Analysis of a Vaccination Model for Carrier Dependent Infectious Diseases with Environmental Effects

We have proposed and analyzed a nonlinear mathematical model for the spread of carrier dependent infectious diseases in a population with variable size structure including the role of vaccination. It is assumed that the susceptibles become infected by direct contact with infectives and/or by the carrier population present in the environment. The density of carrier population is assumed to be governed by a generalized logistic model and is dependent on environmental and human factors which are conducive to the growth of carrier population. The model is analyzed using stability theory of differential equations and numerical simulation. We have found a threshold condition, in terms of vaccine induced reproduction number R(φ) which is, if less than one, the disease dies out in the absence of carriers provided the vaccine efficacy is high enough, and otherwise the infection is maintained in the population. The model also exhibits backward bifurcation at R(φ) = 1. It is also shown that the spread of an infectious disease increases as the carrier population density increases. In addition, the constant immigration of susceptibles makes the disease more endemic.


Introduction
Many infectious diseases are spread by direct contact between susceptibles and infectives. Other diseases are spread in the environment and are transmitted to the human population by insects or other vectors. Here we develop and analyze a model for diseases that are transmitted in both ways. This is the case for typhoid fever and other enteric diseases. There are many carrier dependent infectious diseases which afflict human population around the world. However, the third world countries are most affected by such diseases due to lack of sanitation, wide occurrence of carriers such as flies, ticks, mites, etc. which are generally present in the environment [1]. For example, air-borne carriers or bacteria spread diseases such as tuberculosis and measles; while water-borne carriers or bacteria are responsible for the spread of dysentery, gastroenteritis, diarrhea, etc. [2][3][4]. These carriers transport infectious agents of diseases from infectives to susceptibles and thus spread such diseases in human population. In this paper, we have used the term carriers as a mode of transmission only, which transmit infectious agents of diseases from infectives to susceptibles, without having clinical symptoms.
The modeling and analysis of infectious diseases have been done by many workers, see for example [5][6][7][8][9][10][11]. In particular, Hethcote [10] discussed an epidemic model in which carrier population is assumed to be constant. But, in general, the size of the carrier population varies and depends on the natural conditions of the environment as well as on various human related factors. The effect of variable carrier population has not been considered in these studies, however 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. in the habitat [3,[12][13][14]. In particular, Ghosh et al. [13] studied the spread of carrier dependent infectious diseases with environmental effects using variable carrier population. The density of carrier population further increases as the human population density increases. With increase in human population density, the effects of human population related factors like discharge of household wastes, open sewage drainage, industrial effluents in residential areas, open water storage tanks and ponds etc. leads to further growth of carrier population density. This provides a very conducive environment for the growth of these carriers which enhances the chance of carrying more bacteria from infectives to the susceptibles in the population leading to fast spread of carrier dependent infectious diseases. Thus, unhygienic environmental conditions in the habitat caused by human population become responsible for the fast spread of an infectious disease. It is, therefore, reasonable to assume that the carrier population density is governed by a generalized logistic model. 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 [3,[14][15][16]. In particular, Singh et al. [16] studied the spread of malaria by taking into account mosquito population density governed by a generalized logistic model.
It may be noted that the outbreak of infectious diseases cause mortality of millions of people as well as expenditure of enormous amount of money in health care and disease control. It is, therefore, essential that adequate attention must be paid to stop spreading of such diseases by taking control measures. Vaccination is an important control measure to reduce spreading of such diseases. Various modeling studies have been made to study the role of vaccination on the spread of infectious diseases [17][18][19][20][21][22]. In particular, Shulgin et al. [21] studied a simple SIR epidemic model with pulse vaccination and showed that pulse vaccination leads to epidemic eradication if certain conditions regarding the magnitude of vaccination proportion and on the period of pulses are satisfied. Kribs-Zaleta and Velasco-Hernandez [20] presented a simple two dimensional SIS model with vaccination exhibiting backward bifurcation. Farrington [18] analyzed the impact of vaccination program on the transmission potential of the infection in large populations and derived relation between vaccine efficacy against transmission, vaccine coverage and reproduction numbers. Gumel and Moghadas [19] proposed a model for the dynamics of an infectious disease in the presence of a preventive vaccine considering non-linear incidence rate and found the optimal vaccine coverage threshold needed for disease control and eradication.
In the case of carrier dependent infectious diseases like cholera, measles, etc., the vaccination can be an important tool to help control the spread of such diseases especially when the density of carrier population increases with human population density related factors. It is pointed out here that in above models, vaccination has been studied without considering the effective role of variable carrier population which depends on human population related factors and is responsible for spreading the infectious diseases. In this paper, we extend the model presented by Singh et al. [14] and Ghosh et. al [15] by incorporating the effect of vaccination on the spread of carrier dependent infectious diseases and assuming a generalized logistic model governing the growth of carrier population. In addition, we use more realistic standard mass action type interaction for direct contact between susceptibles and infectives instead of simple mass action. However, we assume that susceptibles are infected by carriers in direct proportion to the density of carrier population (bilinear interaction). The model is analyzed qualitatively to determine the stability of its associated equilibria and the optimal vaccine coverage level needed to control effectively or eradicate the disease. The numerical simulation of the model is also given to see the influence of certain key parameters on the spread of the disease.

