, where See (t) is defined in (11). Let ( S(t), I(t), R(t), B(t)) be a solution of system (3) with initial value (4) and S(0+ ) = S0 > 0, x(t) be the solution of system (11) with initial value x(0+ ) = S0 . By comparison theorem of impulsive differential equations, there exists an integer n1 > 0 such that S(t) ≤ x(t) < xe (t) + ε0 , nT < t ≤ (n + 1)T, n > n1 .

(16)

Thus,

S(t) < See (t)+ε0 ≤

Λ(1 − e−(µ+ω)T ) +ε0 = c1 , nT < t ≤ (n+1)T, n > n1 . µ(1 − (1 − θ)e−(µ+ω)T ) (17) 10

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Furthermore, from the second equation of system (3) and from equation (17), we get

Since

θ1 βηc1 )I(t − τ ) − (µ + d + α)I(t), K(c − p)

∀

θ1 βηc1 < (µ + d + α) K(c − p)

for the following auxiliary system

n > n1 .

θ1 βηc1 )y(t − τ ) − (µ + d + α)y(t), K(c − p)

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y 0 (t) = (

t ≥ nT + τ &

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I 0 (t) ≤ (

for all t ≥ nT + τ and n > n1 . Since ( S(t), I(t), R(t), B(t)) be the solution of system (3) with initial value (4) and by the comparison theorem, we have lim sup I(t) ≤ lim sup y(t) = 0. t→∞

t→∞

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Incorporating into the positivity of I(t), we have lim I(t) = 0.

(18)

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Therefore for any sufficiently small 1 ∈ (0, 1), there exist an integer n2 > n1 ( where n2 T > n1 T + τ or n2 > n1 + Tτ ) such that 0 < I(t) < 1 for all t > n2 T . By the Equation of (2.2), we have

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N 0 (t) > Λ − µN (t) − d1 , t > n2 T.

(19)

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Consider the following comparison system W 0 (t) = (Λ − d1 ) − µW (t), t > n2 T

It is clear that limt→∞ W (t) = an integer n3 > n2 such that

Λ−d1 . µ

N (t) ≥

By the comparison theorem, there is Λ − d1 µ − 1

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(20)

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for all t > n3 T . Since 1 is arbitrary very small and lim supt→∞ N (t) ≤ Λµ . Hence Λ lim N (t) = (21) t→∞ µ

I(t) < 1 ,

N (t) >

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It follows from (18) and (21), that there exists n4 > n3 such that ∀t > n4 T

(22)

When t > n4 T, the fourth and sixth equations of system (3) yield B 0 (t) ≤ η1 − (c − p)B(t).

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Now consider the following comparison system v 0 (t) = η1 − (c − p)v(t),

∀t > n4 T.

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η1 . Then by the comparison theorem, It is easy to obtain limt→∞ v(t) = (c−p) there is an integer n5 > n4 such that η1 + 1 , ∀t > n5 T (23) B(t) ≤ (c − p)

From the second equation of system (1), we have η1 Λθ1 β1 ( (c−p) + 1 )

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E (t) ≤ 0

η1 µ(K + ( (c−p) + 1 ))

− µE(t),

∀t > n5 T.

(24)

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It is easy to obtain that there exists an n6 > n5 such that E(t) < δ1 + 1 ,

Λθ1 β1 (η1 +1 (c−p) , µ2 (K(c−p)+η1 +1 (c−p))

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where δ1 = tion of (3) yields

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S 0 (t) ≥ (Λ +

(25)

∀t > n6 T,

Therefore from (22) and (23), the first equa-

ωΛ − 2ω1 ) − (d1 + µ + ω)S(t)), µ

(26)

η1 where d1 = θ1Kβ1 ( c−p + 1 ). Consider the following comparison impulsive differential equation for t > nT and n > n6

(

u0 (t) = (Λ + ωΛ − 2ω1 ) − (d1 + µ + ω)u(t), µ + u(t ) = (1 − θ)u(t), t = nT, n ∈ N + . 12

t 6= nT

(27)

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By lemma 2.1, we know that periodic solution of system (27) is as follows, uee (t) = Φ + (u∗ − Φ)e−(d1 +µ+ω)(t−nT ) , nT < t ≤ (n + 1)T, Λ+

− 2ω1

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which is globally asymptotically stable, where

(1 − θ)(1 − e−(d1 +µ+ω)T ) . (d1 + µ + ω) (1 − (1 − θ)e−(d1 +µ+ω)T ) By using the comparison theorem of impulsive differential equation, there exist an integer n7 > n6 such that ωΛ µ

,

S(t) > uee (t) − 1 ,

u∗ = Φ

nT < t ≤ (n + 1)T, n > n7 .

