Mathematical Model for the Dynamic Spread and Control of Polio in Nigeria

Mathematical Model for the Dynamic Spread and Control of Polio in Nigeria

Chapter One

Aim and Objectives

Mathematical modelling can be used for several different reasons. How well any objective is achieved depends on the state of knowledge about a system and how well the modelling is done. Examples of the range of objectives are:

1. Developing scientific understanding – through quantitative expression of current knowledge of a system (as well as displaying what we know, this may also show up what we do not know);
2. Test the effect of changes in a system;
3. Aid decision making, including;

(i) tactical decisions by managers,

(ii) strategic decisions by planners.

CHAPTER TWO

LITERATURE  REVIEW

Poliomyelitis, known to most simply as Polio, has afflicted the human race since the late 18th century. It was not until the 1950s that we had any defense against this infectious disease. This was just around the time that Polio was at its peak in the United States, causing paralysis in more than 21,000 Americans in 1952 alone. The first defense against Polio- the injected Polio vaccine (IPV)-was introduced by Jonas Salk in 1955. This was quickly followed by an oral Polio vaccine developed in the early 1960s by Albert Sabin. Each vaccine has its benefits and drawbacks, but these two vaccines continue to be distributed and refined today. Most developed countries have switched to exclusively using IPV because of the vaccine derived infections sometimes caused by OPV, but OPV continues to be used in developing countries because of its cost-effectiveness, reliability for initiating an immune response in the gut, and ease of administration. In 1988, the World Health Organization declared the eradication of Polio by 2000 to be a global goal. Complete eradication was not achieved in this time period, but elimination of Wild-Type Polio 2 (WPV2), one of three wild polio strains, was successful. Today, Polio remains present in areas of Africa and Western Asia, mostly in the form of WPV1, and the WHO has set its sights on 2018 as the new target for eradication. This is a difficult task, since Polio can often remain unnoticed in a population for a long period of time, since the majority of cases are asymptomatic and do not result in the easily detectable Acute Flaccid Paralysis (AFP).

Only three countries have never seen eradication of Polio within their borders: Nigeria, Pakistan, and Afghanistan. Each of those countries presents a different public health challenge to the global community, but the greatest worry is that continued infection in those countries will results in Polio re-infecting communities nearby that have already eliminated the disease. Nigeria in particular is quite close to eradicating Polio, and only four wild Polio cases have been reported this year (though because of Polio’s high prevalence of asymptomatic cases, every detected case is considered a full-scale outbreak. One of Nigeria’s unique roadblocks is that it has one of the fastest growing populations in the world. In this study, we intend to discover how this non-constant population affects spread of the Polio virus in Nigeria. The results of this research can be used to examine strategies for preventing the spread of Polio in Nigeria, and can also be easily extrapolated to suggest vaccination methods for other infectious diseases in populations with non-constant growth.

It is likely that polio has plagued humans for many years. An Egyptian carving from 1400 Bc depicts a young man with a leg deformity similar to the one caused by polio. Polio circulated in human populations at low levels and appeared to be a relatively uncommon disease for most of 1800’s Polio reaches epidemic proportions in the early 1900’s in countries with relatively high standards of living, at the time when other diseases such as diphtheria, typhoid and tuberculosis were declining. Indeed, many scientists think that advances in hygiene paradoxically led to an increase incidence of polio. the theory is that in the past, infants were exposed to polio mainly through contaminated water supplies, then at a very young age infants immune systems aided by maternal antibodies still circulating in their blood, could quickly  defeat polio virus and then to develop lasting immunity to it. Because of widespread vaccination, polio was eliminated from the western hemisphere in 1994. Today it continues to circulate in a handful of countries with occasional spread to neighboring countries (Endemic as of 2015 are Nigeria, Afghanistan and Pakistan) vigorous vaccination programs are being conducted to eliminate these last pockets. Polio vaccination is still recommended worldwide because of imported cases.

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CHAPTER THREE

MATERIAL AND METHODS

MODEL FORMULATION

What is mathematical modeling?

Models describe our beliefs about how the world functions. In mathematical modeling, we translate those beliefs into the language of mathematics. This has many advantages

1. Mathematics is a very precise language. This helps us to formulate ideas and identify underlying assumptions.
2. Mathematics is a concise language, with well-defined rules for manipulations.
3. All the results that mathematicians have proved over hundreds of years are at our disposal.
4. Computers can be used to perform numerical calculations.

There is a large element of compromise in mathematical modeling. The majority of interacting systems in the real world are far too complicated to model in their entirety. Hence the first level of compromise is to identify the most important parts of the system. These will be included in the model, the rest will be excluded.

CHAPTER FOUR

 RESULT AND DISCUSSION

 Result

Since we are particularly interested in minimizing the impact of Polio when it is reintroduced into a disease free population, we want to evaluate whether the vaccination strategies outlined above could be effective in the case of an outbreak, and, if so, to determine what the minimum vaccination rate must be to avoid an epidemic. For our first model that considers vaccination of children in the 1-5 age class, we find that the growing population is an obstacle to satisfaction of inequality (1), and that using the current birth and death rates in Nigeria.