Mathematical model
We consider the human population N (t) at time t with immigration of susceptibles at a constant rate A. The total population is divided into three subclasses: the susceptibles X(t), the infectives Y (t) and the vaccinated individuals V (t). In the modeling process, it is assumed that the susceptibles are infected by the direct interaction with infectives and also by the carrier population of density C(t), which is governed by a generalized logistic model. It is further assumed that the susceptibles 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. It is also considered that the infected individuals, after being recovered, may again become susceptible. The block diagram of the model is given in Fig. 1. Thus, by assuming standard mass action interaction for direct contact between susceptibles and infectives and simple mass action interaction between susceptibles and carrier population density, the model dynamics is governed by following system of nonlinear ordinary differential equations: where β and λ are transmission coefficients due to infectives and carrier population respectively. The parameters φ, ν, and d represent the vaccination coverage (of susceptibles), therapeutic treatment coverage (of infected individuals) and natural deaths respectively, α is the disease related death constant, γ and ν 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 susceptibles get infected i.e., γ ≪ β and ν 1 ≪ λ. The constant s 1 is the death rate coefficient of carriers due to natural factors as well as by control measures. Here, s(N ) denotes the growth rate per capita of the carrier population density such that s(N ) − s 1 is its intrinsic growth rate. It may be noted that if the growth rate and death rate due to natural as well as control measures of carrier population are balanced, then it may tend to zero. Similarly, L(N ) is the modified carrying capacity of the carrier population and its value is ] as compared to usual logistic model.
It has been pointed out in the introduction, that as the human population increases, the effects of human population related factors/activities enhance the chances of growth of carrier population. Thus, in the model, s(N ) and L(N ) are taken to be functions of total human population instead of infective population. Since we assume that the growth rate per capita increases as the human population density increases, we have where s 0 is the value of s(N ) at N = 0 and () ′ denotes the derivative of the function with respect to its argument. We also assume that the modified carrying capacity increases with human population density, so that where L 0 is the value of L(N ) when N = 0. From equations (1), (2) and (3), we see that even if human population related factors are absent, carrier population density increases in its natural environment and it tends to In the model, all the dependent variables and parameters are assumed to be non-negative.

Equilibrium analysis
It is sufficient to consider the reduced system of model (1) The equilibrium analysis of the model system (4) has been carried out and the results are given as follows. There exist following three non-negative equilibria of the system (4).
This equilibrium may be obtained by solving the following algebraic equations, Using equations (6) and (7) in equation (5), we get an algebraic equation in single variable N , i.e., F (N ) = 0, where F (N ) is given by the following equation, Keeping in mind equations (6) and (7), we note that Y and V will be positive only when F (N ) = 0 has a root in the interval ( A α+d , A d ). From equation (8) it is easy to observe Hence, there exists a unique positive root N given by F (N ) = 0. Knowing the value of N , we can compute the values of Y and V from equations (6) and (7), respectively.
Thus there exists a unique carrier-free equilibrium The endemic equilibrium E 2 is given by the solution of the following set of algebraic equations, We may reduce equation (9) in a single variable N i.e., F (N ) = 0 by using equations (10), (11) and (12), where It is clear from equation (13) Knowing the value of N * , the values of Y * , V * and C * can be computed from equations (10), (11) and (12), respectively. Thus, the equilibrium E 2 exists provided s(N * ) − s 1 > 0 and F ′ (N ) > 0. From equation (13) it is easy to note that F (N ) > 0. Keeping in mind the above analysis of equation (13) we have N > N * . From equation (12), it may be noted that dC * /dN * > 0 in view of equations (2) and (3). Thus, the equilibrium value of carrier population density increases with increase in the equilibrium value of human population.

