Analysis of a delayed epidemic model with pulse vaccination

Chaos, Solitons & Fractals 66 (2014) 74–85

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Analysis of a delayed epidemic model with pulse vaccination G.P. Samanta ⇑ Institute of Mathematics, National Autonomous University of Mexico, Mexico D.F. C.P. 04510, Mexico

a r t i c l e

i n f o

Article history: Received 2 January 2014 Accepted 22 May 2014

a b s t r a c t In this paper, we have considered a dynamical model of infectious disease that spread by asymptomatic carriers and symptomatically infectious individuals with varying total population size, saturation incidence rate and discrete time delay to become infectious. It is assumed that there is a time lag (s) to account for the fact that an individual infected with bacteria or virus is not infectious until after some time after exposure. The probability that an individual remains in the latency period (exposed class) at least t time units before becoming infectious is given by a step function with value 1 for 0 6 t 6 s and value zero for t > s. The probability that an individual in the latency period has survived is given by els , where l denotes the natural mortality rate in all epidemiological classes. Pulse vaccination is an effective and important strategy for the elimination of infectious diseases and so we have analyzed this model with pulse vaccination. We have defined two positive numbers R1 and R2 . It is proved that there exists an infection-free periodic solution which is globally attractive if R1 < 1 and the disease is permanent if R2 > 1. The important mathematical findings for the dynamical behaviour of the infectious disease model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Infectious diseases have tremendous influence on human life and are usually caused by pathogenic microorganisms, such as bacteria, viruses, parasites, or fungi. The diseases can be spread directly or indirectly. For certain infectious diseases, there are individuals who are capable to transmit their disease to others without exhibiting any symptoms. These individuals are known as ‘‘carriers’’ and they play an important role in the transmission of the disease in the society. The focus of our discussion is on ‘‘Infectious disease carriers’’ who are asymptomatic (without exhibiting any disease symptoms) and are likely unaware of their conditions and so are more likely to infect others ⇑ Present address: Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India. Tel.: +91 33 26684561; fax: +91 33 26682916. E-mail addresses: [email protected], [email protected]. ac.in http://dx.doi.org/10.1016/j.chaos.2014.05.008 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.

[23]. There are many ‘‘Infectious disease carriers’’ which afflict human population around the world. However, the third world countries are most affected by such diseases due to lack of sanitation [37]. For example, an infectious disease that produces long-term asymptomatic carriers is the Typhoid fever caused by the bacteria Salmonella Typhi. Typhoid fever reached public notoriety at the beginning of the 20th century with the cases of ‘Mr. N. the milker’ in England and ‘Mary Mallon’, or ‘Typhoid Mary’, a cook in New York city, USA. These individuals were first identified healthy carriers (symptom free) and infected hundreds of people over the decades while they worked in the food production industry and private homes. An estimated 21 million people are infected in Typhoid fever and 200,000 died from Typhoid fever worldwide every year [23]. It is believed that asymptomatic carriers play an important role in the evolution and global transmission of Typhi and their presence significantly hinders the eradication of Typhoid fever using treatment and vaccination [35].

G.P. Samanta / Chaos, Solitons & Fractals 66 (2014) 74–85

Hepatitis B (a liver disease caused by the HBV virus of the Hepadnavirus family) is one of the major infectious disease that causes long-term asymptomatic carriage and its symptoms include jaundice, abdominal pain, nausea, fatigue and joint pain. For this infection, susceptible host cells and infected cells are hepatocyte and cannot move under normal conditions, but viruses move freely in the liver. It is estimated that around 30% of people with the HBV virus do not show any of the mentioned symptoms. A significant public-health challenge in the control of hepatitis B infection worldwide is the existence of a large pool of chronic carriers who are responsible for transmitting most of the new infections. Infections of other pathogens are also known to produce asymptomatic carriers, among these the Epstein–Barr Virus (EBV) of the herpes family is one of the most common viruses in humans. It commonly causes glandular fever (infectious mononucleosis) and is characterised by a sore throat, swollen lymph nodes (usually in the neck) and extreme fatigue. The pulse vaccination strategy (PVS) consists of repeated application of vaccine at discrete time with equal interval in a population in contrast to the traditional constant vaccination [15,48]. Compared to the proportional vaccination models, the study of pulse vaccination models is in its infancy [48]. At each vaccination time a constant fraction of the susceptible population is vaccinated successfully. Since 1993, attempts have been made to develop mathematical theory to control infectious diseases using pulse vaccination [1,15]. Nokes and Swinton [34] discussed the control of childhood viral infections by pulse vaccination strategy. Stone et al. [39] presented a theoretical examination of the pulse vaccination strategy in the SIR epidemic model and d’Onofrio [12,13] analyzed the use of pulse vaccination policy to eradicate infectious disease for SIR and SEIR epidemic models. Different types of vaccination policies and strategies combining pulse vaccination policy, treatment, pre-outbreak vaccination or isolation have already been introduced by several researchers [4,14,16,17,19,42,46]. Mathematical epidemiology is the study of the spread of diseases, in space and time, with the objective to identify factors that are responsible for or contributing to their occurrence. Mathematical models are becoming important tools in analyzing the spread and control of infectious diseases. Epidemic models of ordinary differential equations have been studied by a number of researchers [3,7–9,11, 25,27,30,31,33,43]. The basic and important objectives for these models are the existence of the threshold values which distinguish whether the infectious disease will be going to extinct, the local and global stability of the disease-free equilibrium and the endemic equilibrium, the existence of periodic solutions and the persistence of the disease. Stability, persistence and permanence in population biology have been studied by many researchers [40,41]. Hence, as a part of population biology, permanence of disease plays an important role in mathematical epidemiology. Despite their public health significance, the effects of carriers on the transmission dynamics of the disease have not received adequate research attention in the mathematical modelling of epidemiology literature [23]. It is noted

