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Vaccination strategies are designed and applied to control or eradicate an infection from the population. This paper studies three different vaccination strategies used worldwide for many infectious diseases including childhood diseases. These strategies are the conventional constant vaccination strategy, the periodic step (pulse) vaccination strategy and finally the mixed vaccination strategy of both the constant and the periodic one. Simulation of the different vaccination programs is conducted using three parameter sets of measles, chickenpox and rubella. The Poincaré section is playing as a filter of our simulation results to show a wide range of possible behavior of our model. Critical vaccination level is been estimated from the results to prevent severe epidemics.

Vaccination programs are used as a tool to control the spread of epidemics. The simplest vaccination strategy is to vaccinate all susceptible individuals at a constant rate. This may also be combined with vaccination of a fixed fraction of very young children at the smallest possible age where maternal antibodies no longer confound the effect of the vaccine, commonly 9 - 18 months for measles. In the absence of vaccination, many common childhood diseases show a regular periodic oscillation with period of a whole number of years [1,2]. We ignore the effect of maternal antibodies in this paper, so children are vaccinated from birth. Much work has been done analysing seasonal periodic outbreaks of infectious diseases considering seasonal variation in the contact rate [1-3].

Recently it is well known that in some circumstances a periodic vaccination strategy, for example pulse vaccination, can be a more efficient use of limited immunisation resources than continuous constant vaccination effort [4-6]. In this paper we study a general continuous periodic vaccination strategy and extend the results of [

If the combined vaccination strategy is applied in the situation where no disease is present, then the number of susceptibles eventually reaches a unique periodic solution. Our results lead us to conjecture that this combined periodic and fixed vaccination strategy is sufficient to eliminate disease from the population exactly when the weighted time-averaged disease-free susceptible population is less than a certain threshold value.

The conventional strategy is used to vaccinate a fixed proportion of newborns of whom a proportion p, are successfully vaccinated. This constant rate strategy reduces the effective total birth rate of susceptible individuals from to where is the per capita birth rate and is the total population size. So the effect of applying the conventional strategy is to significantly reduce the size of the susceptible population [

Study of a simple SIR model with constant vaccination shows that there is disease free equilibrium point

Examination of the stability of this disease free equilibrium shows that there is a critical level of the proportion of susceptibles who are successfully vaccinated given by

where the basic reproductive number (the number of secondary cases produced by a single infected person entering a population at the disease free equilibrium) is given by

This critical value of p determines the stability of the disease free equilibrium of the the SIR model. The disease free equilibrium is stable if and unstable when To eradicate the infection from the population by applying this constant rate vaccination strategy, it is necessary to keep the susceptible population beneath a certain critical value by vaccinating a large enough proportion p of the susceptibles such that. Therefore the herd immunity must exceed a critical level in order for the reproductive number to be reduced to be less than unity in value [

Pulse vaccination vaccinates susceptibles at discrete points in time, usually at regular intervals. Many recent works studied epidemic models with pulse vaccination strategy. The pulse vaccination was the main aim of these epidemiological investigations [11-16]. One such example is the use of annual immunisation days which were successful in eradicating measles from Gambia between 1967 and 1972 [

Pulse vaccination has also been used in the United Kingdom. In November 1994 a single dose of combined measles and rubella (MR) vaccine was given to children aged 5 to 16 years. In England and Wales an average of 92% of these children were vaccinated. This policy caused a significant fall in the number of cases of measles reported to the Office of Population Censuses and Surveys. It was concluded that application of pulse immunisation to all schoolchildren would probably prevent a large amount of morbidity and mortality and would have a marked effect on measles transmission for several years [

Nokes and Swinton [

Shulgin et al. [

They also found that when the disease transmission rate is a constant, the disease free periodic solution (DFS) is locally stable if the mean value of the susceptible population, over a single pulse period, is below a certain critical value

They used Floquet theory to examine the stability of the DFS when is a non-constant periodic function with the same period as. Floquet theory is a good framework to study the stability of a linear periodic system

where is a periodic matrix such that . This framework starts by calculating the monodromy matrix of this system which is the fundamental matrix at

Floquet multipliers which are the eigenvalues of the monodromy matrix are the critical parameters which determine the stability of the periodic linear system. If all the Floquet multipliers are less than unity in absolute value the trivial solution of the linear system is stable but if one of them is bigger than unity in absolute value the solution will be unstable [

Shulgin et al. [

Here denotes the number of susceptibles at time for the DFS. This is the condition for local stability of the disease free solution (DFS) and under this condition there is no chance of any severe epidemic to occur.

