A Mathematical Model of the Transmission Dynamics of Typhoid Fever and Its Control
Chapter One
AIMS & OBJECTIVES OF THE STUDY
The study is to establish the role control intervention strategies which includes; Treatment, Vaccination and Enlightenment campaign, play on the transmission dynamics of typhoid fever with a view to bringing the disease under control and as such eradicating typhoid fever in our society.
In this direction, the study is aimed at achieving the following objectives.
- Build a mathematical model that can describe the transmission dynamics including the control strategies.
- To establish how the bacteria that causes Typhoid is transmitted.
- To find out the conditions that promotes the spread of the disease.
- To establish conditions that must hold if the disease is to be eradicated from the society under Treatment, vaccination and Enlightenment Campaign as Control Strategies.
- And to make recommendations that will help in the effort at controlling or eradicating the disease.
CHAPTER TWO
LITERATURE REVIEW
LITERATURE REVIEW OF TYPHIOD FEVER MODELS
Mathematical models have played key role in the formulation of Typhoid fever control strategies and the establishment of interim goals for intervention programme. Cvjectanovic et al (1971). A model was developed by Cvjectanovic et al (1971), where the number of newly infected persons was expressed as a function of the infectious and susceptible people in a community within a given time. The age structures of the population are established, which enabled more complicated detailed simulation of the effect of various interventions and strategies to control the disease in different age groups. The study indicated that once the incidence of the infection has fallen below the threshold, it cannot be maintained in a community due to the loss of the main source of infection chronic carriers, as they die out naturally.
Muhammed et al (2005) presented mathematical analysis of Typhoid model with saturated incidence rate. They considered typhoid as one of the communicable diseases where infection causes diarrhea and a rash and it is common mostly due to a type of bacterium called salmonella typhi (S. Typhi).In many developing countries, the disease is endemic and remains one of the public health problems.
They considered a mathematical model of the type SEIR (Susceptible, Exposed, Infected and Removed) to understand the transmission dynamics of the disease. Stability results for both local and global states were analyzed. Numerical solution of the system was investigated using the Range- Kulta method. Their result shows that the endemic equilibrium was stable locally and globally. Their model was given as:
Where, =Growth rate of the population
= Disease contact rate
d = Natural mortality rate
=Rate of flow from class E to class S
=Rate of flow from class I to class S
=Disease related death rate at class E
=The rate at which latent individuals are infected.
= Disease related death rate at class I
=Rate of recovery from infection
q =Proportion of individuals joining the class E
k=Educational adjustment.
Adetunde (2008), stated that typhoid fever is one of the most dangerous human infectious disease caused by a bacterium called Salmonella typhi. It belongs to the family Enterobacteriaceae: members to this family are: Salmonella paratyhpi A, Salmonella paratyphi B, Salmonella choleraesurs and Salmonella Typhi. The bacteria are primarily pathogens of humans and animals. He presented a mathematical model for the dynamics of typhoid fever in Kassena-Nankana District of upper East Region of Ghana. The equilibrium points of the model system were presented and their stability was also investigated. The threshold conditions for the disease free equilibrium for both numerical and qualitative analysis of the model were analyzed. The results of the analysis showed that there exist permanent immunity equilibrium points and were found to be globally asymptotically stable. The formulated model was given as:
Where S(t) = Susceptible class, I(t)= Infected class, C(t)= Carriers,
R(t)= Recovered class.
= the per capital natural mortality rate
= the typhoid fever- indicated mortality rate
= the rate of infection
= Rate of which the infected become carriers
= Rate of recovery for the carrier stage
= Carrier induced mortality rate
b= Rate of recovery for the infected class.
Cook (2010), developed a simple mathematical model on direct and indirect protection by vaccine and benefits of generic vaccination program. The population was divided into vaccinated and the unvaccinated subgroups and its effectiveness redefined. It was found that vaccination reduces the number of susceptible to infection and fewer infected individuals spread the disease among both vaccinated and unvaccinated persons.
Darja & Michael (2011) presented a mathematical model of the effects of carriers on transmission dynamics of infectious diseases. They investigated that infectious diseases could be transmitted through carriers, infected individuals who are contagious but do not show any disease systems. An infectious disease that produces long-term asymptotic carriers is the Typhoid Fever. It was recorded that disease carriage state are infectious while those in the latent period are not. Their model incorporated demography and disease induced death and it allows carriers to become symptomatic over their life time. Mathematical analysis was carried out that completely determined the global dynamics of their model.
Their result showed that, a greater probability to develop carriage will increase the basic reproduction number which makes the infection to persist. Testing and Diagnosis of carriers were seen as effective control measure in a country where infectious diseases persist.