75

here that very little attention has been paid to the mathematical modelling and analysis of such diseases by considering the effect of carrier population to gain insight into its transmission dynamics at population level. One of the earlier attempts was done by Kemper [24], in which a general mathematical model that incorporates disease carriers was developed and analyzed. Medley et al. [29] used a mathematical model for hepatitis B with carriers to analyze the effects of HBV vaccination. Kalajdzievska and Li [23] discussed the effects of carriers on the transmission dynamics of infectious diseases by using SIc IR epidemic model. Several other researchers analyzed by using large-scale computational models with carriers which mainly aimed at hepatitis B and other diseases [18,20,32,44,47]. In the present paper, we propose a general mathematical model for infectious diseases with asymptomatic carriers to investigate the effects of carriers on the transmission dynamics by using the Kermack–McKendrick compartmental modelling framework. We have subdivided the entire high-risk human population into mutually-exclusive epidemiological compartments (based on disease status), to gain insights into the qualitative features of infectious diseases in a human population (with the aim of finding effective ways to control its spread). The main feature of this paper is to introduce time delay, saturation incidence rate with valid pulse vaccination strategy. We have introduced two threshold values R1 and R2 and further obtained that the disease will be going to extinct when R1 < 1 and the disease will be permanent when R2 > 1. The important mathematical findings for the dynamical behaviour of the infectious disease model are numerically verified using MATLAB and also epidemiological implications of our analytical findings are addressed critically in Section 5. The aim of the analysis of this model is to trace the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

2. Model derivation and preliminaries In the following, we consider a dynamical model of infectious disease that spread by asymptomatic carriers (or carriers for short) and symptomatically infectious individuals (or infectious for short) with discrete time delay and pulse vaccination strategy (PVS) which satisfies the following assumptions: The underlying high-risk human population is split up into five mutually-exclusive classes (compartments), namely, susceptible (S), exposed (infected but not yet infectious) (E), asymptomatic carriers (Ic ), symptomatically infectious (showing symptoms of the disease) (I) and recovered (infectious people who have cleared or recovered from infection) (R). The susceptible population increases through birth (a constant influx K of susceptible is assumed) and from recovered hosts and decreases due to direct contact with an infectious individual or a carrier, natural death and pulse vaccination strategy. Standard epidemiological models use a bilinear incidence rate bSI based on the law of mass action [2,3] and

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it is reasonable when the mixing of susceptible with infective is considered to be homogeneous. If the population is saturated with infective, there are three types of incidence forms used in epidemiological model: the proportionate mixing incidence b SIN [3,10,45], nonlinear incidence bSp Iq p

[21,22] and saturation incidence b 1þSIrS [3,28] or b 1þSIrIq [36]. Here incidence rates b1 1þSIrc1 S and b2 1þSIr2 S have been considered since the number of effective contacts between infective and susceptible individuals may saturate at high susceptible levels due to crowding of susceptible individuals. The infected class is increased by infection of susceptible. A fraction ð1  qÞ of the exposed individuals will start to show disease symptoms (and move to the class I), while the remaining fraction q will not (become carriers but still remain capable of infecting others and move to the class Ic ). It is assumed that the rate of transmission b1 for carriers is higher than the rate b2 of symptomatically infected individuals due to the fact that they are more likely to be unaware of their condition and so continue with their regular behaviours. Carriers may become symptomatic at a rate r1 . For infections such as HBV for which carriage can remain life-long, r 1 can be regarded as rate of diagnosis [23]. It is also assumed that there is a time lag to account for the fact that an individual infected with bacteria or virus is not infectious until after some time after exposure. Symptomatically infected individual recover with rate r2 . We incorporate a pulse vaccination strategy in which a fraction p of the susceptible population is vaccinated successfully at discrete time t ¼ T; 2T; 3T; . . .. Thus, the following dynamical model of infectious disease that spread through asymptomatic carriers and symptomatically infectious individuals with discrete time delay and pulse vaccination strategy is formulated: dSðtÞ SðtÞIc ðtÞ SðtÞIðtÞ ¼ K  b1  b2  lSðtÞ þ aRðtÞ; t – nT; dt 1 þ r1 SðtÞ 1 þ r2 SðtÞ dEðtÞ SðtÞIc ðtÞ SðtÞIðtÞ Sðt  sÞIc ðt  sÞ þ b2  b1 els ¼ b1 dt 1 þ r1 SðtÞ 1 þ r2 SðtÞ 1 þ r1 Sðt  sÞ Sðt  sÞIðt  sÞ  lEðtÞ; t – nT; 1 þ r2 Sðt  sÞ   dIc ðtÞ b1 Ic ðt  sÞ b2 Iðt  sÞ ¼ qels Sðt  sÞ þ dt 1 þ r1 Sðt  sÞ 1 þ r2 Sðt  sÞ  b2 els

 ðr1 þ d1 þ lÞIc ðtÞ; t – nT;   dIðtÞ b1 Ic ðt  sÞ b2 Iðt  sÞ þ ¼ ð1  qÞels Sðt  sÞ dt 1 þ r1 Sðt  sÞ 1 þ r2 Sðt  sÞ þ r 1 Ic ðtÞ  ðr 2 þ d2 þ lÞIðtÞ; t – nT;

ð2:1Þ

dRðtÞ ¼ r 2 IðtÞ  lRðtÞ  aRðtÞ; t – nT; dt þ Sðt Þ ¼ ð1  pÞSðtÞ; t ¼ nT; n ¼ 1;2;. ..; Eðtþ Þ ¼ EðtÞ; t ¼ nT; n ¼ 1;2;... ; Ic ðtþ Þ ¼ Ic ðtÞ; t ¼ nT; n ¼ 1; 2; ...; Iðtþ Þ ¼ IðtÞ; t ¼ nT; n ¼ 1;2;.. .; Rðtþ Þ ¼ RðtÞ þ pSðtÞ; t ¼ nT; n ¼ 1;2;. ..;

where all coefficients are positive constants. Here SðtÞ denotes the number of susceptible, EðtÞ denotes the number of exposed, Ic ðtÞ denotes the number of infective in carrier compartment, IðtÞ denotes the number of infective in

symptomatically infected compartment, RðtÞ denotes the number of recovered individuals. The pulse vaccination does not give life-long immunity, there is an immunity waning for the vaccination with the per capita immunity waning rate a, and return to the susceptible class. The influx of susceptible comes from two sources: a constant recruitment K and from recovered hosts (aR). The parameters b1 ; b2 ; l; q; d1 ; d2 ; r 1 ; r 2 ; s; p are: b1 : The coefficient of transmission rate from infective in carrier compartment to susceptible humans (and become exposed) and the rate of transmission of infection is of the form:

b1

SðtÞIc ðtÞ : 1 þ r1 SðtÞ This is a more realistic incidence rate (or, infection rate)

b1 SIc b1 I c because limS!1 1þ r1 S ¼ r1 . Here the so-called saturation

contact rate is

b1 S 1þr1 S

and it tends to the saturation value rb11

for large S. b2 : The coefficient of transmission rate from infective in symptomatically infected compartment to susceptible humans (and become exposed) and the rate of transmission of infection is of the form:

