Modelling Infectiology of Dengue Epidemic
Laurencia Ndelamo Massawe^{1, *}, Estomih S. Massawe^{2}, Oluwole Daniel Makinde^{3}
^{1}Faculty of Science, Technology and Environmental Studies, the Open University of Tanzania, Dar es Salaam, Tanzania
^{2}Mathematics Department, University of Dar es salaam, Dar es Salaam, Tanzania
^{3}Faculty of Military Science, Stellenbosch University, Saldanha, South Africa
Email address:
To cite this article:
Laurencia Ndelamo Massawe, Estomih S. Massawe, Oluwole Daniel Makinde. Modelling Infectiology of Dengue Epidemic. Applied and Computational Mathematics. Vol. 4, No. 3, 2015, pp. 192-206. doi: 10.11648/j.acm.20150403.22
Abstract: In this paper a mathematical model for the transmission dynamics of dengue fever disease is presented. We present a SITR (susceptible, infected, treated, recovery) and ASI (aquatic, susceptible, infected) epidemic model to describe the interaction between human and dengue fever mosquito populations. In order to assess the transmission of Dengue fever disease, the susceptible population is divided into two, namely, careful and careless human susceptible population. The model presents four possible equilibria: two disease-free and two endemic equilibrium.The results show that the disease-free equilibrium point is locally and globally asymptotically stable if the reproduction number is less than unity. Endemic equilibrium point is locally and globally asymptotically stable under certain conditions using additive compound matrix and Lyapunov method respectively. Sensitivity analysis of the model is implemented in order to investigate the sensitivity of certain key parameters of dengue fever disease with treatment, Careful and Careless Susceptibles on the transmission of Dengue fever Disease.
Keywords: Dengue Fever Disease, Careful, Careless, Susceptibles, Equilibrium, Stability, Reproduction Number
1. Introduction
Dengue is a major health problem found in tropical and sub-tropical climates worldwide, mostly in urban and semi-urban areas [1]. Dengue fever disease can cause a severe flu-like illness, and sometimes Dengue fever can vary from mild to severe. The more severe forms of dengue fever include dengue hemorrhagic fever and dengue shock syndrome. Dengue fever (DF) is a vector-borne disease transmitted by female Aedes aegypti and Aedes albopictus mosquitoes because they require blood meal for the development of their eggs. 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. Preventing or reducing dengue virus transmission depends entirely on the control of mosquito. The spread of dengue is attributed to expanding geographic distribution of the four dengue viruses and their mosquito vectors, the most important of which is the predominantly urban species Aedes aegypti [2].
Mathematical models have played a major role in increasing our understanding of the dynamics of infectious diseases. Several models have been proposed to study the effects of some factors on the transmission dynamics of these infectious diseases including Dengue fever and to provide guidelines as to how the spread can be controlled [3].
Mathematical modelling also became considerable important tool in the study of epidemiology because it helped us to understand the observed epidemiological patterns, disease control and provide understanding of the underlying mechanisms which influence the spread of disease and may suggest control strategies [4,5,6,7,8,2,9,10,11]. Moreover in [12] the authors presented a dynamical model that studied the temporal model for dengue disease with treatment. So far no one considered a dynamical system that incorporates the effects of treated individual, Careful and Careless Susceptibles on the transmission of Dengue fever in the society. In this paper, an extension of the model of [12] is presented to include temporary immunity and Susceptibles with different behaviour i.e. the dynamical system that incorporates the effects of Careful and Careless Susceptibles on the transmission of Dengue fever in the society.
Thus, we study and analyse a non-linear mathematical model showing the effect of Treatment, Careful and Careless Susceptibles on the transmission of dengue fever disease in the population.
2. Formulation of the Model
In this section, a deterministic model is developed that describes the dynamics of Dengue fever of population size [13].Two types of population are considered: humans and mosquito. The humans are divided into five mutuall-exclusive compartments indexed by are given by: careful human susceptible, , careless human susceptible, , in which the possibility of careless human Susceptibles contracting the disease is higher than that for careful human Susceptibles, , individuals capable of transmitting dengue fever disease to others; , individual who are treated and , individuals who have acquired immunity at time t. The total number of human is constant, which means that . Similarly, the model has also three compartments for the mosquito (mosquitoes) indexed by are given by: , which represents the aquatic phase of the mosquito (including egg, pupae and larvae) and the adult phase of the mosquito, with and, susceptible and infected, respectively. It is also assumed that.