(28)

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Φ=

Because 0 , 1 → 0 , then it follows from (16), (23) and (28) that Λ θe−(µ+ω)(t−nT ) (1 − ), nT < t ≤ (n + 1)T, µ 1 − (1 − θ)e−(µ+ω)T is globally attractive, i.e.

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See (t) =

lim S(t) = See (t),

lim E(t) = 0,

t→∞

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t→∞

lim B(t) = 0.

t→∞

(29)

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Finally, It follows from (18), (21) and (29) that the infection free periodic solution (See (t), 0, Λµ − See (t), 0) of system (3) is globally attractive. This complete the proof. According to theorem 2.1, we can easily obtain the following result. Corollary 3.1. (i) If Λθ1 βη < µK(c − p)(µ + d + α), then the infection free periodic solution (See (t), 0, Λµ − See (t), 0) is globally attractive. (ii) If Λθ1 βη > µK(c − p)(µ + d + α), then infection-free periodic solution (See (t), 0, Λµ − See (t), 0)is globally attractive provided that θ > θ∗ or T < T ∗ , where A1 =

µK(c − p)(µ + d + α) , Λθ1 βη

T∗ =

1 1 − A1 ln , −(µ + ω) 1 − (1 − θ)A1 13

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θ∗ =

(1 − A1 )(e(µ+ω)T − 1) . A1

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Remark 3.1: Theorem 3.1 determines the global attractivity of system (3) in Ω for the case R1 < 1. It’s epidemiological implication is that the infectious population vanishes in time, i.e., the disease dies out. From theorem 3.1 and corollary 3.1, it is observed that a short period of pulsing ( with T ), a large pulse vaccination rate ( with θ) or a large disinfectant rate (ξ) is the sufficient condition for the global attractivity of an ‘infection-eradication’ periodic solution (See (t), 0, Λµ − See (t), 0). 4. Permanence

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In this section, we say the disease becomes endemic if the infectious population persists above a certain positive level for a long period. Definition 4.1. System (3) is said to be uniformly persistent if there is an m > 0 (independent of the initial conditions) such that every solution (S(t),I(t),R(t),B(t)) with initial condition (4) of system (3) satisfies lim inf S(t) ≥ m, lim inf I(t) ≥ m, t→∞

lim inf R(t) ≥ m, t→∞

M

t→∞

lim inf B(t) ≥ m t→∞

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Definition 4.2. System (3) is said to be permanent if there exists a compact region Ω0 ∈ Ω such that every solution (S(t),I(t),R(t),B(t)) with initial condition (4) of system (3) will eventually enter and remain in the region Ω0 . Denote

and

Λ(µ + ω)(R2 − 1) , Λb∗ + ω(µ + d)R2

CE

I∗ =

PT

Λθ1 βη(1 − θ)(1 − e−(µ+ω)T ) , µK(c − p)(µ + d + α)(1 − (1 − θ)e−(µ+ω)T )

R2 =

where

b∗ =

θ1 βη . K(c − p)

AC

Theorem: 4.1 If R2 > 1 , then there exists a positive constant q such that every positive solution (S(t),I(t),R(t),B(t)) of system (3) satisfies I(t) ≥ q for t large enough. Proof. Let (S(t),I(t),R(t),B(t)) is any positive solution of system (3) with initial conditions (4). The second equation of system (3) may be rewritten as d Z t B(u)S(u) B(t)S(t) I 0 (t) ≤ θ1 β( ) − (µ + d + α)I(t) − θ1 β du. (30) K + B(t) dt t−τ K + B(u) 14