b2

SðtÞIðtÞ : 1 þ r2 SðtÞ This is a suitable incidence rate (or, infection rate)

b2 SI b2 I because limS!1 1þ r2 S ¼ r2 . Here the saturation contact rate b2 S 1þr2 S

and it tends to the saturation value rb22 for large S. It is assumed that b1 > b2 due to the fact that infective in carrier compartment are more likely to be unaware of their condition and so continue with their regular behaviours. l: The coefficient of natural death rate of all epidemiological human classes. d1 : The coefficient of additional disease-related death rate of infective in carrier compartment. d2 : The coefficient of additional disease-related death rate of infective in symptomatically infected compartment. ð1  qÞ: The fraction of the exposed individuals will start to show disease symptoms and move to the class I. The remaining fraction q ð0 < q < 1Þ will not start to show disease symptoms (but still remain capable of infecting others) and move to the class Ic . r 1 : The rate at which carriers become symptomatic and show disease symptoms (move to the class I). r 2 : The rate at which symptomatically infected individuals clear infections and move to the class R. s: The constant latency period from the time of being infected (exposed) to the time of being infectious (capable of infecting others). The probability that an individual remains in the latency period (exposed class) at least t time units before becoming infectious is given by a step function with value 1 for 0 6 t 6 s and value zero for t > s. The probability that an individual in the latency period has survived is given by els . p ð0 < p < 1Þ: The fraction of susceptible who are vaccinated successfully at discrete time t ¼ T; 2T; 3T; . . ., which is called impulsive vaccination rate. is

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The total high-risk human population size NðtÞ ¼ SðtÞþ EðtÞ þ Ic ðtÞ þ IðtÞ þ RðtÞ can be determined by the following differential equation:

Lemma 2.1 [38]. Consider the following equation:

dNðtÞ ¼ K  lNðtÞ  d1 Ic ðtÞ  d2 IðtÞ; dt

where a; b; c; s > 0; xðtÞ > 0, for s 6 t 6 0. We have

ð2:2Þ

which is derived by adding first five equations of system (2.1). Therefore,

dNðtÞ 6 K  lNðtÞ dt K ) 6 lim inf NðtÞ t!1 l þ d1 þ d2 K 6 lim sup NðtÞ 6 :

dxðtÞ ¼ axðt  sÞ  bxðtÞ  cx2 ðtÞ; dt

ð2:7Þ

ab ; c ðIIÞ if a < b; and c P 0; then limxðtÞ ¼ 0:

ðIÞ if a > b; then limxðtÞ ¼ t!1

t!1

K  ðl þ d1 þ d2 ÞNðtÞ 6

l

t!1

Lemma 2.2. Consider the following impulsive differential equation:

ð2:3Þ

Let us simplify the model (2.1) as follows: dSðtÞ SðtÞIc ðtÞ SðtÞIðtÞ  b2  lSðtÞ þ aRðtÞ; t – nT; ¼ K  b1 dt 1 þ r1 SðtÞ 1 þ r2 SðtÞ   dIc ðtÞ b1 Ic ðt  sÞ b2 Iðt  sÞ þ ¼ qels Sðt  sÞ dt 1 þ r1 Sðt  sÞ 1 þ r2 Sðt  sÞ  ðr1 þ d1 þ lÞIc ðtÞ; t – nT;   dIðtÞ b1 Ic ðt  sÞ b2 Iðt  sÞ þ ¼ ð1  qÞels Sðt  sÞ ð2:4Þ dt 1 þ r1 Sðt  sÞ 1 þ r2 Sðt  sÞ þ r1 Ic ðtÞ  ðr2 þ d2 þ lÞIðtÞ; t – nT; dRðtÞ ¼ r2 IðtÞ  lRðtÞ  aRðtÞ; t – nT; dt dNðtÞ ¼ K  lNðtÞ  d1 Ic ðtÞ  d2 IðtÞ; t – nT; dt þ Sðt Þ ¼ ð1  pÞSðtÞ; t ¼ nT; n ¼ 1; 2;. . . ; Ic ðtþ Þ ¼ Ic ðtÞ; t ¼ nT; n ¼ 1; 2; . . .;

where a > 0; b > 0; 0 < p < 1. Then there exists a unique positive periodic solution of system (2.8):

e e ðtÞ ¼ u

a   a bðtkTÞ þ u  e ; b b

kT < t 6 ðk þ 1ÞT;

where u ¼

  að1  pÞ 1  ebT bf1  ð1  pÞebT g

e e ðtÞ is globally asymptotically stable. and u

 d  bt e uðtÞ ¼ aebt : dt Integrating between pulses :

Iðt Þ ¼ IðtÞ; t ¼ nT; n ¼ 1; 2; . . .; Rðt þ Þ ¼ RðtÞ þ pSðtÞ; t ¼ nT; n ¼ 1;2; .. . ;

¼

þ

Nðt Þ ¼ NðtÞ; t ¼ nT; n ¼ 1;2; .. . ;

Z

Z

t

  d ebt uðtÞ

kT t

aebt dt ) uðtÞ ¼

kT

a n ao þ uðkTÞ  ebðtkTÞ ; b b

kT < t 6 ðk þ 1ÞT;

with initial conditions

Sð#Þ ¼ u1 ð#Þ; Ic ð#Þ ¼ u2 ð#Þ; Ið#Þ ¼ u3 ð#Þ; such that

ui ð#Þ P 0 ði ¼ 1; 2; 3; 4; 5Þ; 8# 2 ½s; 0;

ð2:5Þ

where ui ð#Þ P 0 ði ¼ 1; 2; 3; 4; 5Þ are non-negative continuous functions on # 2 ½s; 0. For a biological meaning, we further assume that ui ð0Þ > 0 ði ¼ 1; 5Þ and ui ð0Þ P 0 ði ¼ 2; 3; 4Þ. The solution of system (2.4) and (2.5) is a piecewise continuous function U : Rþ ! R5þ ; UðtÞ is continuous on ðnT; ðn þ 1ÞT; n 2 N and UðnT þ Þ ¼ limt!nT þ UðtÞ exists. There exists a unique solution of (2.4) with initial conditions (2.5) since the right hand sides of (2.4) and the pulse are smooth functions [5,6,26]. From biological considerations, we analyze system (2.4) and (2.5) in the closed set:   K G ¼ ðSðtÞ;Ic ðtÞ;IðtÞ;RðtÞ;NðtÞÞ 2 R5þ : 0 6 S þ Ic þ I þ R;N 6 ;

l

ð2:6Þ

R5þ

ð2:8Þ k ¼ 1; 2; . . . ;

