1. Introduction

Dengue fever is a severe infection, flu-like illness transmitted to humans through the bites of infected female Aedes mosquitoes. Four different serotypes can cause dengue fever. A human infected by one serotype, on recovery, gains total immunity to that serotype and only partial and transient immunity with respect to the other three. Dengue fever can vary from mild to severe; the more severe forms of dengue fever include dengue hemorrhagic fever and dengue shock syndrome. Dengue hemorrhagic fever occurs when a person get infected by different type of dengue virus after being infected by another one sometimes before. Dengue shock syndrome is the most severe form of dengue infection. Dengue is found in tropical and sub-tropical climates worldwide, mostly in urban and semi-urban areas [1].

Mathematical modelling of the population models continues to provide important insights into population behaviour and control. Over the years, it has also become an important tool in understanding the dynamics of diseases, and the decision making process regarding intervention programs for controlling population and disease problems in many countries [2].

Mathematical modelling also became considerable important tool in the study of epidemiology because it helps in understanding the observed epidemiological patterns, disease control and provides understanding of the underlying mechanisms which influence the spread of disease and may suggest control strategies [1-8]. Moreover [9], presented a dynamical model that studied the temporal model for dengue disease with treatment. So far no research has considered a dynamical system that incorporates the control strategies to reduce the spread of the dengue fever disease through the campaign to educate the careless human susceptible, control vector human contact, removing vector breeding areas, insecticides application and control maturation rate from larvae to adult. In this work, we present an extension of the model of [9] to include temporary immunity, control strategies and Susceptibles with different behaviour i.e. the dynamical system that incorporates the effects Careful and Careless Susceptibles on the transmission of Dengue fever in the society with vaccination. In this paper, data reported by the ministry of health in Tanzania is used. In July 2010 for the first time in Dar es Salaam region -Tanzania, an outbreak of dengue fever was reported, over 40 people were infected and then also between May and July 2013,172 were infected with this disease. Moreover in the year 2014, the government of Tanzania announced the dangers of the disease in which people were alerted about the disease and the precaution to be taken. In this 2014, 399 people were infected in which 2 died of the disease in Dar es Salaam region (http://www.wavuti.com/2014/05/wizara-ya-afya-kitengo-cha.html). Data will be obtained from the different literature and estimated since there is no enough data in Tanzania. The purpose of this study is to match the empirical data with the modal simulation. Hence we formulate the SITRS (susceptible, Infected, Treated, Recovered, susceptible) and SVITRS (Susceptible, Vaccinated, Infected, Recovered, Susceptible) models for transmission of dengue fever disease.

2. Formulation of the Model

In this section, we adopt the model presented in [10]. The model is based on two populations: humans and mosquitoes. Human population is divided into five groups such as - Careful human Susceptibles, - Careless human Susceptibles, - infected human, - treated infected human, - recovery infected human, so that we have and the population of female mosquitoes, indexed by is divided into three groups that is -Aquatic phase (that includes the egg, larva and pupa stages), - Susceptibles (mosquitoes that are able to contract the disease), -Infectives (mosquitoes capable of transmitting the disease to human). In formulating the model, the following assumptions are considered:

i. Total human population is constant,

ii. The population is homogeneous, which means that every individual of a compartment is homogeneously mixed with the other individuals,

iii. Immigration and emigration are not considered,

iv. Each vector has an equal probability to bite any host,

v. Humans and mosquitoes are assumed to be born susceptible i.e. there is no natural protection,

vi. The coefficient of transmission of the disease is fixed and does not vary seasonally,

vii. For the mosquito there is no resistant phase, due to its short lifetime,

viii. The possibility of careless Susceptibles contracting dengue fever disease is higher than that for careful Susceptibles.

Considering the above assumptions, we then have the following schematic model flow diagram for dengue fever disease with control:

Figure 1. Model Flow diagram for dengue fever disease with control

From the above flow diagram, the model will be governed by the following equations [10]:

(1)

where is the transmission probability from (per bite), is the maturation rate from larvae to adult (per day), is the control maturation rate from larvae to adult, is the average daily biting (per day) for mosquito susceptible, is the transmission probability from (per bite), is the number of larvae per human, is the number of eggs at each deposit per capita (per day), is the fraction of subpopulation recruited into the population, is the average daily biting (per day) for careful human susceptible, is the average daily biting (per day) for careless human susceptible, is the Positive change in behaviour of Careless individuals, is the campaign of educating Careless human susceptible, is the average lifespan of humans (per day), is the per capita disease induced death rate for humans, is the natural mortality of larvae (per day), is the control of vector human contact , is the reducing vector breeding areas, mean viremic period (per day), average lifespan of adult mosquitoes (per day), insecticide application, portion that moves from compartment to due to loss of immunity and treatment parameter.

