THE HEALTHCARE IMPACT OF COVID-19 EPIDEMIC IN NIGERIA (A STOCHASTIC MATHEMATICAL MODEL)
COVID-19, since its discovery in 2019, has posed a major health problem in the world. It is caused by the SARS-CoV-2 virus and is transmitted via infected respiratory droplets and contaminated surfaces. There is an urgent need to understand the transmission characteristics of the virus in response to social interventions. This is important to evaluate the overall impact of such programs in the management of the disease. We seek to develop a mathematical model that characterizes the transmission dynamics of COVID-19. The model analyzes the impact of preventive practices on the spread of SARS-CoV-2 by incorporating human behavior in modeling disease prevalence depending on contact rates for direct and indirect transmissions and infectious host shedding. The model is also applied to reported data from Wuhan and the state of Tennessee. Our results imply that applying strategically created awareness programs to a geological setting can eradicate COVID-19.
At present, the world faces a pandemic brought about by the SARS-CoV-2 virus. The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), a spike protein virus is the causative agent of the widespread coronavirus disease. It is known to belong to a broad family of viruses known as the coronaviruses. The first severe illness caused by a coronavirus is the 2003 Severe Acute Respiratory Syndrome (SARS) pandemic which started in China. A second flare-up was the Middle East Respiratory Syndrome (MERS) which was discovered in 2012 in Saudi Arabia .The coronavirus disease 2019 was first declared a Public Health Emergency of International Concerns by the World Health Organization on January 30, 2020  and finally a pandemic on March 11, 2020  since the disease was determined to be a public health risk to all nations through the international spread of diseases. The disease is mostly transmitted through contact with infected respiratory droplets from coughs, sneezes and speech[3, 4]. Further research has shown that the disease could be transmitted via airborne transmission . Contact with contaminated surfaces is also a known risk of infection. Some confirmed symptoms of COVID-19 known to appear after 2-14 days of exposure are coughs, fever, windedness, muscle pain, loss of smell, diarrhea sore throat, fatigue and running nose.
As at January 11, 2021 there were 680,908 new infected cases recorded worldwide with
222,921 cases in the USA, 4,255 cases in the state of Tennessee and 134 cases in China [6, 7]. A total of 11,416 deaths were recorded worldwide . The American continent currently has over 40million confirmed cases with the USA leading all other countries in this region and the world with about 25million confirmed cases. While there seem to be a rapidly increasing number confirmed cases, there are also several intervention programs created to combat the current characteristics of the disease. These programs include the current vaccination programs which begun in the last quarter of 2020  , use of disinfectants, social distancing, public health education, use of nose masks and other protective shields, isolation of infected/exposed persons, funding of COVID-19 projects etc.
Awareness programs as defined in this project comprises all actions and measures aimed at the prevention and treatment of COVID-19. These range from individual behavioral change to organizational and worldwide interventions. Social distancing is one of the most utilized COVID-19 preventive strategies. It is observed by limiting face-to-face interactions by remaining in any event 6 feet from others and maintaining a strategic distance from swarmed places. This intervention has resulted in the lockdown of several countries and the closing of organizations such as schools, churches and businesses. Though these interventions appear to be drastic as they have negative impact on productivity, virtual alternatives for social gatherings, meet-ups and workplaces have been discovered. Thus, resulting in the booming economies of the virtual networking industry. Vaccinations is another ongoing major preventive strategy. Presently, 86,452,579 doses of the various vaccines have been administered worldwide with US administering about 32% of the doses. Below are the vaccines currently available with their efficiencies and approvals discussed.
⦁ The Pfizer/BioNtech vaccine is one of the popular administered vaccines known for its high efficiency against the SARS CoV-2 virus which is 95%. It was approved for emergency use in USA, UK, Canada and the EU in December 2020. Storage and transportation of the vaccine requires a temperature of -70oC.
⦁ Moderna- this vaccine like the Pfizer vaccine has a high efficacy rate of 94.1% as they both use a new vaccine approach involving a messenger RNA. It has been approved for use in the USA, Canada, UK and the EU. Vaccine can be stored at 2-8oC. 
⦁ AstraZeneca/Oxford vaccine was approved in India on January 2, 2021 and the UK on Dec 30, 2020 for emergency use. With clinical trial size of 65,000 people, results show that the vaccine is 70% effective at preventing laboratory confirmed COVID-19.[8, 11]
⦁ The Novavax vaccine trials currently show 89.3% efficacy against the SARS CoV-2 virus.
Initial trial phases were conducted in South Africa and the UK whiles further trials are been conducted in the USA and Mexico. This vaccine is yet to receive approvals as it is in its trial stage. 
⦁ The Johnson & Johnson vaccine is the easiest to store and transport as it requires standard refrigeration out of the leading vaccines. The single-dose vaccine’s efficacy rate has currently dropped from 72% in the United States to 66% in Latin America and 57% in South Africa, where a profoundly infectious variation is driving most cases. The vaccine is yet to receive approvals as it is presently in its trial stage. 