Proof. From the first equation of system (2.8) we get,

þ

Rð#Þ ¼ u4 ð#Þ; Nð#Þ ¼ u5 ð#Þ;

duðtÞ ¼ a  buðtÞ; t – kT; dt þ uðt Þ ¼ ð1  pÞuðtÞ; t ¼ kT;

5

where represents the nonnegative cone of R including its lower dimensional faces. It can be verified that G is positively invariant with respect to (2.4) and (2.5). Before starting our main results, we give the following two lemmas which will be essential for study.

where uðkTÞ is the initial value at time kT. Using the second equation of system (2.8) we have the following stroboscopic map:

uððk þ 1ÞTÞ ¼ ð1  pÞ

ha

b ¼ f ðuðkTÞÞ;

n i ao þ uðkTÞ  ebT b ð2:9Þ

   where f ðuÞ ¼ ð1  pÞ ab þ u  ab ebT . Solving the following equation:

 o a þ u  ebT ; b b bT að1  pÞ 1  e : u ¼ bf1  ð1  pÞebT g u ¼ ð1  pÞ

na

we get;

Since jf 0 ðuÞj ¼ ð1  pÞebT < 1, as 0 < p < 1 and b > 0, the system (2.9) has a unique positive equilibrium að1pÞð1ebT Þ u ¼ b 1ð1pÞebT which is globally asymptotically stable. f g Hence the corresponding periodic solution of system (2.8):

a   a bðtkTÞ þ u  e ; kT < t 6 ðk þ 1ÞT; b b   að1  pÞ 1  ebT where u ¼ bf1  ð1  pÞebT g

e e ðtÞ ¼ u

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is globally asymptotically stable. This completes the proof. h

Theorem 3.1. If R1 < 1, then the disease-free periodic solu  K e of system (2.4) with initial tion e S e ðtÞ; 0; 0; K l  S e ðtÞ; l conditions (2.5) is globally asymptotically stable.

3. Global stability of the disease-free periodic solution In this section, we discuss the existence of the diseasefree periodic solution of system (2.4), in which infectious individuals (both in carrier compartment and symptomatically infected compartment) are completely absent, that is, Ic ðtÞ ¼ 0; 8t P 0 and IðtÞ ¼ 0; 8t P 0. Under this circumstances, system (2.4) reduces to the following impulsive system without delay:

dSðtÞ ¼ K  lSðtÞ þ aRðtÞ; dt dRðtÞ ¼ lRðtÞ  aRðtÞ; dt

Sðt þ Þ ¼ ð1  pÞSðtÞ;

t ¼ nT; n ¼ 1; 2; . . . ;

Rðt þ Þ ¼ RðtÞ þ pSðtÞ; Nðtþ Þ ¼ NðtÞ;

ð3:1Þ

zðtþ Þ ¼ ð1  pÞzðtÞ;

From the third and sixth equations of system (3.1), we have limt!1 NðtÞ ¼ Kl. Further, from the second and seventh equations of system (2.1) it follows that

8t P 0 and IðtÞ ¼ 0; 8t P 0:

In the following, we shall show that the susceptible population SðtÞ and recovered population RðtÞ oscillate with period T, in synchronisation with the periodic impulsive vaccination strategy under some condition. Consider the following limit system of system (3.1) as per the previous discussions:

RðtÞ ¼

K

l

 SðtÞ;

  dSðtÞ K ¼ ðl þ aÞ  SðtÞ ; dt l Sðt þ Þ ¼ ð1  pÞSðtÞ;

t – nT;

ð3:2Þ

t ¼ nT; n ¼ 1; 2; . . . :

Using Lemma 2.2, the periodic solution of system (3.2) is given below:

l

t – nT;

ð3:6Þ

t ¼ nT; n ¼ 1; 2; . . .

By (3.2) and (3.3), we know that the periodic solution of system (3.6),

ez e ðtÞ ¼ e S e ðtÞ ¼

K

l

 K ðlþaÞðtnTÞ e þ S  ;

l

nT < t 6 ðn þ 1ÞT; where   Kð1  pÞ 1  eðlþaÞT S ¼ lf1  ð1  pÞeðlþaÞT g

ð3:7Þ

is globally asymptotically stable. Let ðSðtÞ; Ic ðtÞ; IðtÞ; RðtÞ; NðtÞÞ be the solution of system (2.4) with initial conditions (2.5) and Sð0þ Þ ¼ S0 > 0. If zðtÞ be the solution of system (3.6) with initial value zð0þ Þ ¼ S0 > 0, then by the comparison theorem for impulsive differential equation [26] there exists an integer n1 > 0 such that

SðtÞ < zðtÞ < ez e ðtÞ þ ; nT < t 6 ðn þ 1ÞT; n > n1   K 1  eðlþaÞT e ) SðtÞ < z e ðtÞ þ  6 þ lf1  ð1  pÞeðlþaÞT g ¼ n ðsayÞ:

ð3:8Þ

Further, from the second and third equations of system (2.4), we have 8t > nT þ s and 8n > n1 ,

 K K ðlþaÞðtnTÞ e S e ðtÞ ¼ þ S  e ;

l

nT < t 6 ðn þ 1ÞT; where   Kð1  pÞ 1  eðlþaÞT S ¼ lf1  ð1  pÞeðlþaÞT g

t ¼ nT; n ¼ 1; 2; . . .

  dzðtÞ K ¼ ðl þ aÞ  zðtÞ ; dt l

t ¼ nT; n ¼ 1; 2; . . . ;

limEðtÞ ¼ 0 as Ic ðtÞ ¼ 0;

Sðt þ Þ ¼ ð1  pÞSðtÞ;

t – nT;

So, we consider the following comparison impulsive differential system:

t ¼ nT; n ¼ 1; 2; . . . :

t!1

small enough

b1 els ðA þ Þ < h; where r ¼ minfr1 ; r2 g; 1 þ rðA þ Þ   K 1  eðlþaÞT A¼ and h ¼ minfd1 þ l; r 2 þ d2 þ lg > 0: lf1  ð1  pÞeðlþaÞT g ð3:5Þ

  dSðtÞ K 6 ðl þ a Þ  SðtÞ ; dt l

t – nT;

t – nT;

