Analysis of a Vaccination Model For Carrier Dependent Population With A Saturated Incidence Rate

Infectious diseases are spread by carriers which are present in the environment [3]. Carriers are individuals who are able to transmit the disease but do not show any symptoms. Infectious diseases are also known as transmissible disease or communicable disease. Infectious disease can be caused by bacteria, viruses, fungi or parasites. Some infectious diseases can be passed from person to person and some are transmitted by bites from insects or animals. Infectious diseases are also transmitted by contaminated food or water. The spread of such diseases is very much dependent on the carrier population, the density of which increases due to environmental factors such as temperature, humidity, rain, vegetation, etc [13, 15]. The per capita growth rate and the modified carrying capacity of carrier population are taken to be functions of human population density and assumed to increase as the human population density increases [10,11]. Many infectious diseases are prevented by vaccination. The modeling and analysis of infectious disease have been done by many researchers [2, 4, 6, 8, 12, 14]. Infectious diseases model with population dependent death rate and logistic population growth studied by Greenhalth [1] and Gao et al. [7] . Zhou and Hethcote [5] introduced the various kinds of demographics for infectious diseases. Ghosh et al. [9] studied the spread of carrier dependent infectious diseases with environmental effects using variable carrier population.


I. INTRODUCTION
Infectious diseases are spread by carriers which are present in the environment [3]. Carriers are individuals who are able to transmit the disease but do not show any symptoms. Infectious diseases are also known as transmissible disease or communicable disease. Infectious disease can be caused by bacteria, viruses, fungi or parasites. Some infectious diseases can be passed from person to person and some are transmitted by bites from insects or animals. Infectious diseases are also transmitted by contaminated food or water. The spread of such diseases is very much dependent on the carrier population, the density of which increases due to environmental factors such as temperature, humidity, rain, vegetation, etc [13,15]. The per capita growth rate and the modified carrying capacity of carrier population are taken to be functions of human population density and assumed to increase as the human population density increases [10,11]. Many infectious diseases are prevented by vaccination. The modeling and analysis of infectious disease have been done by many researchers [2,4,6,8,12,14]. Infectious diseases model with population dependent death rate and logistic population growth studied by Greenhalth [1] and Gao et al. [7] . Zhou and Hethcote [5] introduced the various kinds of demographics for infectious diseases. Ghosh et al. [9] studied the spread of carrier dependent infectious diseases with environmental effects using variable carrier population.
where ) (t N be the total human population at any time t, which is divided into three subclasses: the susceptible ) (t X , the infectives ) (t Y , and the vaccinated individuals ) (t C represent the density of carrier population, which is governed by a generalized logistic model. It is further assumed that the susceptible are vaccinated at a constant rate and some of them may again become infected while coming in contact with infectives or with carriers due to inefficacy of vaccines.  and  are transmission coefficients due to infectives and carrier population respectively. The parameters  , and d represent vaccination coverage, therapeutic treatment coverage and natural deaths respectively,  is the disease related death constant,  and 1  denote the transmission coefficient of vaccinated individuals due to interaction with infectives and carrier population respectively. However, the rate with which vaccinated persons become infected is very small as compared to the rate with which susceptible get infected i.e.

  
is the modified carrying capacity of the carrier population. It has been pointed out in the introduction, that as the human population increases, the effects of human population related factors enhance the changes of growth of carrier population.

The model
In this study, consider a saturated incidence rate , where a is the psychological effect rate. The model dynamics is governed by following system of nonlinear ordinary differential equations: where a is the psychological effect rate and rest of the parameters are describes in above model (*  (1), (2) and (3) , we see that even if human population related factors are absent, carrier population density increase in its natural environment and it tends to In the model, all the dependent variables and parameters are assumed to be non negative.

II. EQUILIBRIUM ANALYSIS
There exist following three non negative equilibria of the system (4) 1. Disease free equilibrium 0 0, , , 0 () exists, without any condition. The existence of 0 E is obvious.
2. Carrier free equilibrium 0 , , , ( This equilibrium may be obtain by solving the following algebraic equations, Using equation (6) and (7) in equation (5), we get an algebraic equation From equation (6) and (7), we note that Y and V will be positive only when ( ) 0 FN  has a root in the interval , AA dd      . From equation (8) it is easy to observe that,  (6) and (7), respectively.
Thus there exists a unique carrier free equilibrium

Endemic equilibrium
The endemic equilibrium, The endemic equilibrium is given by the solution of the following set of algebraic equations, We may reduce equation (9) in a single variable N i.e. () FNby using equations (10), (11) and (12), where (1 ) (1 ) Proof: the general variational matrix M for the system is given as follows: (1 From equation (5), (6) and (7), we get  as a function of Y alone  is the threshold vaccination rate given by the term 1 c  . We have already shown the uniqueness of Y for ( ) 1 R   in section 3, Now here we show the bifurcation analysis for the disease free equilibrium 0 We note that for ( ) 1 Local Stability of endemic equilibrium will be negative under the following conditions 2 ** * * * * ** 1 0 Hence dt dU 1 is a negative definite under the conditions as stated in the statement of the theorem, showing that 2 E is locally asymptotically stable.
To study the non linear asymptotic stability of endemic equilibrium 2 E , we require the bounds of dependent variables. For this , we state the following lemma giving the region of attraction, with out proof.
it is clear from (19) that in the absence of human related factors, i.e. 0 pq , the inequality is automatically satisfied. This implies that human population related factors, conductive to the growth of carrier population, have a destabilizing effect on the system. Here we also note that due to presence of a vaccinated class, a condition (20) is required for the nonlinear stability which further destabilizes the system.

Proof.
Then by using assumptions of the theorem and the mean value theorem ,we have , The above theorem implies that under appropriate conditions, if the carrier population density increases, then the number of infectives in human population also increases leading to fast spread of carrier dependent infectious diseases.

VI. CONCLUSIONS
In this paper, we have analyzed a vaccination model for carrier dependent population with a saturated incidence rate. We have shown that there are three non-negative equilibrium, namely disease free (DFE), carrier free (CFE) and the endemic equilibrium. The stability analysis has shown that the disease free and endemic equilibriums are locally asymptotically stable under certain condition by using Routh Hurwitz criterion.