Stability analysis
Now, we analyze the stability of equilibria E 0 , E 1 and E 2 . The local stability results of these equilibria are stated in the following theorem: The equilibrium E 0 is unstable whenever E 1 or E 2 exists, E 1 is unstable whenever E 2 exists and the equilibrium E 2 is locally asymptotically stable provided the following conditions are satisfied, Proof. The general variational matrix M for the system (4) is given as follows: (4) is given by, Since all the model parameters are assumed to be nonnegative, it follows that ψ 2 , ψ 3 < 0. Thus, the stability of E 0 will depend on the sign of ψ 1 and ψ 4 .
From equations (5), (6) and (7), we get φ as a function of Y alone From equation (16), we note that The expression for φ(0) is the threshold vaccination rate given by the term φ C1 in equation (33). We have already shown the uniqueness of Y for R(φ) > 1 in Section 3. Now here we show the bifurcation analysis for the disease free equilibrium E 0 when R(φ) = 1. We note that for R(φ) = 1, one eigenvalue of M 0 is 0, whereas other eigenvalues are negative provided s(A/d) − s 1 < 0.
From equation (16), we have φ ′ (0) > 0 provided When d is very small, ν and α are large, R 1 is far from both 0 and 1 and R 0 is large. Then there is a backward bifurcation at R(φ) = 1 for the model (4) as above inequality (17) is satisfied.
For one set of parameters we have made a graph of Y * versus R(φ) showing backward bifurcation. In Fig. 2 we have shown the backward bifurcation of the model (4) at R(φ) = 1 for the parameter values A = 1.0, α = 1.0, β = 5.0, d = 0.1, ν = 0.35, γ = 0.6. Fig. 2 also shows that there may exist two endemic equilibria of system (4) for R(φ) < 1. Out of these two equilibria one will be stable (solid line), whereas second one will be unstable (dashed line).
To establish the local stability of endemic equilibrium E 2 , we consider the following positive definite function, where k i (i = 0, 1, 2, 3) are positive constants to be chosen appropriately and y, v, n and c are small perturbations about E 2 , as follows Y = Y * + y, V = V * + v, N = N * + n and C = C * + c.
Differentiating (18), with respect to t, using the linearized system corresponding to E 2 , we get, Now dU1 dt will be negative definite under the following conditions, +λC * , k 2 = 1, we can choose k 1 and k 3 such that The stability conditions are then obtained as given in the theorem. Hence, dU1 dt is a negative definite under the conditions (14) and (15) as stated in the statement of the theorem, showing that E 2 is locally asymptotically stable.
To study the nonlinear asymptotic stability of endemic equilibrium E 2 , we require the bounds of dependent variables. For this, we state the following lemma giving the region of attraction, without proof.  (2) and (3), let s(N ) and L(N ) satisfy 0 ≤ s ′ (N ) ≤ p and 0 ≤ L ′ (N ) ≤ q for some positive constants p and q in Ω, then E 2 is nonlinearly asymptotically stable in Ω provided the following inequalities are satisfied:

Theorem 2. In addition to assumptions
It is clear from (22) that in the absence of human related factors, i.e., p = q = 0, the inequality is automatically satisfied. This implies that human population related factors, conducive 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 (23) is required for the nonlinear stability which further destabilizes the system.
Proof. Consider the following positive definite function, where the coefficients m 0 , m 1 , m 2 and m 3 can be chosen suitably. Differentiating (24) with respect to t and using (4), we get, where f (N ) and g(N ) are defined as follows, Then by using the assumptions of the theorem and the mean value theorem, we have, After choosing m 0 = 1, m 2 = 1 α , we choose m 1 and m 3 such that: The stability conditions can then be easily obtained, as given in the statement of the theorem. Thus, dU1 dt is negative definite under the conditions (21)- (23). Hence proof. 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.

Vaccine induced reproduction number
We define R(φ), the vaccine induced reproduction number as, (see Section 4) . where Thus, R(φ) is a decreasing function in φ ≥ 0. This indicates the impact of vaccination in reducing the vaccine induced reproduction number. Moreover, in the absence of vaccination i.e., From the definition of R(φ) and R 0 , it is clear that the introduction of vaccination implies R(φ) ≤ R 0 and, consequently, if R 0 < 1 then R(φ) < 1 when φ > 0. Thus E 0 is locally asymptotically stable as long as R(φ) is less than one. As indicated earlier that β ≫ γ, therefore, we can write γ = σβ, where 0 ≤ σ < 1. From equation (30), we can deduce that . This implies that R(φ) > 1 and, therefore, no amount of vaccination can bring R(φ) below one. Hence σ c defines the critical value for the vaccine-related reduction rate of infection.
From equations (10)-(12), we also find that, The critical vaccination level that ensures disease eradication, when carrier population remains at its equilibrium is obtained as, Therefore, if critical vaccination level φ is such that φ > max(φ C1 , φ C2 ), then disease eradication is possible in the population.