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Let us consider the following differential function V(t), Z t

t−τ

B(u)S(u) du. K + B(u)

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V (t) = I(t) + θ1 β

The derivative of V(t) along solutions of system (3), it follows from (30) V 0 (t) = I(t)[

θ1 βη S(t)) − (µ + d + α)], K(c − p)

(t 6= nT, (t − τ ) 6= nT ) (32)

c2 =

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Since R2 > 1, we easily see that I ∗ > 0, then there exists a sufficiently small  > 0 such that θ1 βη (c2 ) > 1, (33) K(c − p)(µ + d + α) where, ηI ∗

[Λ − ω + ωµ (Λ − (µ + d)I ∗ )][(1 − θ)(1 − e−[(µ+ω)+θ1 β1 ( c−p +)]T )] ηI ∗

ηI [(µ + ω) + θ1 β1 ( c−p + )][1 − (1 − θ)e−[(µ+ω)+θ1 β1 ( c−p +)]T ] ∗

.

M

We claim that it is impossible that I(t) ≤ I ∗ for all t ≥ t0 ( t0 is nonnegative constant). Suppose the contrary i.e., I(t) > I ∗ then as t ≥ t0 (34)

B 0 (t) ≤ ηI ∗ − (c − p)B(t).

(35)

ED

N 0 (t) ≥ (Λ − dI ∗ ) − µN (t)

PT

and

CE

By (34) and (35), there exists T1 ≥ τ such that

Λ dI ∗ Λ − dI ∗ N (t) ≥ −= − − µ µ µ 0

AC

and

B 0 (t) ≤

ηI ∗ + , c−p

∀t ≥ t0 + T1 .

Hence, when t ≥ t0 + T1 and t 6= nT (n ∈ N ), then from the first equation of system (3), we have S 0 (t) ≥ [Λ − ω +

ω ηI ∗ (Λ − (µ + d)I ∗ )] − [(µ + ω) + θ1 β1 ( + )]S(t) µ c−p 15

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Consider the following comparison impulsive differential equation for t > nT and n > n5 u01 (t) = g1 − g2 u1 (t), u1 (t+ ) = (1 − θ)u1 (t),

By lemma 2.1, we obtain, ue1 e(t) =

t 6= nT t = nT, n ∈ N + .

(36)

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(

g1 g1 + (u∗1 − )e−g2 (t−nT ) , nT < t ≤ (n + 1)T g2 g2

is globally asymptotically stable periodic solution of system (36). Where

u∗1 =

ω (Λ − (µ + d)I ∗ ), µ

g1 (1 − θ)(1 − e−g2 T ) . g2 (1 − (1 − θ)e−g2 T )

ηI ∗ + ), c−p

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g1 = Λ − ω +

g2 = (µ + ω) + θ1 β1 (

M

Thus, there exists a T ∗ greater than T1 satisfying

ED

S(t) > ue1 e(t) −  = c2 , θ1 βη (c ) K(c−p)(µ+d+α) 2

PT

From (33), we have

V 0 (t) > (µ+d+α)I(t)[

CE

Set

∀t ≥ t0 + T ∗ , t1 .

(37)

> 1. By (32) and (37), we have

θ1 βη c2 −1], (t 6= nT, (t−τ ) 6= nT ), ∀t ≥ t1 . (38) K(c − p) Il =

min

t∈[t1 ,t1 +τ ]

I(t) > 0.

AC

We will show that I(t) ≥ Il for all t ≥ t1 . Suppose if it is not true. Then there is a T0 ≥ 0 such that I(t) ≥ Il for all t1 ≤ t ≤ t1 + τ + T0 , I(t1 + τ + T0 ) = Il and I 0 (t1 + τ + T0 ) ≤ 0. However, the second equation of system (3) implies that 1 +T0 )S(t1 +T0 ) ) − (µ + d + α)I(t1 + τ + T0 ) I 0 (t1 + τ + T0 ) = θ1 β( B(tK+B(t 1 +T0 ) θ1 βη ≥ Il ( K(c−p) c2 − (µ + d + α)) > 0