3. Model Analysis

We study the solutions of System (1) in the closed set

The set is positively invariant with respect to Equation (1) [10].

3.1. Disease Free Equilibrium (DFE)

For the disease free equilibrium, it is assumed that there is no infection for both populations of human and mosquitoes i.e. and , denoted by . Thus of the model system (1) is obtained as

where

3.2. The Basic Reproduction Number, R₀

The basic reproduction number, denoted by , is defined as the average number of secondary infections that occurs when one infective individual is introduced into a completely susceptible population [11].

The basic reproduction number of the model (1) is calculated by using the next generation matrix of an ODE [11]. Using the approach of [11]. is obtaining by taking the largest (dominant) Eigen value (spectral radius) of

where, is the rate of appearance of new infection in compartment is the transfer of individuals out of the compartment by all other means and is the disease free equilibrium.

Using the linearization method, the associated matrix at DFE is given by

This implies that

With

,

we have

The transfer of individuals out of the compartment is given by

Using the linearization method, the associated matrix at DFE is given by,

This gives

With

Therefore

(2)

Then eigenvalues of the equation (2) is given by

This gives

where

consequently

It follows that the Basic Reproductive number which is given by the largest Eigen value for model system (1) denoted by is given by

(3)

If , the disease cannot invade the population and the infection will die out over a period of time, and also, if , then an invasion is possible and infection can spread through the population. Generally, the larger the value of , the more severe, and possibly widespread the epidemic will be, [10].

3.3. Local Stability of Disease Free Equilibrium Point

To determine the local stability of the disease free equilibrium, the variation matrix of the model system (1) corresponding to the disease free is obtained as

(4)

where

Therefore the stability of the disease free equilibrium point can be clarified by studying the behaviour of in which for local stability of DFE we seek for its all eigenvalues to have negative real parts. It follows that, the characteristic function of the matrix (4) with being the eigenvalues of the Jacobian matrix, by using Mathematica software gives the following values:

The other eigenvalues are given as

when is not a real number,

when is not a real number,

when is not a real number, and finally.

when is not a real number, where

Hence under certain conditions the system is stable since all the eight eigenvalues are negative. These imply that at the Disease Free Equilibrium point is locally asymptotically stable.

3.4. Global Stability of Disease Free Equilibrium Point

In this subsection, the global behaviour of the equilibria for system (1) is analysed. The following theorem provides the global property of the disease free equilibrium of the system. The results are obtained by means of Lyapunov function. In choosing the Lyapunov function, we adopt the idea of [12].

Theorem1: If , then the infection-free equilibrium is globally asymptotically stable in the interior of .

Proof: To establish the global stability of the disease-free equilibrium, we construct the following Lyapunov function:

(5)

Calculating the time derivative of along (4), we obtain

Then substituting from system (1), we get

it follows that

where

Thus, is negative if and if and only if is reduced to the DFE. Consequently, the largest compact invariant set in when is the singleton . Hence, by LaSalle’s invariance principle, it is implied that is globally asymptotically stable in Ω [13]. This completes the proof.

3.5. Existence of Local and Global Asymptotic Stability of Endemic Equilibrium

Since we are dealing with presence of dengue fever disease in human population, we can reduce system (1) to a 4-dimensional system by eliminating respectively, in the feasible region . The values of can be determined by setting to obtain

(6)

3.5.1. The Endemic Equilibrium and Its Stability

Here, we study the existence and stability of the endemic equilibrium points. If then the host-vector model system (6) has a unique endemic equilibrium given by

in , with

But from (3)

where

3.5.2. Local Stability of the Endemic Equilibrium

In order to analyse the stability of the endemic equilibrium, the additive compound matrices approach is used, using the idea of [14]. Local stability of the endemic equilibrium point is determined by the variational matrix of the nonlinear system (6) corresponding to to get

(7)

From (7) the second additive compound matrix is given by

(9)

where

The following lemma is stated and proved by [15] to demonstrate the local stability of endemic equilibrium point .

Lemma 1: Let be a real matrix.

If and are all negative, then all eigenvalues of have negative real parts.

Using the above Lemma, we will study the stability of the endemic equilibrium.

Theorem 2: If the endemic equilibrium of the model (1) is locally asymptotically stable in .

Proof: From the Jacobian matrix in (7), we have

.

But from (3)

Let

It follows that where

Thus, from the lemma 1, the endemic equilibrium of the model system (7) is locally asymptotically stable in .

3.6. Global Stability of Endemic Equilibrium Point (EEP)

Theorem 3: If the endemic equilibrium of the model system (1) is globally asymptotically stable.