>0

From the first and sixth equations of (2.4), it follows that

t – nT;

dNðtÞ ¼ K  lNðtÞ; dt

Proof. Since R1 < 1, we can choose such that

d b nels fIc ðtÞ þ IðtÞg 6 1 fIc ðt  sÞ þ Iðt  sÞg dt 1 þ rn ð3:3Þ

and e S e ðtÞ is globally asymptotically stable. b1 els A Denote R1 ¼ ; where r ¼ minfr1 ; r2 g; fð1 þ rAÞhg   K 1  eðlþaÞT A¼ and h ¼ minfd1 þ l;r 2 þ d2 þ lg > 0: lf1  ð1  pÞeðlþaÞT g ð3:4Þ

 hfIc ðtÞ þ IðtÞg:

ð3:9Þ

Consider the following comparison equation:

dyðtÞ b1 nels ¼ yðt  sÞ  hyðtÞ: dt 1 þ rn

ð3:10Þ

From (3.5), we have

b1 nels < h ) limyðtÞ ¼ 0; by Lemma 2:1: t!1 1 þ rn

ð3:11Þ

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Set ðSðtÞ; Ic ðtÞ; IðtÞ; RðtÞ; NðtÞÞ be the solution of system (2.4) with initial conditions (2.5) and Ic ð#Þ ¼ u2 ð#Þ P 0; Ið#Þ ¼ u3 ð#Þ P 0; 8# 2 ½s; 0 where ui ð0Þ > 0 ði ¼ 2; 3Þ; yðtÞ be the solution of (3.10) with initial condition yð#Þ ¼ u2 ð#Þ þ u3 ð#Þ P 0; 8# 2 ½s; 0 where u2 ð0Þ þ u3 ð0Þ > 0. By the comparison theorem of differential equation and the positivity of solution (with Ic ðtÞ P 0; IðtÞ P 0), we have

limfIc ðtÞ þ IðtÞg ¼ 0 ) limIc ðtÞ ¼ limIðtÞ ¼ 0:

t!1

t!1

t!1

ð3:12Þ

Hence for any 1 > 0 (sufficiently small), there exists a positive integer n2 , where n2 T > n1 T þ s, such that 0 < Ic ðtÞ; IðtÞ < 21 ; 8t > n2 T. Using the fifth equation of system (2.4), we get

dNðtÞ > K  lNðtÞ  ðd1 þ d2 Þ1 ; 8t > n2 T: dt

) limNðtÞ ¼ t!1

K

IðtÞ <

2

;

Making

nT < t 6 ðn þ 1ÞT; n > n6 :

ð3:21Þ

1 ! 0, it follows from (3.8) and (3.21) that  peðlþaÞðtnTÞ ; 1  ð1  pÞeðlþaÞT ð3:22Þ

NðtÞ >

l

 1 ;

8t > n4 T:

8t > n4 T:

ð3:15Þ

where A1 ¼

it follows

ð3:24Þ

Using (3.12), (3.14), (3.23), (3.24) and from the restriction NðtÞ ¼ SðtÞ þ EðtÞ þ Ic ðtÞ þ IðtÞ þ RðtÞ, we have

lim RðtÞ ¼

t!1

K

l

e S e ðtÞ:

ð3:25Þ

Therefore, we conclude that if R1 < 1, then the disease  e S e ðtÞ; 0; 0; K  e S e ðtÞ; K of system

free periodic solution

l

l

(2.4) with initial conditions (2.5) is globally asymptotically stable. This completes the proof. h

ð3:16Þ

It is clear that there exists an integer n5 > n4 such that

EðtÞ < A1 þ 1 ; 8t > n5 T;

1 ! 0,

t!1

Therefore, from the second equation of system (2.1), we have

dEðtÞ Kb1 1 6  lEðtÞ; dt l þ rK

ð3:23Þ

lim EðtÞ ¼ 0:

1 > 0 is arbitrarily smallÞ:

K

lim SðtÞ ¼ e S e ðtÞ:

t!1

By the positivity of EðtÞ and making from (3.17) that

8t > n 3 T

It follows from (3.12) and (3.14) that there exists an integer n4 > n3 such that

0 < Ic ðtÞ;

ð3:20Þ

which is globally asymptotically stable. By the comparison theorem for impulsive differential equation [26], there exists an integer n6 > n5 such that

l

ð3:14Þ

1

 3a1 and b1 1 þ l þ a   ð1  pÞ 1  eðb1 1 þlþaÞT z2 ¼ U ; f1  ð1  pÞeðb1 1 þlþaÞT g

U¼

is globally attractive and so

 1 ;

ðas

l

where

K þ alK  aA1

nT < t 6 ðn þ 1ÞT

So, by the comparison theorem, there exists an integer n3 > n2 such that

l

nT < t 6 ðn þ 1ÞT;

 K e S e ðtÞ ¼ 1

l

K  ðd1 þ d2 Þ1

  ez 2e ðtÞ ¼ U þ z2  U eðb1 1 þlþaÞðtnTÞ ;

SðtÞ > ez 2e ðtÞ  1 ; ð3:13Þ

dz1 ðtÞ Now; ¼ fK  ðd1 þ d2 Þ1 g  lz1 ðtÞ ) limz1 ðtÞ t!1 dt K  ðd1 þ d2 Þ1 : ¼

NðtÞ P

By Lemma 2.2, we know that the periodic solution of system (3.19) is

Kb1 1 : lðl þ rKÞ ð3:17Þ

So, from the first and sixth equations of system (2.4) we get

 dSðtÞ aK P Kþ  aA1  3a1  ðb1 1 þ l þ aÞSðtÞ; t – nT; dt l Sðtþ Þ ¼ ð1  pÞSðtÞ; t ¼ nT; n ¼ 1;2;...: ð3:18Þ Let us consider the following comparison impulsive differential system 8t > n5 T and 8n > n5 :

 dz2 ðtÞ aK ¼ Kþ  aA1  3a1  ðb1 1 þ l þ aÞz2 ðtÞ; t – nT; l dt z2 ðt þ Þ ¼ ð1  pÞz2 ðtÞ; t ¼ nT; n ¼ 1; 2; . . .: ð3:19Þ

4. Permanence In this section, we wish to discuss the permanence of the system (2.4), this means that the long-term survival (i.e., will not vanish in time) of all components of the system (2.4) with initial conditions (2.5). It demonstrates how the disease will be permanent (i.e., will not vanish in time) under some conditions. Definition. The system (2.4) is said to be permanent, i.e., the long-term survival (will not vanish in time) of all components of the system (2.4), if there are positive constants mi ði ¼ 1; 2; 3; 4Þ such that:

lim inf SðtÞ P m1 ; lim inf fIc ðtÞ þ IðtÞg P m2 ; t!1

t!1

lim inf RðtÞ P m3 ; t!1

lim inf NðtÞ P m4 ; t!1

hold for any solution ðSðtÞ; Ic ðtÞ; IðtÞ; RðtÞ; NðtÞÞ of (2.4) with initial conditions (2.5). Here mi ði ¼ 1; 2; 3; 4Þ are independent of (2.5).