Aron [

This strategy instantaneously introduces vaccination of a constant proportion of newborns individuals at time. She found numerically that periodic endemic solutions coexist for the post-vaccination SEIR model. She also found that the period of the endemic solution depends on the pre-vaccination reproduction number. The vaccination strategy reduces the reproduction number by a factor [

Finally in the mixed vaccination policy both the conventional and the vaccination pulse strategies are applied simultaneously. Because the conventional strategy is currently used in many countries world wide, it is useful to combine both the pulse and the conventional strategy to provide a good comparison with the conventional strategy [

Shulgin et al. [

The SEIR model of the spread of infectious diseases makes the following assumptions:

1) The total population size is and the per capita birth rate is a constant. As births balance deaths we must have that the per capita death rate is also.

2) The population is uniform and mixes homogeneously.

3) The population is divided into susceptible, exposed, infective and recovered individuals. The total number of individuals in each of these classes are respectively and.

4) The infection rate is defined as the total rate at which potentially infectious contacts occur between two individuals. A potentially infectious contact is one which will transmit the disease if one individual is susceptible and the other is infectious, so the total rate at which susceptibles become exposed is Biological considerations mean that is continuous. We also assume that either (i) is not identically zero, positive, non-constant and periodic of period or (ii) is a constant.

5) The susceptibles move from the exposed class to the infective class at a constant rate where is the average latent period conditional on survival to the end of it.

6) The infectives move from the infective class to the recovered class at a constant rate where is the average infectious period conditional on survival to the end of it.

7) A fraction of all new-born children are vaccinated. In addition all susceptibles in the population are vaccinated at a time dependent periodic rate. This is the periodic vaccination strategy. We shall suppose that is periodic with period for some integer numbers including.

Our SEIR model with time dependent vaccination strategy can be written as a set of four coupled non-linear ordinary differential equations as follows:

and

with

Here the disease transmission rate and the vaccination rate are non-zero, positive, continuous periodic functions. The system (1)-(5) has no equilibrium points but a disease free solution (DFS), with is still possible.

Consider the region in defined by

The system of differential equations (1)-(4) with initial conditions in obviously starts off in the region. The right-hand sides of these equations are differentiable with respect to and with continuous derivatives. It is straightforward to show using standard techniques [

In the case that is a non-constant bounded continuous periodic function, there is no equilibrium point for the system (1)-(5). So there is no disease free equilibrium point. But still there is a periodic DFS corresponding to the case that In this case (1) becomes

If, (6) has a solution for . We shall examine the behaviour of this solution. Integrating (6) we find that

Hence

Equation (8) gives a recursive relationship between the number of susceptibles at time If we let then (8) defines a mapping such that

If and are different values of then we have that

So is a contraction mapping [

Hence So is a periodic function of Differentiating (9) is continuously differentiable with respect to and and is a disease free periodic solution of the system (1)-(5) which repeats itself every years. We have the following result:

Theorem 1. Equations (1)-(5) have a disease free periodic solution of period which is continuously differentiable and this is the only disease free periodic solution to (1)-(5), and any disease free solution to (1)-(5) approaches this one as time becomes large.

Proof see [

Recall that, the basic reproduction number of the disease is defined as the expected number of secondary cases caused by a single infected case entering the disease-free population at equilibrium [

Define

and

It is proved that with both of the inequalities being strict if is non-constant on [

In this paper the simulations of the SEIR model with three different vaccination strategies have been conducted using the XPPAUT package and data estimated from the literature. Parameter values corresponding to the childhood diseases of measles, chickenpox and rubella have been used.