⦁ Sinovac Biotech vaccine is the least effective vaccine with trial-based efficacy rate of
50.38% in Brazil. Though it has a low efficacy rate, it has been approved in Indonesia and Turkey where is known to have higher efficacy rates of 65.3% and 91.25% respectively. Vaccination requires two doses whiles storage and transportation can be done in standard refrigeration conditions. [9, 11]
⦁ Sinopharm- it was approved in China on Dec 31st, 2020 for general use. Clinical trials conducted on the vaccine shows 79% efficiency. It is currently been administered in Morocco, China, Hungary and the United Arab Emirates.[9, 11]
⦁ The CanSino Biologics’ vaccine utilizes an innocuous cold virus to transfer its genetic payload. Only a shot of the vaccine is required. Though its efficacy is yet to be determined due to delay in clinical trials, it was cleared in June 29, 2020 for China military use as it was the first COVID-19 vaccine to enter clinical trials.
Notable side effects of these vaccines are pain at the injection site, tiredness, headache, fever and chills.
1.2 PROBLEM STATEMENT
The 2019 coronavirus disease is a current health problem which was first discovered in Wuhan, China. Due to its high mortality and infection rates, and the lack of potent treatment available, it has claimed over 2.6million lives with over 118million infected cases worldwide.
Since its emergence, there has been a rise on intervention programs created with the aim of mitigating its transmission. However the extent of impact of these intervention programs is unknown. To date, there is no mathematical model that predicts the efficiency of these programs. So, the impact of the awareness programs is hard to analyze especially on a global scale. Thus, there is an urgent need for a fit-for-purpose model to probe into the impact of these awareness programs.
The study was designed to achieve the following;
⦁ To formulate a mathematical model that measures the impact of current awareness programs created in efforts of minimizing the spread of COVID-19.
⦁ Analyze the stability of the model developed at the Disease-Free Equilibrium and Endemic Equilibrium.
⦁ To develop a model to accurately predict the spread of COVID-19.
⦁ Investigate and predict the transmission dynamics of COVID-19 in Wuhan, China and the state of Tennessee.
1.4 THESIS LAY OUT
Chapter one of the study gives a detailed background of the study, the problem statement, objectives, and thesis lay out. In Chapter 2, we shall put a pertinent related literature on COVID-19 and the SEIR models. These include publications, journals and seminars. Chapter 3 discusses the model description, model analysis and equilibrium analysis of the model. Chapter 4 is devoted to numerical simulations and results. Chapter 5, the final chapter presents the discussion of the results, conclusions, and recommendations for further studies.
Coronavirinaea, the scientific name of the coronaviruses is a subfamily of enveloped positive-sense-single stranded RNA viruses known to infect mammals and birds[13, 14] .They usually cause infectious bronchitis and enteritis in birds, while they cause diarrhea in cows, rabbits and other mammals alike with a high mortality rate in rabbits. In humans, these viruses are responsible for several respiratory diseases . Seven strains of the coronaviruses are known to have infected the human populace[14, 15]. The viruses include;
1. Human Coronavirus OC43 (HCoV-OC43)
2. Human Coronavirus HKU1 (HCoV-HKU1)
3. Human Coronavirus HKU1 (HCoV-NL63) which is also known the New Haven Coron-
4. Human Coronavirus HKU1 (HCoV-229E)
5. Middle East Respiratory Syndrome-related Corona Virus (MERS-CoV)
6. Severe Acute Respiratory Syndrome
7. Severe Acute Respiratory Syndrome Corona Virus 2 (COVID-19)
Other than the current COVID-19 vaccines, there are no other antiviral treatment to prevent or cure infections caused by the coronaviruses. Thus there is the need to measure the impact of the current measures put in place to eradicate the coronavirus disease 2019.
Mathematical modeling has become an increasingly important area as computers are expanding our innate ability in translating mathematical equations and formulations into precise conclusions. These models by using basic assumptions and collected statistics can project and predict the progress of infectious diseases, show the likelihood of an epidemic, and aid in deciding which interventions are required.
This chapter focuses on the various studies done in attempt to model the epidemiology of the coronavirus disease 2019. Several researches have been conducted to aid in the understanding of the intrinsic bacterial behavior and transmission dynamics of the causative agents of COVID-19. However, none has investigated the impact of the intervention programs using a COVID-19 SEIR deterministic model with compartments for awareness programs and virus concentration despite the vaccination and preventive measures we have today.
A deterministic model also known as compartmental model, describes the transportation of materials in a system consisting of a collection of groups that are connected by material flow. Each compartment comprises of characterized materials and can exchange these materials with other compartments following some strict principles set for each compartment. The compartments are assumed to be homogeneous entities within which the materials being modelled are equivalent. Materials can also move into a compartment from outside through a source and be removed to the outside of the biological system under study through a sink or drain. 
Deterministic models help to account for the movement of material under study by following some conservational laws. In the deterministic model, the compartments are developed based on the law of conservation of energy which states that energy cannot be created nor destroyed but can change from one form to another. Thus, the differences in energy from one compartment to another can be calculated since no energy is lost during the exchange of materials.
Most deterministic models, in mathematical epidemiology have more than one compartment represented as equations. These models are generated by following the conservational law of energy for each equation. Before proposing a model for the transmission behavior of COVID19, we briefly examine the earlier proposed models for the study of COVID-19 and their results. We also in this chapter assess the basic SEIR model and the various ways it has been modified in disease modelling..