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This is a contradiction. Hence, I(t) ≥ Il for all t ≥ t1 . Consequently, ∀t ≥ t1 , we have that θ1 βη c2 − 1)Il > 0, (t 6= nT, (t − τ ) 6= nT ). K(c − p)

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V 0 (t) ≥ (µ + d + α)(

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Since V(t) is continuous on [0, +∞) and these points at which V(t) is not derivable are at most countable, this implies V (t) → +∞ as t → +∞. This is a contrary to the boundedness of V(t). Hence, the claim is proved. From the claim, we will discuss the following two possibilities. (i) I(t) ≥ I ∗ for all large t. (ii) I(t) oscillates about I ∗ for all large t. Finally, we will show that I(t) ≥ I ∗ e−(µ+d+ω)(T

∗ +τ )

,q

as t is large sufficiently. Evidently, we only need consider the case (ii) Let t1 and t2 be large sufficiently times satisfying: I(t1 ) = I(t2 ) = I ∗ ,

M

I(t) < I ∗ ∀t ∈ (t1 , t2 ).

AC

CE

PT

ED

If t2 − t1 ≤ T ∗ + τ, since I 0 (t) ≥ −(µ + d + ω)I(t) and I(t1 ) = I ∗ which implies I(t) ≥ q for all t ∈ [t1 , t2 ]. If t2 − t1 > T ∗ + τ, then it is clear that I(t) ≥ q for all t ∈ [t1 , t1 + T ∗ + τ ]. Thus proceeding exactly as the proof for above claim, we see that I(t) ≥ q for all t ∈ [t1 , t1 + T ∗ + τ ] and S(t) > c2 for all t ∈ [t1 + T ∗ , t2 ]. Next, we will prove that I(t) ≥ q is still valid for all t ∈ [t1 +T ∗ +τ, t2 ]. If it is not true, then there is a T1 ≥ 0 such that I(t) ≥ q for all t ∈ [t1 , t1 +T ∗ +τ +T1 ], I(t1 +T ∗ +τ +T1 ) = q and I 0 (t1 +T ∗ +τ +T1 ) ≤ 0. Using the second equation of system (3), as t = t1 + T ∗ + τ + T1 , we further obtain B(t − τ ) )S(t − τ ) − (µ + d + α)I(t), I 0 (t) = θ1 β( K + B(t − τ ) ≥(

θ1 βη c2 − (µ + d + α))q > 0. K(c − p)

This is a contradiction. So I(t) ≥ q is valid for all t ∈ [t1 , t2 ]. The proof of theorem (4.1) is complete. Theorem: 4.2 Suppose R2 > 1. Then the system (3) is permanent.

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Proof. Let (S(t),I(t),R(t),B(t)) be any solution of system (3). From the first equation of system (3), we have θ1 β1 Λη )S(t) µk(c − p)

Consider the following auxiliary comparison system (

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S 0 (t) ≥ Λ − (µ +

θ1 β1 Λη x0 (t) = Λ − (µ + µk(c−p) )x(t), t 6= nT + x(t ) = (1 − θ)x(t), t = nT, n ∈ N + .

By lemma (2.1), we can easily prove that

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lim inf S(t) ≥ q1 ,

(40)

t→∞

where

θ1 β1 Λη

q1 =

(µ +

Λ(1 − θ)(1 − e−(µ+ µk(c−p) )T )

θ1 β1 Λη )(1 µk(c−p)

θ1 β1 Λη

− (1 − θ)e−(µ+ µk(c−p) )T )

− ,

ED

It is easy to see that

M

 is sufficiently small. In view of theorem (4.1) and third equation of system (3), one has R0 (t) ≥ αq − (µ + ω)R(t), lim inf R(t) ≥ q2 , t→∞

where q2 = Now moreover, it follows from the fourth and eight equation of the system (3), we have

PT

αq . (µ+ω)

CE

(

B 0 (t) ≥ ηq − (c − p)B(t), t 6= nT B(t+ ) = (1 − ξ)B(t), t = nT, n ∈ N + .

(41)

AC

Consider the following comparison system (

z 0 (t) = ηq − (c − p)z(t), t 6= nT z(t+ ) = (1 − ξ)z(t), t = nT, n ∈ N + .

