Mathematical Model of Infectious Disease with Multistage Vaccine

Assefa Erba Bikila, College of Natural and Computational Science, Department of Mathematics, Wolaita Sodo University, Sodo ,Ethiopia P.O.BOX 138 Email: asebikila@gmail.com Abstract Many diseases, such as the seasonal influenza, tetanus, and smallpox, can be vaccinated against with a single dose of a vaccine. However, some diseases require multiple doses of a vaccine for immunity. Diseases requiring a multistage vaccine such as Hepatitis B can have extra complications with its vaccination program, as some who start the doses may forget to complete the program or could become infected before completing the program. This thesis concerns the setup and analysis of a model for developing a mathematical model to describe the dynamics of an infectious disease with a multistage vaccine. In this thesis, we considered Susceptible-Infected-Removed (SIR) epidemic models and discussed the mathematical analysis and simulation study is conducted. We discuss an epidemic model which represents the direct transmission of infectious disease. The model assumes that individuals are equally likely to be infected by the infectious individuals in a case of contact except those who are immune. We formulated SIR epidemiological model to determine the transmission disease by using compartmental model approach to using a system of nonlinear differential equations. We study about basic reproduction number and equilibrium point for compartmental mathematical models of infectious disease transmission. The basic reproduction number R0, which is a threshold quantity for the stability of equilibrium point is calculated. If R0 < 1 then the disease-free equilibrium point is globally asymptotically stable and it is the only equilibrium point. On the contrary, if R0 > 1 then an endemic equilibrium point appears which is locally asymptotically stable.

that there exists an infection related death rate. They also show the existence of nonnegative solutions of the model, and also give a detailed stability analysis of disease free and positive fixed points.

Design and Methodology
A research needs a firm design or structure before commencement of data collection and general research process for the functions of enabling the researcher in order to obtained evidences answer the initial research in unambiguous way. Therefore, this research work was undertaken with design of the study instruments.
We develop a mathematical model which describes the dynamics of infectious disease and its transmission using a system of non-linear ordinary differential equations. The analysis of the model was done using different tools.
The method used in this study divides the population into three compartments (S, I, R) consisting of susceptible, Infected and Recovered class. Determination of equilibrium points and their stability is then performed as well as basic reproduction number is found. The Jacobean matrix is used to find sufficient conditions for the local stability of the equilibrium points of the system in the model equations. The Lyapunov function is used to show global stability of a positive equilibrium point of the system of the model equations. To supplement the analytical solutions of the model equations we used MATLAB and mathematica for the numerical simulations.