A constant population size of has been considered. We also supposed that corresponding to an average human lifetime of 50 years [2,21]. We chose this value to be consistent with previous studies even though the actual value of the average lifetime in many countries is higher. For example the average lifetime in the UK is around years. We do not feel that this will have much effect on the results of our simulations as we are mainly considering childhood diseases and the proportion of individuals who catch the disease at 50 years or later is negligible. Mainly the following specific values of and have been taken as in [3,10,25-27] for our models:

1) Measles: and;

2) Chickenpox: and;

3) Rubella: and.

We have taken also as estimated from the literature, for our simulations results for all of the bifurcation diagrams presented and for the three diseases under investigation as follows: for measles, for chickenpox and for rubella respectively.

The key parameter in the analytical results was the basic reproduction number [22,24]. So the computer simulations of our models were performed using values of to insure that the disease is in the endemic state. The values of were determined by the value of the mean level of the disease transmission function, and which determines the amplitude parameter of the periodic transmission rate

This paper targeted the long term behaviour of the system in response to changes in the vaccination parameter, (the value the vaccination rate of the conventional strategy, the amplitude of the vaccination function of both the pulse and the mixed one), which is our bifurcation parameter. The basic idea of this study is simply that, given a set of parameter values compound with appropriate initial values then the endemic equilibrium solution is obtained by running the system for a long time to eliminate transient solutions. Filtering the equilibrium solutions by looking at Poincaré sections of them taken every year (recall that the underlying seasonal variation in the contact rate has period one year). So in this paper the vaccination parameter is used as a filter of the long term equilibrium solution. By plotting the sections of the long term endemic equilibrium solutions against the vaccination parameter we obtain a number of points in a vertical line corresponding to each value chosen for the vaccination parameter. These points on the filter, represent the period of the stable long term periodic solution of our model. For example a single point indicates a solution of period one year, two points a solution of period two years, points a solution of period years and an infinite number of points a chaotic solution. In the following simulation results which represent global bifurcation diagrams for SEIR model with vaccination using our filter are given. We say global because the filter described above is used to plot the bifurcation diagrams for a large range of values of the vaccination parameter. The comparison of the simulation results of our model show that the type of vaccination parameter affecting the pattern of the dynamics of the disease. The pattern of the mixed vaccination is the simplest pattern and the most controllable one.

This paper looked at bifurcation diagrams for three different vaccination functions one of which is the constant strategy. These three vaccination programs are applied for the SEIR model with the seasonally periodic transmission function the more realistic reparameterised step function as, , with mean value, of period one year [22,24], where

and is the largest integer number less than

Our three different vaccination strategies are of the following forms:

1) The constant vaccination function to vaccinate the newborns as many as possible all the time.

2) The periodic binary step vaccination function with period one year to vaccinate the susceptible population, where

3) The mixed vaccination strategy which is composed of the periodic function, combined with the constant vaccination one In all of our simulations we have taken.

The simulation study is been designed to start off just before school opening days. It means that the disease transmission rate is at its highest value. Therefore at this critical moment we start our vaccination to control the disease dynamics or possibly prevent severe epidemics to occur. We simulate our model with the three different vaccination strategies under consideration by varying the vaccination parameter from to in value. Then plotting the long term solution against the vaccination parameter to have wide range of possible behaviour of the disease under consideration. From the obtained patterns we can decide easily which vaccination strategy is more effective than the others.

We start off our simulation with the most studied disease, measles, with the three different vaccination strategies under consideration.

Therefore we can claim that the mixed vaccination strategy is the most effective policy to control measles disease. Moreover using this mixed vaccination strategy reduces the number of the susceptibles in the system by a fraction which is the rate at which the newborns are vaccinated.

nated newborns should exceed 95% to prevent severe epidemics to occur. This proportion of newborns is very difficult to be vaccinated for different reasons [

_{1 }tends to one in value. Similar to measles disease, the pattern of chickenpox with the mixed vaccination strategy shows neither long period solutions nor any chaotic behaviour.