By lemma (2.1), it is easy to prove that lim inf B(t) ≥ q3 , t→∞

18

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where q3 =

ηq(1 − ξ)(1 − e−(c−p)T ) − , (c − p)(1 − (1 − ξ)e−(c−p)T )

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 is sufficiently small. Set

4 Ω0 = {S, I, R, B ∈ R+ : S ≥ q1 , I ≥ q, R ≥ q2 , B ≥ q3 ,

Λ Λ ≤ S+I+R ≤ }. µ+d µ

µK(c − p)(µ + d + α) , (1 − θ)Λθ1 βη

T∗ =

1 1 − A2 ln , −(µ + ω) 1 − A2 (1 − θ)

θ∗ =

(1 − A1 )(1 − e−(µ+ω)T ) . 1 − e−(µ+ω)T + A1 e−(µ+ω)T

ED

M

A2 =

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By theorem 4.1 and above discussions, we know that the set Ω0 is a global attractor in Ω, i.e., every solution of system (3) with initial conditions (4) will eventually enter and remain in region Ω0 . Therefore, system (3) is permanent. The proof of theorem 4.2 is complete. Denote

AC

CE

PT

Corollary 4.1. (i) If Λθ1 βη > µK(c − p)(µ + d + α), then the disease will be endemic and system (3) is permanent provided θ < θ∗ . (ii) (1 − θ)Λθ1 βη > µK(c − p)(µ + d + α), then the disease will be endemic and system (3) is permanent provided T > T∗ . A1 has been given in Section 3. Remark 4.1: From Theorem 4.2 and Corollary 4.1, we can see that a long period of pulsing ( with T ) or a small pulse vaccination rate ( with θ) or a small disinfectant rate (ξ) is the sufficient condition for the permanence of model. 5. Numerical Simulations In this section, we present some numerical simulations to demonstrate our theoretical results established in this paper. If we set Λ = 0.7, ω = 0.5, µ = 19

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CE

PT

ED

M

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0.25, β1 = 0.20509(i.e., β = 0.2), α = 0.5, η = 0.3, p = 0.3, c = 0.35, d = 0.25, θ1 = 0.1, θ = 0.5, ξ = 0.2, τ = 0.1, T = 2, then R1 = 0.839448 < 1, R2 = 0.419724 < 1. According to theorem 3.1, we know that the disease will disappear, (see Fig. 1). If we set Λ = 0.7, ω = 0.5, µ = 0.25, β1 = 0.20509(i.e., β = 0.2), α = 0.5, η = 0.3, p = 0.3, c = 0.35, d = 0.25, θ1 = 0.25, θ = 0.5, ξ = 0.02, τ = 0.1, T = 2, then R1 = 2.09862 > 1, R2 = 1.04931 > 1. According to theorem 4.1, disease will be permanent, (see Fig. 2). If we use the same parameters as in, Fig. 2 except choosing ξ = 0.02, and ξ = 0.2, then, Fig. 3 shows the periodical disinfectant is an effective method to prevent the disease. Now letting Λ = 0.7, ω = 0.5, µ = 0.25, β1 = 0.20509(i.e., β = 0.2), α = 0.5, η = 0.3, p = 0.3, c = 0.35, d = 0.25, θ1 = 0.25, θ = [0.1, 0.5], ξ = 0.02, τ = 0.1, T = 2. The result shows that the disease fades away when the proportion of those vaccinated successfully θ = 0.5; but the disease will exists everlasting when the proportion of those vaccinated successfully θ = 0.1. So, this verifies the results in corollary (3.1), (see Fig. 4). Also it can be seen that with increase of θ which typically causes oscillation bigger of the susceptible. If we use the same parameters as in, Fig. 2 except choosing θ1 = 0.1 and θ1 = 0.25, if we decrease of θ1 ( i.e., increases in ρ), number of infected individuals is decreasing , so these results indicates the effectiveness of sanitation to prevent the disease, (see Fig. 5). Remark 5.1: In this paper, we have analytically studied two cases under the impulsive vaccination with respect to two thresholds R1 and R2 : (i) R1 < 1 and (ii) R2 > 1. Since R2 = (1−θ)R1 , hence R1 ≥ R2 . The numerical results is also support the analytical findings. The dynamics behavior of model has not been studied analytically when R2 ≤ 1 ≤ R1 . In this case the dynamical behavior of the system explored through numerical experimentation for the following set of parameters: (i) Λ = 0.6, ω = 0.5, µ = 0.25, β1 = 0.20509(i.e., β = 0.2), θ = 0.5, α = 0.5, η = 0.3, p = 0.3, c = 0.35, d = 0.25, θ1 = 0.25, ξ = 0.02, τ = 0.1, T = 2, then R1 = 1.79882 > 1 and R2 = 0.899409 < 1. From the numerical results in fig. 6, it is observed that I(t) and B(t) does not tend to zero, hence disease will be uniformly persistent. Again for the set: (ii) Λ = 0.7, ω = 0.5, µ = 0.25, β1 = 0.20509(i.e., β = 0.2), θ = 0.5, α = 0.5, η = 0.3, p = 0.3, c = 0.35, d = 0.25, θ1 = 0.15, ξ = 0.02, τ = 0.1, T = 2, we have R1 = 1.25917 > 1 and R2 = 0.629586 < 1; shows that the disease will extinct from the environment (see Fig. 7). Therefore, we can conclude that if R2 ≤ 1 ≤ R1 , then the dynamics of the system is undetermined. 20

