Analysis of a Mathematical Model to Investigate the Dynamics of Dengue Fever

: In this paper, we formulated a compartmental model to investigate the dynamics of dengue fever in a population with some measure of disease control. We qualitatively and quantitatively analyzed the model and found that the model has a disease free equilibrium (DFE), an endemic equilibrium point and undergoes the phenomenon of backward bifurcation. It was also discovered that Dengue can be eliminated irrespective of the initial size of the infected population whenever the effective reproduction number is less than one. Numerical simulations were carried out on the model and effective control measures were proposed that will result in reducing the burden of the disease in the population. © JASEM

Dengue, a mosquito-transmitted disease caused by any of four closely-related virus serotypes (DEN-1-4) of the genus Flavivirus, is endemic in at least 100 countries in Africa, the Americas, the Eastern Mediterranean and subtropical regions of the world, inhibited by over 2.5 billion people (Garba, et al., 2008) . In developing countries population growth is an important factor that contributes to the increase in the incidence of communicable diseases which affects mainly the urban poor, with infants and children among the groups particularly at risk (Nuraini et al, 2009). Urbanization and population growth increase the demand on the basic essential services such as housing, water supply, etc., and at the same time induce conditions that increase the transmission of some vector-borne diseases (Nuraini et al., 2009). Dengue is a viral, vector borne disease, spread by the Aedes Aegypti mosquito. It was estimated that about 50 million infections occur annually in over 100 countries. There is no specific treatment for curing dengue patients (Nuraini et al., 2009). Hospital treatment, in general, is given as supportive care which includes bed rest and analgesics (Nuraini et al., 2009).
Dengue virus is one of the most difficult arboviruses to isolate (Nuraini et al., 2009). There are four serotypes of the dengue virus; Den-1, Den-2, Den-3, Den-4, and each of the serotypes has numerous virus strains (Nuraini et al., 2009). Infection with one dengue serotypes may provide long life immunity to that serotype, but there is no complete crossprotective immunity to other serotype (Gubler, 1998). Identification of the primary target cells of dengue viruses' replication in the infected human body has proven to be extremely difficult (Nuraini et al., 2009).
The incubation period of the disease in an infected host is 3-14 days (average 4-7 days) (Nuraini et al., 2009). At the end of the incubation period, the patient may experience a sudden onset of fever (Nuraini et al, 2009). Viraemia is the presence of the virus in the blood stream (Nuraini et al., 2009). It is detected using the mosquito inoculation technique. Viraemia is assumed to become detectable on the second or the third day before the onset of symptoms and ends on the last days of illness (Nuraini et al., 2009). It usually peaks at the time of or shortly after the onset of illness (Gubler et al., 1981). Susceptible mosquitoes can be infected when they bite dengue infected hosts during the febrile viremic stage (Nuraini et al., 2009). It is usually believed that dengue viruses quickly clear in human body within approximately 7 days after the day of sudden onset of fever (Vaughn et al., 1994). Naturally this clearing process is done by the immune system which is as a result of complex dynamics reactions (Nuraini et al., 2009). Over the last decade mathematical models have been formulated to evaluate the dynamics of Dengue Fever. In this paper, a mathematical model is formulated and analysed to investigate the dynamics of Dengue Fever in a population in order to reduce the public health burden of the disease.

MATERIALS AND METHODS
Let N H (t) and N V (t) denote the total number of humans and vectors at time t, respectively. The model sub-divides these populations into a number of mutually-exclusive compartments, as given below.

ANDRAWUS JAMES; EGUDA, FELIX YAKUBU
The total population of human and vectors is divided into the following mutually exclusive epidemiological classes, namely, susceptible humans (S H (t)), humans with dengue in latent stage (E 1 (t)), humans with dengue (I I (t)), humans treated of dengue (R 1 (t)), susceptible vectors (S V (t)), vectors with latent dengue (E V (t)), vectors with dengue (I V (t)), Hence, we have that, , Where < , this accounts for the relative infectiousness of humans with latent dengue E 1 compared to humans in the I 1 class.

Derivation of Model Equations:
Singly infected individuals with latent dengue progress to active dengue at a rate 1 γ . Natural human death occurs at a rate H µ in the classes ,, , , , , respectively and those in class undergo an additional dengue induced death, at rate . Natural vector death occurs, at a rate , in the classes , "#$ , while the vectors in the class undergoes additional dengue induced death, at a rate , although this is negligible as infected vectors are not deemed to be suffering dengue. Exposed vectors progress to the infectious stage at the rate % .
The above assumptions result in the following system of nonlinear ordinary differential equations:

Lemma 1 The region D 2 is positively invariant for the system (1)
Proof: The rate of change of the total human population is given as By standard comparison theorem, (4) Using the integrating factor method By standard comparison theorem, Similarly, using the integrating factor method, we have In So, 7 8 is a positively invariant set under the flow described in (1). Hence, no solution path leaves through the boundary of 7 8 . Also, since solution paths cannot leave 7 8 , solutions remain non-negative for non-negative initial conditions. Solutions exist for all time t. In this region, the model (1) is said to be well posed mathematically and epidemiologically.

Positivity of Solutions
Similarly, we can show that (J) > 0,  (1) has a disease-free equilibrium, obtained by setting the right hand side of the model to zero and also setting the disease classes to zero we obtain ( ) The stability of j 8 is established using the next generation operator method on the system (1). Using the notation in van den Driessche and Watmough (2002) And, Where,

Lemma 3 The DFE of the system (1) is locally asymptotically stable if R D < 1 and unstable if R D > 1.
The threshold quantity R D is the effective or control reproduction number for the Dengue model. By Lemma 3, biologically speaking, Dengue is eliminated from the population when R D < 1 if the initial sizes of the subpopulations of the model are in the region of attraction of j 8 . However, the disease free equilibrium may not be globally asymptotically stable even if R D < 1 in the case when a backward bifurcation occurs. That is, there is the presence of a stable EEP co-existing with the DFE.

Analysis of a Mathematical
Similarly, "(j 8 ) … * has a left eigenvector, ( ) 7  6  5  4  3  2  1   ,  ,  , ,  5  4  3  3   3   3  4  1  2  1 , 0 , 0 , 0 The above eigenvectors were obtained by solving (44) (49) that the bifurcation coefficient, ", is positive whenever, K 1 >K 2 +K 3 Thus, the model (1) undergoes a backward bifurcation at R D =1 whenever the inequality (55) holds.  (1) showing the force of infection as a function of the control reproduction number with all the parameters used as stated in Table (2) except o = 2 and o = 1 so that < 1.  (1) showing the force of infection as a function of the control reproduction number with all the parameters used as stated in Table (2) except o = 2 and o = 1 so that < 1. Table 3: Parameter Information Using the parameter values in Table 3, we carried out some simulations of model (1).