Dengue fever is a re-emergent disease affecting more than 100 countries. Its incidence rate has increased fourfold since 1970 with nearly half the world’s population now at risk. In the chapter, a mathematical model with immunization is proposed to simulate the succession of 2 epidemics with variable human populations. Stability of the equilibrium points is carried out and simulation is given for different parameters settings.
At the dawn of the third millennium, the world population is facing a double burden of non communicable diseases (NCDs) and infectious diseases (Boutayeb, 2006). NCDs, once known as the disease of “the rich”, are now also affecting developing countries where Cardio-Vascular Diseases (CVDs), cancer and diabetes are flourishing (WHO, 2003; Boutayeb, 2005; Parkin, 1999). In parallel, infectious diseases continue to be the major causes of mortality and morbidity in low and middle income countries, where, well known existing, emerging and re-emerging diseases like tuberculosis, cholera, meningitis, hepatitis, malaria, dengue, yellow fever, AIDS, Ebola, SARS and others are causing suffering and mortality to a wide population. Among the infectious diseases, dengue fever, especially known in Southeast Asia, is now endemic in more than 100 countries world-wide. Its incidence has increased fourfold since 1970 and nearly half the world population (2.5-3 billion) is now at risk. It estimated that more than 50 million people are infected every year of which half a million of Dengue Haemorrhagic Fever (DHF) (DengueNet, 2007; Reprot, 2002; Teixeira, 2002). The two recognised species of the vector transmitting dengue are Aedes aegypti and Aedes albopictus. The first is highly anthropophilic, thriving in crowded cities and biting primarily during the day while the later is less anthropophilic and inhabits rural areas. Consequently, the importance of dengue is two- fold:
With increasing urbanisation, crowded cities, poor sanitation and lack of hygiene, environmental conditions foster the spread of the disease which, even in the absence of fatal forms, breeds significant economic and social costs (absenteeism, immobilisation, debilitation, medication).
The potential risk of evolution towards the haemorrhagic form and the dengue shock syndrome with high economic costs and which may lead to death.
Many authors have presented the disease as a major health problem either for the last decades of the 20th century or for the third millennium (Gubler, 1997; Gubler, 2002). The need for research and surveillance is often dealt with and many authors have stressed that DF/DHF is still perceived as unimportant and receives little attention despite its social and economic impact being similar to some of the most visible infectious diseases (Meltzer, 1998; Coleman, 2004).
Key Terms in this Chapter
Simulation: The use of a mathematical model to recreate a situation, or to imagine different scenarios with various parameters settings.
Incidence: The frequency with which a disease appears in a particular population or area in a given period of time.
Dengue: An acute, infectious tropical disease caused by an arbovirus transmitted by mosquitoes, and characterized by high fever, rash, headache.
Adequate Contact Rate: The average number of contacts between a person and a mosquito.
Stability Analysis: Analysis indicating how a model reacts to perturbations and changes.
Epidemic: A disease (infection) that is spread (in general by transmission) affecting a large part of the population over a wide area.
Immunization: natural or acquired protection against infection.
Compartmental Model: A model subdivided into different classes.