Analysis of the fractional order dengue transmission model: a case study in Malaysia
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Abstract
Dengue is one of the important infectious diseases in the world. In Malaysia, dengue occurs nationally and has been endemic for more than a decade. Hence, the modeling of dengue transmission is of great importance to help us understand the dynamical behavior of the disease. In this paper, we developed a compartmental model of the dengue transmission using the fractional order differential equation. It consists of six compartments representing the human and mosquito dynamics. The diseasefree and the positive endemic equilibrium point are obtained. The stability analysis of the equilibria is presented. A sensitivity analysis of the model is performed to determine the relative importance of the model parameters to the transmission. Numerical simulations are given for different parameter settings. A case study, using the outbreak dengue data in the state of Selangor, Malaysia, in 2012, is presented.
Keywords
Fractional system Dengue Stability analysis Reproduction number Sensitivity analysis1 Introduction
Dengue fever, commonly known as dengue, is a painful, debilitating mosquitoborne tropical disease caused by the dengue virus. It is a viral disease transmitted by the bite of an Aedes mosquito infected with any of the four serotypes, denoted by DENI, DENII, DENIII, and DENIV, respectively. In recent decades, the spread of the dengue virus has increased rapidly and according to the World Health Organization (WHO) there are millions of dengue cases reported every year worldwide [1]. A human that gets infected by any of these dengue serotype produces permanent immunity to it, but only a temporary crossimmunity to the other serotypes [2].
In Malaysia, dengue occurs nationally with increased risk in urban and periurban areas. The number of dengue fever cases reported in our nation continues to increase since 2010 [3]. There is no specific vaccine available for dengue in Malaysia. Preventing and controlling the dengue virus depends solely on the control of the mosquito vector or interruption of humanvector contact. Hence, a reliable mathematical model is essential to give a deeper understanding of the mechanism of dengue transmission and on how to control the spread of the disease.
The wellknown susceptible–infected–recovered (SIR) epidemic model introduced by Kermack and McKendrick in 1927 [4] has been used by many researchers to study the dynamics of the infectious disease. Later, Bailey [5] developed a simple vectorhost dengue transmission model for a single serotype based on the SIR epidemic model. The development of the dengue transmission model has become attention ever since, parallel to the increase in the number of dengue cases reported all around the world. Most of the model is based on the ordinary differential equation [6, 7, 8, 9, 10].
In the last few years, fractional order calculus is found to be more interesting in modeling a real problem in comparison to a classical integer order as it provides a tool for the description of memory effects and genetic properties of various materials [11, 12, 13, 14]. In this paper, the proposed dengue epidemic model is derived using the generalized fractional order derivative.
Pooseh et al. [15] were the first that fractionalized the dengue ODE model using the Riemann–Liouville definition. Later, Diethelm [16] proposed a more sophisticated way of fractional dengue model by using the Caputo definition. Diethelm solved the problem of mismatched dimension in the earliest model by Pooseh et al. Sardar et al. in [17, 18] showed the significance of order incorporated with memory in the dengue transmission. In this paper, we will not only be considered the adult stage of mosquito, but also the dynamic of the aquatic stage of mosquito. We will be considered the work of Diethelm [16] in fractionalize the ODE system, but with the assumption that any parameter related to the birth and death in both human and mosquito population is assumed to be memory independent.
The rest of the paper is organized as follows. In Sect. 2, the formulation of the model is presented. The stability analysis of the equilibrium points is given in Sect. 3. In Sect. 4, the sensitivity analysis is performed. Section 5 consists of the numerical experiments done to verify the theoretical analysis shown in Sects. 3 and 4. Lastly, the conclusion is in Sect. 6.
2 Mathematical model
For many years, fractional operators have been interpreted in several ways that fit the concept of their integrals or derivatives. For instance, the most common definitions of the fractional derivative are the Riemann–Liouville derivative, the Caputo derivative, and the Grünwald–Letnikov derivative. In this paper, the Caputo derivative is used, since the classical initial conditions can be used without encountering any problem during obtaining the solutions.
In the formulation of the model, we assume that the total human and vector population to be constant. We also assume that the infection is produced by only one serotype of dengue virus. The dynamics of the female \(Aedes\) mosquito are included aquatic stage (\(A_{m}\)) and adult stage. The adult female mosquito (M) is divided into two compartments that are susceptible \(M_{s}\) and infected \(M_{i}\). The total human population, \(N_{h}(t)=H\) is partitioned into three compartments: susceptible, \(H_{s}\), infected, \(H_{i}\), and recovered, \(H_{r}\) individuals.
Description of the dengue model (3) parameters and their possible feasible ranges
Parameter  Biological meaning  Range of values  References 