It is important to simulate our model with exposed or latent class and with different vaccination strategy, to evaluate which strategy is more efficient. We have simulated the control of the dynamics of three childhood infectious disease by using three different types of vaccination strategies. We perform these simulations for an SEIR model with a seasonally varying disease transmission rate. Using a periodic vaccination strategy in such an SEIR model seems to lead to periodicity in the disease dynamics [

possibly eradicate diseases by applying the most efficient vaccination strategy. Efficiency means sometimes fewer number of vaccinated individuals leads to perfect control of the disease. This work can be summarised as follows: In Section 1 a short introduction to common practically used vaccination strategies was given. Section 2 outlined the SEIR model which we studied and gave the assumptions formulating the model. Section 3 addressed the results of the previous work by [22,28] which showed that there is a unique DFS for our SEIR epidemic model and this solution is periodic with period equal to that of the vaccination function. Also a conjectured expression for, the basic reproduction number of the disease, is given when the vaccination campaign is used. Lower and upper bounds, and respectively, for this expression were also defined. In this section the stability of the DFS of our model is stated. We found that the DFS was GAS when and in this case the infection will ultimately fade out of the population [22, 28].

Section 4 comprises the computer simulations model for different infectious diseases including measles. The simulation results presented in this paper were conducted when the basic reproduction number However we tried to simulate our model with a seasonal variations in the incidence of childhood infectious diseases due to the opening and closing of schools using our suggested reparameterised periodic step function . Also we use a periodic step vaccination strategy that starts to give the susceptibles a pulse of vaccine at the opening of schools, using the vaccination function. These new results for this periodic contact rate in combination with the vaccination strategy, are totally original for any set of parameter values of childhood infectious diseases.

For a highly infectious disease such as measles, using vaccination at birth only, approximately 91% - 94% of newborn individuals must be vaccinated to guarantee elimination of the disease [

We conjectured that to control or eradicate the disease it was both necessary and sufficient to keep the mean value of the product of the disease transmission rate and the susceptible population at the DFS beneath a critical threshold value. If this is true then it is possible for a few individuals to be vaccinated, provided only that the weighted mean value of the number of susceptibles at the DFS over the period of the vaccination function does not exceed the threshold value, the disease will still be eradicated. This contrasts with the strategy of constant vaccination where a critical fixed level of immunisation effort must always be applied to guarantee eradication, and this is an advantage of a periodic vaccination strategy over a constant one.

However the simulation results have indicated that using different functional forms of vaccination strategies generates different patterns of solutions for each disease parameter. The bifurcation diagrams show that the simplest patterns are those of the mixed vaccination strategy. Apart from some of the results these diagrams show a one year solution for the whole diagram except the first quarter of range of the vaccination parameter. The most complicated diagrams are those of the constant vaccination parameter which show a wide range of periodic and aperiodic solutions all over there patterns for all of the three diseases.

It is interesting to note the difference between the bifurcation diagrams in the case of using a periodic pulse vaccination function and the constant conventional vaccination strategy for all of the three diseases, (measles, chickenpox and rubella) studied. The bifurcation diagram for the periodic step vaccination shows that the disease reaches the DFS at a vaccination level of less than 60% of the total number of the susceptible population. On the other hand the constant vaccination strategy failed to control the disease before the vaccination rate exceeds 95%.

Finally it is important to note the difference between the bifurcation diagrams, when using the periodic pulse vaccination function and using the mixed vaccination strategy for all of the three diseases under investigation. The patterns show that, the level of vaccinated population at which the disease starts to be controlled, in the case of using the mixed vaccination strategy is much fewer than that of the only periodic vaccination strategy. The diagrams show that, the effective vaccination parameter p_{1} in the case of the mixed vaccination is about a third of the that of the only periodic pulse vaccination strategy for all of the three diseases. Using a continuous periodic vaccination strategy in conjunction with vaccination of a fixed proportion of newborn individuals, reduces the proportion of newborns who need to be immunised to a more realistic level. Moreover from (9) one can see easily that using such a mixed vaccination strategy uniformly reduces the level of fluctuation of susceptible in the DFS compared with a purely periodic vaccination function. This agrees with our simulation results. As the diagrams show that at the end of range of the vaccination parameter, the number of susceptible population in the system when the mixed vaccination strategy is used, is approximately half its corresponding number of susceptible population in the system when the purely periodic vaccination strategy is used. Hence it is more optimal to use a combined vaccination approach in order to prevent major outbreaks of infectious disease occurring.