MODEL FORMULATION AND ANALYSIS
To formulate our model, we first assume that the disease we are studying will break up our population into three groups. These are susceptible (S) those individuals who are healthy and can be infected, infective (I) those individuals who are infected and are able to transmit the diseases, and recovered (R) those individuals who are immune because have been infected and now have recovered.
The possible flow of individuals from one group to another is S→I→R. The SIR model is used for modeling general epidemics and to know how the spread of a disease is in a particular population and some possible ways of controlling such a disease. This description of the SIR model was made more mathematical by a formulating differential equation for the rate of proportion of individuals in each class.
To use the classic Kermack and McKendrick formulation of the SIR model, we further assume that the population we are studying is relatively small (say, a small, relatively isolated town), where every contact between two randomly selected individuals is equally likely.
The infectious diseases spread from an infected individual to other susceptible individuals in the surroundings.
There is no treatment failure, a patient will either recover or die. The total population at time t represented by N(t) is considered as a constant and is the sum of the populations in the compartments S(t), I(t) and R(t), that is N = S + I + R. The population is closed in the sense that the immigrations, new births and deaths of people are not considered.
Here we consider SIR mathematical model assuming that the numbers of both births and deaths are equal since the time duration under consideration is quite short and as a result population size remains to be a constant Every person in our population is susceptible to the infectious disease. The population is homogeneously mixed.
We will also assume that the per capita infection rate as well as the rate of recovery remains constant. We then introduce a multistage vaccine that can result in susceptible becoming immune. Hence, those who complete the vaccination program before becoming sick can go right into the recovered group. For this model, we will assume that those who receive immunity via vaccination will be placed in a separate group, V.
To administer the vaccine, we initially assume that a proportion of individuals will embark on the vaccination program, and others will choose not to. After administering these first doses (this will be built into the initial conditions for the model), we specify that any subsequent dose must occur at a time , i = 1, 2… n-1 after the first dose is administered. However, practically it is not feasible to administer all of the vaccines during a single instant of time, and so we will also allow a grace period to receive each vaccine. Thus, represents the deadline upon which an individual must receive the th dose, and there is an interval of length that ends at such that someone may receive that dose any time in that interval. Specifically, if we refer to the example of the 2009 H1N1 influenza vaccine (inactivated virus) for those children aged 6 months to 9 years, the second dose should be given 28 days after the first dose, but can potentially be given as early as 21 days after the first dose (Centers for Disease Control and Prevention 2009). So, this would allow an individual to receive the second dose any time between days 21 and 28, with 1 = 28 days and 1 = 7 days. We also assume that the grace periods do not overlap; this is to prevent someone from potentially receiving two doses at the same time, and for a clear distinction as to who should be in which group at a given time.
Next, we can incorporate births and deaths into the population. We specifically say that the birth rate and the per capita death rate are constant, and that all of the births are assumed susceptible. Thus, the births would be placed into the S group, with the per capita death rate being the same for all groups.
Finally, for this first formulation of the model, we will only administer the full vaccine, or all n doses, once.
Thus, we will assume that those who do not complete the vaccination program, but do not get sick, will simply remain in which ever susceptible group represents how many total doses they received unless they become sick.
Based on these assumptions, we define the following variables and parameters Integrating both sides of the inequalities, gives Integrating both sides of the inequalities, Integrating both sides of the inequalities, gives Hence, all variables are positive for all time t > 0.

Boundedness of solution
In this section, we show a description of some basic properties of the model equations, such as feasible solution.
The feasible solution shows the region in which the solutions of the equations of the system are biologically meaningful. To proof this consider the following steps: From the system of equations (1 -3), the total population is given by N = S +I+R. Therefore, adding the differential equations (1)  This implies that N0 ≤ N(t) ≤ . So, N ∈ [ 0 , ]. Therefore, Ω is boundedness.

Equilibrium Points
The equilibrium points of the system can be obtained by equating the rate of changes to zero. provided that I* > 0 and R* > 0

The Basic Reproduction Number Ro
In mathematical epidemiology an important concept is related to the basic reproduction number R0 as it serves as a threshold parameter that governs the spread of infectious diseases in a population. The basic reproduction number, R0 is a measure of the potential for disease spread in population. Mathematically, R0 is a threshold for stability of a disease free equilibrium and is related to the peak and final size of an epidemic. The reproductive number can provide significant insight into the transmission dynamics of a disease and can guide strategies to control its spread.
The basic reproduction number R0 has been defined as the average number of secondary infections that occur when one infective is introduced into a completely susceptible host population. These determine whether an epidemic will persist or die out. The disease persists when > 0. So, to find the basic reproduction number Thus, the basic reproduction number is R0 = ( + ) .

Local Stability Analysis
We discuss the local stability of an endemic equilibrium and a disease free equilibrium points of the system of equations (1 -3). The disease free equilibrium point E0 is given by E0 = ( ,0, 0).
Theorem: -If R0 < 1, then the disease free equilibrium E0 is locally asymptotically stable.
Proof: To determine the stability of the disease free equilibrium point, we turn to the Jacobean matrix of the system evaluated at the disease free equilibrium point. The Jacobean matrix at the disease free equilibrium point is given by The characteristic equation of the Jacobean matrix at the disease free equilibrium point is given by The eigenvalue of the Jacobean matrix are 1 = − , 2 = − , and 3 = − − . Since having all eigenvalues negative guarantees local asymptotic stability, then we need < + to guarantee local asymptotic stability of the disease free state.
All the eigenvalues being negative means that the disease free equilibrium is asymptotically stable.