CHAPTER FIVE

Summary, Conclusions and Recommendations

Summary

In this paper, an epidemic model on Polio with the effect of vaccination is considered. Two threshold parameters 0R and 1 R corresponding to interaction of susceptible with infective and exposed class respectively are found. The sum of these two threshold values, denoted by R, is proved to be a sharp threshold value which completely determines the stability dynamics and the outcome of the disease. It is found that if 1 R, the disease free equilibrium is stable and the disease dies out, however if 1 R , there exists a unique endemic equilibrium which is locally asymptotically stable. Persistence of the disease is shown for 1 0 R and 11R.

To substantiate the analytical findings, the model is studied numerically and for which the system of differential equation is integrated using fourth order Runge-Kutta method, which satisfy the stability conditions. Further, to illustrate the global stability of the equilibrium point, numerical simulation is performed for different initial starts and the results are displayed. It is found that the system exhibits steady state bifurcation for some parameter values. It is concluded that endemic level of infective population increases with the increase in rate of transmission of infection from infective to susceptible class, which can further be enhanced if transmission of infection occurs from latent hosts during incubation period. However, for a particular value of disease transmission coefficient r, exposed and infective population die out. Variation of infective equilibrium size of the population with basic reproduction number is determined and it is found that endemic infective level first rises with the increase in basic reproduction ratio and then becomes constant.

Conclusion

It is found that although vaccination helps in eradicating polio by decreasing endemic equilibrium level yet careful administration of vaccination is desired because if vaccine is administered in an individual during incubation period of polio, endemic equilibrium level increases and disease spreads faster than usual pace. Like any other IM injection, it can precipitate paralysis in a patient who is already in incubation period of polio, as can occur during polio epidemics. Hence, IPV or any other IM injection should be avoided in an unimmunized sick child with fever especially during season of polio epidemics. It is found that the periodic outbreak of the disease occurs when infection due to exposed and infective class occurs at the same rate. It is pointed out from our study that population movement also contributes to the spread of disease; constant migration in human population makes the disease endemic. Themovement of infected people from areas where polio is still endemic to areas where the disease has been eradicated led to resurgence of the disease. It is further observed that transmission of infection due to population in exposed state of polio plays an important role in the spread of Polio and hence some measures must be adopted to trace the population in exposed class as it is very difficult to trace them out because of absence of symptoms of disease in them.

Recommendations

1. People should be educated on the mode of transmission of the disease and on home-care strategies for people infected by the disease.
2. There should be more enlightenment campaign on the dangers of polio and on its symptoms
3. Polio tests and treatment should continue to be free-of-charge to enable poor people assess them.
4. More effort should be made to encourage people to voluntarily go for polio tests by discouraging stigmatization of people infected by the disease.
5. The conditions that promote rapid spread of polio should be discouraged. Such conditions include: overcrowded accommodation, high level of illiteracy, lack or inadequate medical facilities, the tradition of giving birth to many children.
6. All National and International agencies in charge of polio control should be better funded to enable them to continue to provide vaccines, test kits, and drugs for polio prevention, detection and treatment.
7. People who are on treatment for polio should be educated on the need to complete their treatment course. This will help reduce the incidence of drug resistance currently on the increase.
8. There should be provision of more trained personnel and more polio laboratory microscopy services.
9. There should be increase in the number of polio care detection. This can be achieved by providing more test kits and by encouraging people to go for polio tests.
10. Assistance of more international organizations should be sort to join those currently at the forefront of the fight for the control of polio.

REFERENCES

• Anderson, R.M. and May, R.M. (1979). Population biology of infectious diseases: part1, Nature, Vol. 280, No. 5721, pp. 361-367
• Abdulraheem IS, Saka MJ (2004). The immune response to polio virus after natural infection and immunization with oral polio vaccine:  panoramic view of the issues and problems. Nig. Med. Pract. 46 (3): 50-55.
• DuintjerTebbens RJ, Pallansch MA, Kew OM, Cáceres VM, Sutter RW, Thompson KM.  “A dynamic model of poliomyelitis outbreaks: Learning from the past to help inform the future.” American Journal of Epidemiology, 2005a; In press.
• Francis T Jr, Korns RF, Voight RB, Boisen M, Hemphill FM, Napier JA, Tolchinsky E. “An evaluation of the 1954 poliomyelitis vaccine trials.” Am J Public Health. 1955;45(5, Part 2):1-63.
• Salk J (1952) One dose immunizations against paralytic poliomyelitis using a noninfectious vaccine. Rev. Infect. Dis. 6: S444-S450. Science Africa (2004).
• Sabin AB, Boulger LR (1973). History of Sabin attenuated poliovirus oral live vaccine strains. J. Biol. Stand. 1: 115-118.

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