Considering the above considerations and assumptions, we then have the following schematic model flow diagram for dengue fever disease with treatment, Careful and Careless Susceptibles:
Figure 1. Model Flow diagram for dengue fever disease with treatment, Careful and Careless human susceptible.
From the above flow diagram, the model is described by an initial value problem with a system of eight differential equations given as follows:
(1)
where,,, ,,,,, for all.
3. Mathematical Analysis of the Model
The dynamics of dengue fever disease is determined by the basic reproduction number which is a key concept and is defined as the average number of secondary infection arising from a single infected individual introduced into the susceptible class during its entire infectious period in a totally susceptible population [14,15],for if the result is disease free-equilibrium and if means that there exists endemic equilibrium point. The model system of equations (1) will be analysed qualitatively to get a better understanding of the effects of treated individual, careful and Careless human Susceptibles of Dengue fever disease.
3.1. Disease Free Equilibrium (DFE)
3.2. The Basic Reproduction Number,’’
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
consequently
or
It follows that the Basic Reproductive number which is given by the largest Eigen value for model system (1) denoted by is given as
But
It is follows that
or
where
Model System (1) has infection-free equilibrium if, otherwise endemic equilibrium.
3.3. Sensitivity Analysis of Model Parameters
In this subsection we would like to know difference factors for disease transmission where this helps to reduce mortality and morbidity due to dengue fever disease.
In order to determine how best human mortality and morbidity due to dengue fever disease is reduced, we calculate the sensitivity indices of the reproduction number to each parameter in the model using the approach of [14]. These indices tell us which parameters have high impact on and should be targeted by intervention strategies [13].
Definition 1: The normalised forward sensitivity index of a variable ‘p’ that depends differentiable on a parameteris defined as:
in [16]. (3)
Having an explicit formula for in equation (3), we derive an analytical expression for the sensitivity of as
where
Then analytical expression for the sensitivity of with respect to each parameter can be calculated using a set of reasonable parameter values. Parameter values are obtained from the different literatures like [12], [13] and [17] (http://www.wavuti.com/2014/05/wizara-ya-afya-kitengo-cha.html). Other parameter values are estimated to vary within realistic means and given as, ,,,,,,,,,,,,,, ,.
The sensitivity indices ofwith respect to and are given by and respectively.
Other indices ,, , , , , ,,,,,and are obtained following the same method and tabulated as follows:
Table 1. Sensitivity Indices of Model Parameters to .
S/N | Parameter Symbol | Sensitivity index |
1 |
| +0.506702261 |
2 |
| +0.500000191 |
3 |
| +0.499999846 |
4 |
| +0.499999617 |
5 |
| +0.499998468 |
6 |
| +0.244896308 |
7 |
| +0.016085811 |
8 |
| +0.010204166 |
9 |
| +0.00002580447057 |
10 |
| +0.000002867004084 |
11 |
| -0.00005539116931 |
12 |
| -0.001495356938 |
13 |
| -0.006702415127 |
14 |
| -0.498452312 |
15 |
| -1.016085624 |
The parameters are ordered from most sensitive to the least.
3.3.1. Interpretation
By analysing sensitivity indices of model parameters to ,it is observed that the following parameters , , ,,, ,,,and when each one increases keeping the other parameters constant they increase the value of implying that they increase the endemicity of the disease as they have positive indices. While parameters such as, , ,and when each one increases while keeping the other parameters constant they decrease the value of implying that they decrease the endemicity of the disease as they have negative indices.
But individually, the most sensitive parameter is maturation rate from larvae to adult (per day) , followed by the transmission probability from (per bite), average daily biting (per day) for mosquito susceptible, transmission probability from (per bite) , number of larvae per human , Fraction of subpopulation recruited into the population, number of eggs at each deposit per capita (per day), average daily biting (per day) for careful human susceptible , average daily biting (per day) for careless human susceptible, Positive change in behaviour of Careless individuals , average lifespan of humans (per day), Per capita disease induced death rate for humans , natural mortality of larvae (per day), mean viremic period (per day) and finally the least sensitive parameter is average lifespan of adult mosquitoes (Per day) .
3.4. Local Stability of Disease Free Equilibrium Point
Local stability of the disease free equilibrium is determined by the variation matrix of the model system (1) corresponding to the disease free as
(4)
where ,
Thus the stability of the disease free equilibrium point is 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,. The Jacobian matrix has the following eigenvalues:
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
Therefore the system is stable since all the eight eigenvalues are negative. This implies that at the Disease-free Equilibrium point is locally asymptotically stable.