Global Stability of Endemic Equilibrium
We now study the properties of the endemic equilibrium points and derive the stability condition for this equilibrium point. The globally asymptotic stability of the endemic steady state is proved by constructing a Lyapunov function. Where W1 and W2 are positive constants to be chosen letters. Take the derivative of the above function.  increases. This decrease may be possibly because of the high rate of recovery due to mass vaccination, since individual become permanently immune upon recovery. The contact rate also has large impact on the spread of a disease through a population.
The higher the rates of contact, the more rapid the spread of the disease, it is also observed that as the contact rate decreases, the fraction of individuals infected decreases, the fraction of individuals infected decreases at a faster rate as would be expected logically.
We observed that that the populations of infected individuals at the very beginning rise sharply as the rate increases and the fall uniformly as time increases. This rapid decline of the infected individuals may be due to detection of the infectious disease. This graph also demonstrates that the contact rate has large impact on the spread of the disease through population. If the contact rate is observed to be high then the rate of infection of the disease will also be high as would be expected logically.
It is realized that the number of individuals Recovered rise steadily as rate increases. This may be due to early detection of the disease as well as education about the disease transmission. It can also be observed that the population of the recovered individuals rise up steadily for some number of days and then drops and remains nearly a constant. This could be due to the greater number of infectious individuals who have been treated and also acquired education about the dynamics of infectious disease transmission.

Mass Vaccination
If a vaccination programme causes the proportion of immune individuals in a population to exceed the critical threshold for a significant length of time, the transmission of the infectious disease in that population will gradually decrease. From figure 4.4 the immediate effect of the vaccine is that the peak number of infections is 40 significantly reduced. Thus, the vaccine has actually not only helped those who received all of the doses of the vaccine, but some that did not. Ideally, the vaccine is given to enough individuals to completely destroy the disease, which does not need to be the entire population, a concept referred to as herd immunity. Here, herd immunity may occur due to enough susceptibles receiving partial immunity from receiving some of the doses of the vaccine. In the period of vaccination we consider three cases: In the first case,we say that a vaccine would be effective if the administration of a vaccine results in the number of infections changing from an increasing number to a decreasing number once the final dose of the vaccine is administered. We would say that a vaccine would be ineffective if the number of infections is still increasing after the final dose of the vaccine is administered, and a vaccine would be unnecessary if the number of infections is already decreasing before the final dose of the vaccine is administered.
Generally, an effective vaccine was chosen based on our assumption that, if a vaccine is effective, its administration should cause the number of infections to go from increasing before the final dose of the vaccine to decreasing after the final dose of the vaccine.

DISCUSSION
Infectious diseases continue to have a major impact on individuals, populations and the economy, even though some of them have been eradicated. The spread of infectious diseases crucially depends on the pattern of contacts between individuals. Mathematical model has long been an important tool for understanding and controlling the spread of infectious diseases. Infectious disease models help us not only understand the dynamics of spreading pathogens but also design effective strategies for controlling outbreaks.
The main objective of this thesis is to develop a mathematical model which describes the dynamics of infectious disease and its transmission. In this study, an epidemic model is presented and analyzed to that effect. The basic reproductive number R0 has been computed to determine the stability of the disease. The biological meaning of the reproductive number is the average number of secondary cases produced by one infected individual during the infected individual's entire infectious period when the disease is first introduced. It characterizes the threshold behavior such that if R0 < 1, the modeled disease will die out if a small number of infected individuals are introduced into a susceptible population, and if R0 > 1, the disease will spread in the population. A good estimate of the reproductive number can provide significant insight into the transmission dynamics of the disease and can lead to effective strategies to control and eventually eradicate the disease. The basic reproductive number R0, can also be used to establish effective vaccination program. Effects of different vaccination programs on R0 are useful in setting the programs. So, we can say that reproductive ratios turned out to be an important factor in determining targets for vaccination coverage.
It was also realized that, in the absence of mass vaccination programme as well as early detection and supervised treatment, the transmission of the disease cannot be eradicated from the population. The introduction of proper and education about the infectious disease transmission as well as early detection of the disease can help reduce the disease in a population.
The results has also shown that the domain Ω is positively invariant, because no solution paths leave through any boundary. Based on results for the SIR models, we expect that all paths in Ω with infectives go to the endemic Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.9, No.10, 2019 42 equilibrium if R0 > 1. Then we have the usual behavior for an endemic model, in the sense that the disease dies out below the threshold, and the disease goes to a unique endemic equilibrium above the threshold.