Numerical simulation
It is noted here that our aim is to study, through a non-linear model and its qualitative analysis, the role of vaccination on the spread of carrier dependent infectious diseases. It is, therefore, desirable that we must show the existence of equilibrium values of variables of the model as well as the feasibility of stability conditions numerically for a set of parameters.
To study the dynamical behaviour of the model, numerical simulation of the system (4) is done by MAPLE 7.0 using the parameters [9,25] Since all the eigen values are found to be negative, therefore, endemic equilibrium is locally asymptotically stable for the above set of parameters. The computer simulations are performed for different initial starts in the following four cases and displayed graphically in Figs. 4, 5. In these figures the variation of infectives and vaccinated population with total human population respectively is shown. The trajectories starting with different initial starts reach equilibrium point E 2 . Hence we infer that the system (4) may be nonlinearly asymptotically stable about this equilibrium point E 2 for the above set of parameters.   The results of numerical simulation are displayed graphically in Figs. 6-14. In Figs. 6, 7, the variation of infective and vaccinated population is shown respectively for different values of vaccination rate φ. It is found that as vaccination rate (φ) increases, the infective population decreases, whereas the vaccinated population increases tremendously. This signifies that only by increasing the vaccination rate, spread of carrier dependent infectious disease cannot be significantly controlled.  Figs. 8, 9 depict the role of decay coefficient (s 1 ) of carrier population on carriers and infectives. When there is a rise in the decay coefficient either due to natural factors or control measures, carrier population density decreases significantly, and consequently the infective population declines. This decline in infective population is not much significant. It seems, it is due to the fact that disease spreads not only through carriers but also through direct interaction of susceptibles with infectives. It is, therefore, speculated that not only the growth of carrier population be curbed using effective control mechanism but the direct interaction of susceptibles with infectives be also restricted. This is also clear from    As is pointed out earlier that due to inefficacy of vaccines, some of the vaccinated population may again become infected due to interaction with infectives as well as with carriers. In Figs. 11, 12, the effect of γ and ν 1 , the contact rate of vaccinated population with infectives and with carrier population density respectively, is shown on vaccinated and infective populations. It is found that with increase in transmission rate γ and ν 1 , the vaccinated population decreases, which in turn, increases the infective population. Thus, we conclude that vaccine efficacy should be high enough so that vaccinated individuals do not get infected either by direct interaction with infectives or with carrier population density (see Figs. 11, 12 for γ = 0, ν 1 = 0). In Fig. 13, the effect of disease induced death rate is shown and it is found that with increase in α, the infective population also decreases. Fig. 14 shows the effect of immigration rate of susceptibles and it is seen that the infective population increases with increase in the rate of immigration. Thus, the constant migration of susceptibles make the disease more endemic.
Finally from the above discussion, we infer that in order to keep the spread of carrier dependent infectious diseases under control, a proper vaccination campaign (as discussed in Section 5) dependent on the critical vaccination rate, be introduced. Moreover, the vaccine efficacy be high enough to ensure that vaccinated persons while coming in contact with infectives or carriers, are not infected again. A suitable control mechanism like elimination of carrier breeding sites, larvaciding, adulticiding, etc. may be devised so that the carrier population density is diminished which otherwise increases due to human population related factors and other unhygienic environmental conditions leading to fast spread of infectious diseases.

Conclusions
In this paper, a nonlinear vaccination model is proposed and analyzed to study the spread of carrier dependent infectious diseases with vaccination in a population with variable size structure. It is assumed that the disease spreads by direct contact of susceptibles with infectives and by carrier population density present in the environment. The density of carrier population, which increases by environmental and human population related factors, is assumed to be governed by a generalized logistic model. The growth rate per capita and the modified carrying capacity of the carrier population are also assumed to increase as the human population density increases. Some inferences have been drawn regarding the spread of the disease by establishing local and global stability results and numerical simulation. The model exhibits three equilibria, namely disease free, carrier free and endemic equilibrium. The first two equilibria are found to be unstable, whereas third equilibrium is locally stable. It is also found to be nonlinearly asymptotically stable under certain conditions. We have found a threshold condition in terms of vaccine induced reproduction number R(φ) which is, if less than one, the disease dies out, in the absence of carrier population otherwise the infection is maintained in the population. And at R(φ) = 1, the model exhibits backward bifurcation. The results show that in the absence of carrier population into the community, if vaccination rate is above the critical vaccination level (to ensure R(φ) < 1) then the disease can be eradicated. The presence of human population related factors, causing the growth of carrier population, have a destabilizing effect on the system. Moreover, if the vaccination rate is above a critical level φ C2 , then carrier dependent infectious disease does not take a form of epidemic provided the density of carrier population remains at its equilibrium level. Thus, three main control strategies against the spread of carrier dependent infectious diseases in the human population are carrier (such as flies, ticks, mites etc.) reduction policy, vaccination against the disease and the vaccine efficacy. The carrier reduction strategies may include elimination of carrier breeding sites, larvaciding, adulticiding, keeping surroundings clean and hygienic. On the other hand, vaccinating the susceptibles such that vaccination level is maintained above the critical vaccination level φ C i.e., max(φ C1 , φ C2 ) and the vaccine efficacy be high enough so that vaccinated individuals are not infected again either by infectives or by carriers.