Proof: To establish the global stability of endemic equilibrium we construct the following positive Lyapunov function as follows;

Direct calculation of the derivative of along the solutions of (1) gives,

(8)

where

Thus from equation (8), if Then will be negative definite, implying that . It then follows that if and only if . Therefore the largest compact invariant set in is the singleton where is the endemic equilibrium of the model system (1). By LaSalle’s invariant principle, then it implies that is globally asymptotically stable in if . This completes the proof.

4. Numerical Simulations

Here, we illustrate the analytical results of the study by carrying out numerical simulations of the model system (1). Parameter values are obtained from the different literatures like (http://www.wavuti.com/2014/05/wizara-ya-afya- kitengo- cha.html), [9], [10] and [16]. Other parameter values are estimated to vary within realistic means and given as shown below.

Figures 2 show the distribution of population with time in all classes of human and mosquito when no control is applied.

Figures 2 show the human and mosquito populations in the absence of any control. The human infection reaches a peak between the 2th and the 20th day. The infection of the mosquitoes reaches a peak between the 10th and the 30th day. The total number of infected humans obtained from System (1) is higher than observations in Tanzania. The difference is due to the absence of the data in the whole country of Tanzania [17].

Figure 2. Distribution of population with time in all classes of human and mosquito when no control is applied i.e.

Figures 3 (i)-(ii) show the variation of infected human and mosquito populations with combine use of all five controls as shown:

Figure 3. (i)-(ii): Variation of infected human and mosquito populations with combine use of all five controls

From figure 3 (i)-(ii), it is observed that when all the controls are used, the disease is eradicated.

5. Formulation of the Model with Vaccination

In this section, we develop a deterministic model that describes the dynamics of Dengue fever under application of Vaccination and treatment for humans where is the careful human susceptible population, is the careless human susceptible population, is the vaccinated human population, is the infected human population, is the treated human population and is the recovered human population, is the fraction of the vaccinated careful human susceptible, is the proportional rate at which vaccinated careful human susceptible loses effect, is the reaction of the vaccinated careless human susceptible, is the proportional rate at which vaccinated careless human susceptible loses effect, is the proportion of the vaccinated new born, is the infection rate of vaccinated careful human susceptible and is the infection rate of vaccinated careless human susceptible. Susceptible individuals acquire Dengue fever through the bite of female Aedes mosquito with force of infections given by , and where .

Considering the above clarification, we then have the following schematic model flow diagram for dengue fever disease with vaccination:

From the above flow diagram, the model will be governed by the following equations [17]:

(10)

Mass vaccination generates the possibility of eliminating or eradication the infectious disease [18]. The more vaccinated people, the less likely a susceptible person will come into contact with the infection. With the introduction of a vaccine, the model related to the human population changes to the model. Vaccination is continuous with a constant proportion of vaccinated new born. A fraction and of careful and careless susceptible is vaccinated respectively. The vaccination reduces but does not eliminate susceptibility to infection. For this reason, we consider the infection rate of vaccinated people: when the vaccine is perfect and when the vaccine has no effect at all. The vaccination loses effectiveness at a rate and careful and careless susceptible respectively [17].

5.1. Model Simulation

Here, we perform numerical simulations of the model system (11) using the set of estimated parameter values. Parameter values are obtained from the different literatures like (http:/e/www.wavuti.com/2014/05/wizara-ya-afya-kitengo-cha.html), [9], [10], [16], other parameter values are estimated to vary within realistic means and given as ,

Figures 4 show the variation of infected human populations with different levels of infection rate of vaccinated careful human susceptible , infection rate of vaccinated careless human susceptible , fraction of the vaccinated careful human susceptible and fraction of the vaccinated careless human susceptible .

From figure 4 we vary infection rate of vaccinated careful susceptible , infection rate of vaccinated careless human susceptible , fraction of the vaccinated careful human susceptible and fraction of the vaccinated careless human susceptible , and it is observed that the effectiveness of the vaccine reduces the disease spread.

Figure 4. Variation of infected human populations with different levels infection rate of vaccinated careful human susceptible , infection rate of vaccinated careless human susceptible , fraction of the vaccinated careful human susceptible and fraction of the vaccinated careless human susceptible

Figure 5. Model Flow diagram for dengue fever disease with vaccination

6. Conclusions

A compartmental model for Dengue fever disease was presented, The was model based on campaign of educating Careless human susceptible , control vector human contact , reducing vector breeding areas , and insecticide application , control maturation rate from larvae to adult and treatment. The results show that Treatment and the controls on the transmission of dengue fever disease will have a positive effect on decreasing the growth rate of dengue fever disease. Then also shows that when the infected human population decrease and when the infected human population increase.