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Theorem 4.1. If R2 > 1, then there exists a positive constant m such that each positive solution ðSðtÞ; Ic ðtÞ; IðtÞ; RðtÞ; NðtÞÞ of the system (2.4) with initial conditions (2.5) satisfies ðIc ðtÞ þ IðtÞÞ P m for sufficiently large time t, where

 

ls  Kð1  pÞ 1  elT b2 e  r0 ; 0 lf1  ð1  pÞelT g h ¼ maxfr1 ; r2 g; h0 ¼ maxfd1 þ l; r 2 þ d2 þ lg:

R2 ¼

r0

ð4:1Þ

Proof. From the second and third equations of system (2.4), we have

dDðtÞ Sðt  sÞDðt  sÞ P b2 els  h0 DðtÞ dt 1 þ r0 Sðt  sÞ   SðtÞ ¼ DðtÞ b2 els  h0 0 1 þ r SðtÞ Z t d SðuÞDðuÞ du; where  b2 els dt ts 1 þ r0 SðuÞ DðtÞ ¼ Ic ðtÞ þ IðtÞ: Define; VðtÞ ¼ DðtÞ þ b2 els

Define D ¼

8t P t 2 :

ð4:9Þ

D0 ðt2 þ s þ T 0 Þ > h0 D1



b2 els n0 1 h ð1 þ r0 n0 Þ 0



> 0:

ð4:10Þ

So, we have got a contradiction and hence DðtÞ P D1 ;

SðuÞDðuÞ du 0 ts 1 þ r SðuÞ

8t P t2 . As a consequence of (4.3), (4.4), (4.8) and (4.9),

l

we get

V 0 ðtÞ > h0 D1



b2 els n0 1 h ð1 þ r0 n0 Þ

b2 els n0 > 1; where h ð1 þ r0 n0 Þ    Kð1  pÞ 1  eðb1 D þlÞT >0 n0 ¼  ðb1 D þ lÞf1  ð1  pÞeðb1 D þlÞT g ð4:4Þ for a sufficiently small  > 0. Let us claim that there exists a t 1 > 0 such that DðtÞ P D ; 8t P t 1 . Otherwise, there exists a t 1 > 0 such that DðtÞ < D ; 8t P t 1 . It follows from the first and sixth equations of (2.4):

ð4:5Þ

Let us consider the following comparison impulsive differential system 8t P t 1 :

ð4:6Þ

By Lemma 2.2, we know that the periodic solution of system (4.6) is



0

> 0;

8t P t 2

) VðtÞ ! 1 as t ! 1:

ð4:11Þ

This is a contradiction because

VðtÞ ¼ DðtÞ þ b2 els 6 DðtÞ þ b2 els

0

dz3 ðtÞ ¼ K  ðb1 D þ lÞz3 ðtÞ; t – nT; dt z3 ðt þ Þ ¼ ð1  pÞz3 ðtÞ; t ¼ nT; n ¼ 1; 2; . . . :

ð4:8Þ

Otherwise, there exists a T 0 > 0 such that DðtÞ P D1 ; where Dðt 2 þ s þ T 0 Þ ¼ D1 and D0 ðt 2 þ s þ T 0 Þ 6 0. However, from (4.2), (4.4) and (4.8), we get

t

dSðtÞ > K  ðb1 D þ lÞSðtÞ; t – nT; dt Sðt þ Þ ¼ ð1  pÞSðtÞ; t ¼ nT; n ¼ 1; 2; . . .

8t P t2 :

8t 2 ½t 2 ; t 2 þ s þ T 0 ,

ðR2  1Þ > 0; ðsince R2 > 1Þ b1  ) D ! 0þ as  ¼ ðR2  1Þ ! 0þ

)

SðtÞ > ez 3e ðtÞ   ) SðtÞ > z3   ¼ n0 ; t2½t2 ;t 2 þs

) V 0 ðtÞ   SðtÞ 0 ; ðusing ð4:2ÞÞ P DðtÞ b2 els  h 1 þ r0 SðtÞ   b els SðtÞ ¼ h0 DðtÞ 0 2 1 : h ð1 þ r0 SðtÞÞ ð4:3Þ 

ð4:7Þ

which is globally asymptotically stable. By the comparison theorem for impulsive differential equation [26], there exists t 2 > t1 þ s such that the followings hold:

Next; let D1 ¼ min DðtÞ ) DðtÞ P D1 ;

ð4:2Þ Z

   K K eðb1 D þlÞðtnTÞ ; þ z3    b1 D þ l b1 D þ l nT < t 6 ðn þ 1ÞT; where    Kð1  pÞ 1  eðb1 D þlÞT z3 ¼ ;  ðb1 D þ lÞf1  ð1  pÞeðb1 D þlÞT g

ez 3e ðtÞ ¼

Z

t

ts Z t

SðuÞDðuÞ du 1 þ r0 SðuÞ SðuÞDðuÞdu

ts

Z t 2 K þ b2 els du l l ts   K Ksb2 els : 1þ ¼ 6

K

l

l

ð4:12Þ

Therefore, we conclude for any t1 > 0, the inequality DðtÞ < D cannot hold for all t P t1 . Thus we are left to consider the following two cases: (i) DðtÞ P D for sufficiently large t; (ii) DðtÞ oscillates about D for sufficiently large t. It is clear that if DðtÞ P D for sufficiently large t, then our desired result is obtained. So, we only need to consider the case (ii). Let

m ¼ min

   D 0 ; D eh s ; 2

where h0

¼ maxfd1 þ l; r 2 þ d2 þ lg:

ð4:13Þ

Now, we will show that DðtÞ P m for sufficiently large t. Let t > 0 and t 0 > 0 satisfy Dðt  Þ ¼ Dðt  þ t 0 Þ ¼ D and DðtÞ < D for t  < t < t  þ t0 , where t  is sufficiently large such that SðtÞ > n0 for t  < t < t  þ t0 . It is clear that DðtÞ is uniformly continuous since the positive solution of