CONCLUSION
Epidemic modeling is an increasingly important tool in the study of the infectious diseases. In the SIR model, the total population is divided into three classes, susceptible, infected and recovered. The model has two equilibrium states; an infection free equilibrium and endemic equilibrium state. Many models for the spread of infectious diseases in population have been analyzed mathematically and applied to specific diseases.
Many diseases that are possible to be vaccinated against only require one dose of that vaccine, and then most of those people are immune to that disease. However, some diseases, such as Hepatitis B, require multiple doses of a vaccine, spread apart by a designated period of time, for immunity. Hepatitis B requires three doses of a vaccine, with the second occurring about 1 month after the first and the third occurring about 5 months after the second, for immunity. Thus, we wanted to look at how a multistage vaccine could work at providing population herd immunity. Specifically, we set up a model based on the classic SIR model from Kermack and McKendrick and then studied the effects of a multistage vaccine.
We found that a vaccine could be effective, ineffective, or unnecessary, depending on whether the vaccine caused a decrease in the number of infected individuals or not. The effect of a grace period on a vaccine is important. It allows for more individuals to be fully vaccinated and thus immune, potentially increasing the effectiveness of the vaccine if a sufficiently long grace period can be provided.
We were able to derive a necessary condition for a vaccine to be effective based only on parameter values as well as a test condition for effectiveness should one want to, solve the system numerically. The model has shown success in attempting to predict the causes of infectious disease transmission within a population. The model strongly indicated that the spread of a disease largely depend on the contact rates with infected individuals within a population.
In this study, however, we found that one administration of the vaccine is not enough. The disease will eventually resume whatever behavior it would have had without the vaccine. Thus, if the disease would become endemic in the population without a vaccine, it would also do so once the vaccine administration was complete.
So, we then found that, pending the birth and death rates were biologically reasonable in value, and the first administration of a vaccine was effective, it would be possible to continually cause the number of infections to decrease by repeating the administration of the vaccine. This could then suppress the disease for as long as vaccines were administered and potentially for many time units after they were ceased. If a multistage vaccine is required, it should be continued to be administered for as long as the disease needs to be suppressed, which will effectively result in herd immunity for the population for a long time after the vaccines are ceased.
One of the changes we have considered is allowing for the proportions of individuals that get vaccinated to Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.9, No.10, 2019 43 change based on how the first administration of the vaccine goes. If it is effective, then we would increase the proportions of people who get vaccinated in the next administration, but, if it is ineffective or unnecessary, then we would decrease the proportions of people receiving the vaccination.
Another change is to make the model more individual based, where, instead of vaccinating individuals uniformly throughout a grace period, we would allow individuals to get vaccinated at any time during that grace period, if they choose to do so. This would be much more akin to what happens in reality, so the hope is that, upon implementing this change, we would still see that a multistage vaccine could be effective for a population.
If the proportion of the population that is immune exceeds the herd immunity level for the disease, and then the disease can no longer persist in the population. Thus if this level can be exceeded by mass vaccination, then the disease can be eliminated. The model also pointed out that early detection has a positive impact on the reduction of infectious disease. People should be educated in order to create awareness to the disease transmission so that society will be aware of this deadly disease.