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6. Conclusions

AC

CE

PT

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M

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In this paper, we introduce a SEIRB epidemic model for waterborne disease with a latent period of infection. Also included three control strategies, namely impulsive disinfection, sanitation and pulse vaccination for disease eradication. Two thresholds have established, one for global attractivity of the infectious-free state and another for the persistence of the endemic state in terms of R1 and R2 respectively. Using Floquet theorem and comparison theorem of the impulsive differential equation, we prove that there exists a disease-free periodic solution, which is globally attractive when R1 < 1, and obtained a sufficient condition for the permanence of epidemic disease when R2 > 1. We analyze in theory that the eradication of pathogen V. cholerae by using disinfectant impulsively from the aquatic environment is one of the important strategies for preventing the prevalence of cholera. Also, sanitation and pulse vaccination bring effects on infection-eradication and the permanence of epidemic disease. Theorems 3.1 and 4.1 imply that the disease dynamics is completely determined by R1 < 1 and R2 > 1. It is observed from corollary 3.1, that a short period of pulsing ( with T < T ∗ ), a large pulse vaccination rate ( with θ > θ∗ ) or a large disinfectant rate (ξ) is sufficient condition for the global attractivity of an infection-eradication periodic solution (See (t), 0, Λµ − See (t), 0). Further it is established in corollary 4.1, that a small pulse vaccination rate ( with θ < θ∗ ), a long pulsing period ( with T > T ∗ ) or a small disinfectant rate (ξ) of the disease implies that the disease remains permanent, i.e., the disease cannot be eradicated from the environment. In this paper, we introduced two threshold R1 and R2 (where R1 ≥ R2 ) and discussed two cases: (i) R1 < 1 ( or θ > θ∗ ), (ii) R2 > 1 ( or θ < θ∗ ). When R2 ≤ 1 ≤ R1 , the dynamical behavior of system (1) has not been clear, in future it can be explored analytically. Again, we have considered same period of vaccination and water inoculation, one can study this system with two different periods for vaccination and water inoculation. Acknowledgement We are very grateful to the reviewers for their valuable suggestions and careful reading of the manuscript.

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[33] C. Browne, L. Bourouiba, R. Smith, From regional pulse vaccination to global disease eradication: insights from a mathematical model of poliomyelitis, Journal of Mathematical Biology 71(1) (2015) 215–253.

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[34] Y. Kuang, Delay differential equation with application in population dynamics, Academic Press, New York (1993) 67–70.

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Figure 1: Time series graph of all population densities using the parameter values Λ = 0.7, ω = 0.5, µ = 0.25, β1 = 0.20509(i.e., β = 0.2), α = 0.5, η = 0.3, p = 0.3, c = 0.35, d = 0.25, θ1 = 0.1, θ = 0.5, ξ = 0.2, τ = 0.1, T = 2, then R1 = 0.839448 < 1, R2 = 0.419724 < 1.