3.5. Global Stability of Disease Free Equilibrium Point
In this subsection, we adopt the idea of [8], to analyse the global behaviour of the equilibria for system (1). The following theorem provides the global property of the disease free equilibrium of the system. The results are obtained by means of Lyapunov function.
Theorem 1: If, then the infection-free equilibrium is globally asymptotically stable in the interior of
Proof:
To determine the global stability of the disease-free equilibrium point, we construct the following Lyapunov function:
(5)
Calculating the time derivative of along (5), we obtain
Then we substitute from system (1) to obtain
Consequently
But it follows that
implying that
Therefore
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 implies that is globally asymptotically stable in Ω [18]. This completes the proof.
3.6. Existence and 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 Ω. The values of can be determined by setting to obtain
(6)
3.6.1. 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
where
3.6.2. 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 [19]. Local stability of the endemic equilibrium point is determined by the variational matrix of the nonlinear system (6) corresponding toand get the matrix
(7)
From (7) the second additive compound matrix is given by
where
,
The following lemma was stated and proved by [20] to demonstrate the local stability of endemic equilibrium point.
Lemma 3.2:
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 3.3: If the endemic equilibriumof the model (6) is locally asymptotically stable in
Proof:
From the Jacobian matrix in (7), we have
where ,
,
Thus, from the lemma 1, the endemic equilibrium of the model system (6) is locally asymptotically stable in .
3.6.3. 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;
(8)
Direct calculation of the derivative of along the solutions of (8) gives,
Consequently
which gives
(9)
where
Thus from equation (9), if then will be negative definite, meaning that . It follows that if and only if , , ,, and .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
In this section, we illustrate the analytical results of the study by carrying out numerical simulations of the model system (1) using a set of reasonable parameter values. Parameter values are obtained from the different literatures like [12], [13] and [17] (http://www.wavuti.com/ 2014/05/wizara- ya- afya- kitengo-cha.html). Other parameter values are estimated to vary within realistic means and given as shown below.
(10)
Figures 2 (i)-(vi) show the proportion of Dengue fever disease infectives, treated and recovery proportion all plotted against the proportion of susceptible population. This shows the dynamic behaviour of the endemic equilibrium of the model system (1) using the parameter values in (10) for different initial starting values in three cases as shown below [16].
(i) (ii)
(iii) (iv)
(v) (vi)
The equilibrium point of the endemic equilibrium was obtained as,, and then,and
It is observed from figures 2(i)-(vi) that for any starting initial value, the solution curves tend to the equilibrium. Therefore we conclude that the model system (1) is globally stable about this endemic equilibrium point for the parameters displayed in (10).
Figure 3 shows the variation of population in different classes.
From figure 3, it is observed that careful human susceptible population increases in time reaching its equilibrium position due to treatment and change of behaviour of careless susceptible. Moreover, careless human susceptible population decreases with time, due to careless individual moving to other classes. Dengue fever disease infected population decreases in time then reaches equilibrium due to the increase in the number of population changing behaviour and become careful and increase of recovered population. Treated infected population decrease due to the increase of the recovered population. Furthermore aquatic phase increases and then reach the equilibrium point due to its short life span and other move to susceptible class. Mosquito susceptible increases with time and reaches its equilibrium point due to its short life span and others move to infected class.
Figures 4(i)-(iv) show the variation of careful and careless human susceptible, infected human and infected mosquito population for different values of maturation rate from larvae to adult (per day)
(i)
(ii)
(iii)
(iv)
Figure 4. (i)-(iv) variation of careful and careless human susceptible, infected human and infected mosquito population for different values of maturation rate from larvae to adult (per day) .
From figure 4(i)-(iv) the maturation rate from larvae to adult (per day) is varied, and it is observed that when maturation rate from larvae to adult (per day) increases, careful human susceptible increases and then decreases with time due to the increase of production of infected mosquito. Moreover careless human susceptible decrease with time while infected human and mosquito population increase.
5. Conclusion
A compartmental model for Dengue fever disease was presented, based on two populations, humans (with temporary immunity, careful and careless susceptible) and mosquitoes with treatment. Sensitivity analysis revealed that the most sensitive parameter is maturation rate from larvae to adult (per day). Simulation shows that when maturation rate from larvae to adult (per day) increase, the number of infected individual increase while careful and careless susceptible decrease. This indicates that on the reduction of maturation rate from larvae to adult (per day), it is possible to maintain the basic reproduction number below unity and the disease can be eradicated from the community.
References