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G.P. Samanta / Chaos, Solitons & Fractals 66 (2014) 74–85

(2.4) is ultimately bounded and DðtÞ ¼ ðIc ðtÞ þ IðtÞÞ is not affected by impulsive effects. Hence there exists a constants T 1 , where 0 < T 1 < s and T 1 is independent of t , such that DðtÞ > D =2 for t  6 t 6 t þ T 1 . If t 0 6 T 1 , the required result is obtained. If T 1 < t0 6 s, since D0 ðtÞ > h0 DðtÞ and Dðt Þ ¼ D , it follows that 0 DðtÞ P D eh s for t < t < t þ t0 . If t 0 > s; we have 0 DðtÞ P D eh s for t < t < t þ s and by using the same 0 arguments we can obtain DðtÞ P D eh s for     t þ s < t < t þ t0 as the interval ½t ; t þ t0  can be chosen arbitrarily. So, we can conclude that DðtÞ P m for sufficiently large t. On the basis of the previous discussions, the choice of m is independent of the positive solution of (2.4) and hence any positive solution of (2.4) satisfies DðtÞ P m for t large enough. This completes the proof. h Theorem 4.2. If R2 > 1, then the system (2.4) with initial conditions (2.5) is permanent. Proof. Suppose ðSðtÞ; Ic ðtÞ; IðtÞ; RðtÞ; NðtÞÞ be any solution of system (2.4) with initial conditions (2.5). From the first and sixth equations of system (2.4), we have

dSðtÞ P K  b1 SðtÞIc ðtÞ  b2 SðtÞIðtÞ  lSðtÞ dt

 b K P K  1 þ l SðtÞ; t – nT; as

l

Sðt þ Þ ¼ ð1  pÞSðtÞ; t ¼ nT; n ¼ 1; 2; . . . :

b1 > b2 ;

ð4:14Þ Let us consider the following comparison impulsive differential system:

 dz4 ðtÞ b K ¼ K  1 þ l z4 ðtÞ; dt l þ

z4 ðt Þ ¼ ð1  pÞz4 ðtÞ;

t – nT;

ð4:15Þ

t ¼ nT; n ¼ 1; 2; . . . :

By Lemma 2.2, we know that the periodic solution of system (4.15) is

ez 4e ðtÞ ¼

lK b1 K þ l

 þ z4  2

lK

   b K  1l þl ðtnTÞ e ; 2

b1 K þ l

nT < t 6 ðn þ 1ÞT; where

b K   1 lKð1  pÞ 1  e l þl T   z4 ¼ b K   ; 1 ðb1 K þ l2 Þ 1  ð1  pÞe l þl T

ð4:16Þ

which is globally asymptotically stable. By the comparison theorem for impulsive differential equation, there exists sufficiently small 1 > 0 such that the following holds:



lKð1  pÞ 1  e limSðtÞ P

t!1

b

1 Kþ

l

 

l T

 b K    1 > 0: 1 ðb1 K þ l2 Þ 1  ð1  pÞe l þl T ð4:17Þ

From the fourth equation of system (2.4) and using Theorem 4.1, we have

dRðtÞ P r 2 m  ðl þ aÞRðtÞ ) limRðtÞ t!1 dt r2 m  >0 P lþa 2

ð4:18Þ

for a sufficiently small 2 > 0 (m is given by (4.13)). Hence system (2.4) with initial conditions (2.5) is permanent and this completes the proof. h Note: The fate of the disease is not clear when R2 6 1 6 R1 . 5. Numerical simulations and biological interpretations Here we have done some numerical simulations using MATLAB. We first consider the case when R1 ¼ 0:9577 < 1 using the parameter values given in Table 1. Using these parameter values, the movement paths of SðtÞ; Ic ðtÞ; IðtÞ and RðtÞ are presented in Fig.1(a). This figure shows that the disease dies out when R1 < 1, which supports our analytical result given in Theorem 3.1. Its epidemiological implication is that the infectious population vanishes, i.e., the disease dies out when R1 < 1 (see Fig. 1(a)). In Fig. 1(b), the effects of pulse vaccination ðpÞ on the threshold value R1 is presented using the parameter values given in Table 1. It shows that the threshold values R1 gradually decrease when the pulse vaccination rate ðpÞ increases. This implies that the strategy of pulse vaccination is very effective to eradicate the infectious disease. Next, we consider the case when R2 ¼ 2:3490 > 1 using the parameter values given in Table 2. Using these parameter values, the movement paths of SðtÞ; Ic ðtÞ; IðtÞ and RðtÞ are presented in Fig. 2(a). This figure shows that the disease will be permanent when R2 > 1, which supports our analytical result given in Theorem 4.2. In Fig. 2(b), the effects of pulse vaccination ðpÞ on the threshold value R2 is presented using the parameter values given in Table 2. It shows that the threshold values R2 gradually decrease when the pulse vaccination rate ðpÞ increases. This also implies that the strategy of pulse vaccination is very effective to eradicate the infectious disease. We also consider the case when R1 ¼ 2:1979 > 1 and R2 ¼ 0:1142 < 1 with parameter values given in Table 3. Using these parameter values, the movement paths of Table 1 Parameter values for Figs. 1(a) and (b), 4(a) and (b). Parameter

Values

K b1 b2

0.1 0.09 0.1 0.3 0.4 0.01 0.2 0.4 0.4 0.2 0.2 0.15 0.8 1 5

r1 r2 l a q r1 r2 d1 d2 p

s T

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G.P. Samanta / Chaos, Solitons & Fractals 66 (2014) 74–85 1.08

4 3.5

1.06

3

1.04

R(t)

1.02 R1

Population

2.5 2

1 1.5 0.98

1 0.5 0 0

I(t)

0.96

S(t) Ic(t) 5

10

15

20

25 Time(t)

30

35

40

45

0.94 0

50

0.1

0.2

0.3

0.4 p

0.5

0.6

0.7

0.8

(b)

(a)

Fig. 1. (a) Movement paths of SðtÞ; Ic ðtÞ; IðtÞ and RðtÞ for R1 ¼ 0:9577 < 1, (b) the effects of pulse vaccination ðpÞ on the threshold value R1 , with parameter values given in Table 1.