CHAPTER THREE
THE ORGANISM OF TYPHOID FEVER
Typhoid fever is caused by Salmonella typhi, a gram-negative bacterium. The nomenclature for these bacteria is confused because the criteria for designating bacteria as individual species are not clear. Two main views on the nomenclature of the genus salmonella have been discussed.
(Popoff & Minor, 1997) suggested the two species as:
Salmonella bongori and Salmonella enterica. S. enterica included 6 subspecies of which subspecies 1(one) contained all the pathogens of warm-blooded animals. S. typhi was a serotype within subspecies 1(one): This proposal was rejected by the international judicial commission because the name was not well known to clinicians and its uses might cause accidents endangering health or life. The original rules therefore remain in force, WHO, (2003) noted that the correct nomenclature for the causal agent of typhoid fever is Salmonella typhi and have requested that the current sub specific status of serotype paratyphi A should be raised to specific status, i.e salmonella paratyphi A.
- typhi can be classified in the laboratory by several biochemical and serological tests. One of the most specified is that of polysaccharide capsule Vi, which is present in about 90% of all freshly isolated S.typhi and has a protective effect against the bacteriadal action of the serum of infected patients. This capsules provides the basis for one of the commercially available vaccines. Vi antigen is present in some other bacteria (citrobacter freundii, Salmonella paratyphi C and Salmonella Dublin) but not in exactly the same genetic context. The ratio of disease caused by S. typhi to that caused by S. paratyphi is about 10 to 1 in most of the countries where this matter has been studied WHO, (2003).
Further understanding of the host-pathogen dynamics that occur during invasive Salmonella disease is required to recognize factors affecting disease transmission and human immune response to infection. Recently, a program of typhoid and paratyphoid human challenge studies has been undertaken by the Oxford Vaccine Group at the University of Oxford based on previous typhoid challenge work performed by the University Of Maryland School Of Medicine. These studies aim to address several critical knowledge gaps, thereby informing disease transmission models and accelerating the development of public health improvement strategies, Malick et al (2015)
CHAPTER FOUR
MODEL FORMULATION
- We assume that the susceptible populations are recruited by birth at a level of π
- We assume that the susceptible populations (S) become exposed to the bacteria at a level (β) =ψBS, where ψ is the interaction rate between the Susceptible individuals (S) and the Bacteria (B).The population of the susceptible is also increased due to the coming in of the Recovered individuals who lost Immunity and become susceptible again at the rate (α). This class is decreased by natural death denoted by (µ), or by those vaccinated against the disease at the rate of (Ө) and also as a result of progression of the Susceptible class (S) to the Exposed class (E) because of failure to hearken to enlightenment control measure to avoid being exposed which is denoted by (x) and lastly as a result of the acceptance of the enlightenment control measure to go for vaccination at the rate of (z). This of course increases the rate of vaccination.
- The Exposed class (E) is increased by the progression of individuals from the susceptible class and by the progression of some vaccinated individuals who through carelessness come in contact with the Bacteria (B) at the contact rate (δ) which is subject to the waning rate of the vaccine at the rate of (w). This population reduces as a result of the progression of the exposed class to the infective class at the rate (λ) and as a result of natural death at the rate of (µ).
- (4) The Infective compartment (I) increases due to the progression of the Exposed individuals (E) to the infective class at the rate (λ). This class reduces as a result of those that are being treated at the rate (γ), which is subject to the rate at which the infective class accepts the enlightenment control measure to go for treatment at the rate (y). This class is then finally reduced by natural death at the rate (µ) and disease induced death on the infective class at rate (d1).
CHAPTER FIVE
CONCLUSION
Based on the results of this study we conclude that the most effective way to control typhoid disease within a population is to use effective educational enlightenment campaign on the radio, television, newspapers, in churches, mosques, even in schools on the need for individuals to make use of safe water, and better food hygiene practices. A typhoid patient should be encouraged to go for treatment always so as to protect their lives and as this will reduce mortality rate due to the disease. Lastly vaccination of the susceptible population should be encouraged, as protection is better than cure.
RECOMMENDATIONS
Having conducted this research, we therefore recommend the following:
- Enlightenment campaign program on the disease should be re-emphasized; Educators should use appropriate local media, such as radio, television, newspaper in disseminating health education messages to the general public.
- The need for typhoid patient to always go for treatment should be re-emphasized to avoid complication at the long run.
- Vaccination of the susceptible class should be encouraged within the population against the disease.
- Government should ensure that her citizens have access to a safe water supply.
- Appropriate facilities for waste disposal must be available for all community by the Government.
- Typhoid centres for screening and treatment should be established across the countries.
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