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10

20

30

40

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time t

60

0.2

0.1

70

80

90

0

100

(c)

0

20

40

60

80

100

time t

(d)

Figure 2: Time series graph of all population densities using the parameter values Λ = 0.7, ω = 0.5, µ = 0.25, β1 = 0.20509(i.e., β = 0.2), α = 0.5, η = 0.3, p = 0.3, c = 0.35, d = 0.25, θ = 0.5, θ1 = 0.25, ξ = 0.02, τ = 0.1, T = 2, then R1 = 2.09862 > 1, R2 = 1.04931 > 1.

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0.5

0.5

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I(t)

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(a)

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S(t)

1.6

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PT

0.3

0.2

ξ=0.2

M

ξ=0.02

0.6

AC

0 0.4

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(b)

0.7

0 0.4

AN US

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B(t)

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1.8

2

2.2

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(c)

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0.8

1

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1.4

S(t)

(d)

Figure 3: Dynamical behavior of the systems (3) with the parameters Λ = 0.7, ω = 0.5, µ = 0.25, β1 = 0.20509(i.e., β = 0.2), α = 0.5, η = 0.3, p = 0.3, c = 0.35, d = 0.25, θ1 = 0.25, θ = 0.5, τ = 0.1, T = 2. (a) Effect of disinfectant (ξ) on susceptible population against the infected population with ξ = 0.02. (b) Effect of disinfectant (ξ) on susceptible population against the infected population with ξ = 0.2 (c) Effect of disinfectant (ξ) on susceptible population against the bacteria population with ξ = 0.02. (d) Effect of disinfectant (ξ) on susceptible population against the bacteria population with ξ = 0.2

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0.5

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(a)

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M

θ = 0.1

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ED

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B(t)

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S(t)

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theta = 0.5

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0 0.4

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(b)

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1.8

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2.2

2.4

0 0.4

0.6

0.8

1

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1.4

S(t)

(d)

Figure 4: Dynamical behavior of the systems (3) with the parameters Λ = 0.7, ω = 0.5, µ = 0.25, β1 = 0.20509(i.e., β = 0.2), α = 0.5, η = 0.3, p = 0.3, c = 0.35, d = 0.25, θ1 = 0.25, ξ = 0.02, τ = 0.1, T = 2.(a) Effect of pulse vaccination (θ) on susceptible population against the infected population with θ = 0.1. (b) Effect of pulse vaccination (θ) on susceptible population against the infected population with θ = 0.5 (c) Effect of pulse vaccination (θ) on susceptible population against the bacteria population with θ = 0.1. (d) Effect of pulse vaccination (θ) on susceptible population against the bacteria population 29 with θ = 0.5

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0.7

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θ1=0.1

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B(t)

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0.5

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time t

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ED

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(a)

0.1

0

0

20

40

60

80

100

time t

120

140

160

180

200

(b)

AC

CE

PT

Figure 5: Dynamical behavior of the systems (3) with the parameters Λ = 0.7, ω = 0.5µ = 0.25, β1 = 0.20509(i.e., β = 0.2), α = 0.5, η = 0.3, p = 0.3, c = 0.35, d = 0.25, θ = 0.5, ξ = 0.02, τ = 0.1, T = 2. (a) Effect of θ1 on infected population I(t) with θ1 = 0.25, 0.1 respectively. (b) Effect of θ1 on the bacteria population B(t) with θ1 = 0.25, 0.1 respectively.

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AN US

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500

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(b)

M

Figure 6: This figure shows that movement paths of I and B as functions of time t. R1 = 1.79882 > 1 and R2 = 0.899409 < 1. The disease is permanent.

0.5

0.4

0.35

0.5

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0.2

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B(t)

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I(t)

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(a)

0

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100

time t

120

140

160

180

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(b)

Figure 7: This figure shows that movement paths of I and B as functions of time t. R1 = 1.25917 > 1 and R2 = 0.629586 < 1. The disease dies out.

31