SðtÞ; Ic ðtÞ; IðtÞ and RðtÞ are presented in Fig. 3(a). This figure shows that the disease dies out. For R1 ¼ 2:3401 > 1 and R2 ¼ 0:9690 < 1 where p ¼ 0:3 and other parameter values are given in Table 3, the movement paths of SðtÞ; Ic ðtÞ; IðtÞ and RðtÞ are presented in Fig. 3(b). This figure shows that the disease is still permanent though the level of disease is very low. From the figures it is observed that a large pulse vaccination rate will lead to eradication of the disease.

Table 2 Parameter values for Fig. 2(a) and (b). Parameter

Values

K b1 b2

0.1 0.8 0.9 0.2 0.3 0.01 0.2 0.4 0.04 0.02 0.02 0.015 0.8 1 5

r1 r2 l a q r1 r2 d1 d2 p

s T

Remark 1. When R2 6 1 6 R1 , the dynamical behaviour of model (2.4) and (2.5) has not been clear. So, in this case the fate of the disease is not clear.

2.5

200 180

2

160

1.5

120 R2

Population

140 I(t)

1

80 60

Ic(t) 0.5

40 R(t)

20

S(t) 0

100

0

10

20

30

40

50 Time(t)

(a)

60

70

80

90

100

0

0

0.1

0.2

0.3

0.4 p

0.5

0.6

0.7

0.8

(b)

Fig. 2. (a) Movement paths of SðtÞ; Ic ðtÞ; IðtÞ and RðtÞ for R2 ¼ 2:3490 > 1, (b) the effects of pulse vaccination ðpÞ on the threshold value R2 , with parameter values given in Table 2.

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G.P. Samanta / Chaos, Solitons & Fractals 66 (2014) 74–85

Remark 2. Our mathematical results are also valid for simpler models by setting r1 ¼ r2 ¼ 0 (for standard mass action mixing) and a ¼ 0 (for lifelong immunity) which are numerically presented in Fig. 4(a) and (b). For R2 > 1, the same qualitative behaviours can be obtained in these cases (the levels of recovered population will be higher for a ¼ 0). Our model is also well defined for T ¼ s and the same qualitative behaviours can be obtained.

Table 3 Parameter values for Fig. 3(a) and (b). Parameter

Values

K b1 b2

0.1 0.6 0.8 0.3 0.35 0.01 0.2 0.4 0.4 0.2 0.6 0.4 0.8 1 5

r1 r2 l a q r1 r2 d1 d2 p

s T

6. Conclusions In this paper we have considered a dynamical model of infectious disease that spread by asymptomatic carriers (or carriers for short) and symptomatically infectious individuals (or infectious for short) with discrete time delay, pulse

3.5

2 1.8

3

1.6 R(t)

1.4 Population

Population

2.5

2

1.5

S(t)

1.2 1 0.8 0.6

1

R(t)

0.4 0.5

I(t)

0 0

5

S(t)

I(t)

0.2

Ic(t) 10

15

20

25 Time(t)

30

35

40

45

0

50

0

Ic(t) 10

20

30

40

50 Time(t)

60

70

80

90

100

(b)

(a)

Fig. 3. Movement paths of SðtÞ; Ic ðtÞ; IðtÞ and RðtÞ (a) for R1 ¼ 2:1979 > 1 and R2 ¼ 0:1142 < 1 with parameter values given in Table 3, (b) for R1 ¼ 2:3401 > 1 and R2 ¼ 0:9690 < 1 where p ¼ 0:3 and other parameter values are given in Table 3. 4

4.5

3.5

4 3.5

3

3 Population

Population

2.5 R(t)

2 1.5

0 0

R(t)

2 1.5

1 0.5

2.5

1 I(t)

S(t)

0.5

5

10

S(t)

I(t) Ic(t)

Ic(t) 15

20

25 Time(t)

(a)

30

35

40

45

50

0 0

5

10

15

20

25 Time(t)

30

35

40

45

50

(b)

Fig. 4. Movement paths of SðtÞ; Ic ðtÞ; IðtÞ and RðtÞ (a) for R1 ¼ 0:9886 < 1 with r1 ¼ r2 ¼ 0; b1 ¼ 0:03 and other parameter values are given in Table 1, (b) for R1 ¼ 0:2164 < 1 with a ¼ 0 and other parameter values are given in Table 1.

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vaccination strategy and saturation incidence rate. The entire high-risk human population is split up into five mutually-exclusive epidemiological compartments (based on disease status), namely, susceptible (S), exposed (infected but not yet infectious) (E), asymptomatic carriers (Ic ), symptomatically infectious (showing symptoms of the disease) (I) and recovered (infectious people who have cleared or recovered from infection) (R). The susceptible population increases through birth (a constant influx K of susceptible is assumed) and from recovered hosts and decreases due to direct contact with an infectious individual or a carrier, natural death and pulse vaccination strategy. The infected classes are increased by infection of susceptible. A fraction of the exposed individuals will start to show disease symptoms (and move to the class I), while the remaining fraction will not (become carriers but still remain capable of infecting others and move to the class Ic ). It is assumed that there is a time lag to account for the fact that an individual infected with bacteria or virus is not infectious until after some time after exposure. A fraction of the asymptomatically infectious individuals (belong to carrier compartment) show disease symptoms and move to the class I. The infective in symptomatically infected compartment are decreased through recovery from infection, by disease-related death and by natural death. The infective in carrier compartment are decreased by showing disease symptoms (and move to the class I), by disease-related death and by natural death. The most basic and important questions to ask for the systems in the theory of mathematical epidemiology are the persistence, extinctions, the existence of periodic solutions, global stability, etc. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have introduced two threshold values R1 and R2 and further obtained that the disease will be going to extinct when R1 < 1 and the disease will be permanent when R2 > 1. The important mathematical findings for the dynamical behaviour of the infectious disease model are also numerically verified using MATLAB. It is observed that a large pulse vaccination rate will lead to eradication of the disease and when R2 6 1 6 R1 , the dynamical behaviour of the infectious disease is not clear. Our mathematical results are also valid for simpler models by setting r1 ¼ r2 ¼ 0 (for standard mass action mixing) and a ¼ 0 (for lifelong immunity). The same qualitative behaviours can be obtained in these cases (the levels of recovered population will be higher for a ¼ 0). Our model is also well defined for T ¼ s and the same qualitative behaviours can be obtained. The aim of the analysis of this model is to trace the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness. Acknowledgements The author is grateful to the anonymous referees and the Editor-in-Chief (Dr. Paolo Grigolini, Ph.D.) for their careful reading, valuable comments and helpful suggestions, which have helped him to improve the presentation of this work significantly. He likes to thank TWAS, UNESCO

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