q  Proportion of eggs  0–1  [8] 
ϕ  Oviposition rate  0–11.2 per day  [8] 
\(\sigma_{A}\)  Transition rate from aquatic to adult  0–0.19 per day  [8] 
\(\mu_{A}\)  Average aquatic mortality rate  0.01–0.47 per day  [8] 
\(1/\mu_{m}\)  Average lifespan of adult mosquito  11–56 days  [21] 
\(1/\mu_{h}\)  Average lifespan of human  73–75 years  [22] 
b  The biting rate  0–1 per day  [23] 
\(\beta_{m}\)  Transmission probability from human to vector  0.375  [6] 
\(\beta_{h}\)  Transmission probability from vector to human  0.375  [6] 
\(\gamma_{h}\)  Recovery rate in the host population  0.328833 per day  [21] 
\(\rho_{h}\)  Diseaseinduced death rate for humans  10^{−3}  [24] 
3 Existence and stability of equilibrium points
Suppose that the set \(\varOmega=\{(A_{m},M_{s},M_{i},H_{s},H_{i},H_{s}) \in\mathbb {R}^{5}_{+}:H_{s}+H_{i} \leq K; 0\leq M_{s}+M_{i} \leq Q_{1}, \mbox{and } Q_{1}\geq\frac {\sigma_{A} A}{\mu_{m}}; 0\leq A_{m} \leq Q_{2}, \mbox{and } Q_{2} \geq q\phi M \} \) is the region of biological interest, that is, positively invariant to the system (4). The proof is similar to the proof in [20].
Proposition 1

if\(R_{m}\leq0\), there is a diseasefree equilibrium (DFE), known as trivial equilibrium, \(E_{0} = (0,0,0,H,0)\);

if\(R_{m}>0\), there is a biologically realistic diseasefree equilibrium (BRDFE), \(E_{1} = (\bar{A_{m}}, \bar{M_{s}},0,H,0)\).
In modeling the infectious disease, the basic reproduction number \(R_{0}\) is important. This threshold quantity value represents the expected number secondary cases produced in a completely susceptible population, by a typical infected individual during its entire period of infectious [25]. In this paper we use the nextgeneration matrix approach to obtaining the \(R_{0}\). By following [26], we are thus lead to the following proposition.
Proposition 2
If\(R_{m}>0\), then the basic reproduction number associated to the system (4) is\(R_{0}^{2}= \frac{b^{2\alpha}\beta_{m} \beta_{h}}{(\gamma _{h} + \mu_{h}+\rho_{h})\mu_{m}} \frac{\bar{M_{s}}}{H}\). The BRDFE is locally asymptotically stable if\(R_{0}<1\)and unstable otherwise.
By the Routh–Hurwitz stability criterion, \(E_{1}\) is locally asymptotically stable if and only if \(R_{0}<1\), otherwise the BRDFE is unstable.
Therefore, \(R_{0}>1\), it implies \(R_{0}^{2}>1\), and, as a result, from (10) there exists a unique positive endemic equilibrium \(E_{2}=(A_{m}^{*},M_{s}^{*},M_{i}^{*},H_{s}^{*},H_{i}^{*})\). Thus, we have the following result.
4 Sensitivity analysis
Sensitivity analysis helps us to identify the parameters that have a big impact on the disease transmission. Such information is important not only for experimental design but also for data assimilation and reduction to complex nonlinear models [27]. Normally, in the epidemiological model, the analysis is used to discover parameters that have greatest influence on the basic reproduction number \(R_{0}\) and should be targeted by the control strategies.
The sensitivity indices of the \(R_{0}\) are determined to allow us to measure which parameter has the greatest influence on the changes of \(R_{0}\) and, hence, the greatest effect in determining whether the disease can be eliminated in the population. The normalized forward sensitivity index of a variable (\(R_{0}\)) with respect to a parameter is the ratio of the relative change in the variable (\(R_{0}\)) to the relative change in the parameter. When the variable is a differentiable function of the parameter, the sensitivity index can be defined using the partial derivatives [27].
Definition 1
(cf. [28])
Sensitivity indices of \(R_{0}\) evaluated at the baseline parameter values for \(\alpha=0.9\)
Parameter  Sensitivity index 

q  −0.0097 
ϕ  −0.0097 
\(\sigma_{A}\)  +0.5073 
\(\mu_{A}\)  +0.0073 
\(\mu_{m}\)  −1.0097 
\(\mu_{h}\)  −5.5798 × 10^{−5} 
b  +0.90 
\(\beta_{m}\)  +0.50 
\(\beta_{h}\)  +0.50 
\(\gamma_{h}\)  −0.5092 
\(\rho_{h}\)  −0.0015 
Sensitivity indices of \(R_{0}\) evaluated at the baseline parameter values for \(\alpha=0.6\)
Parameter  Sensitivity index 

q  −0.0097 
ϕ  −0.0097 
\(\sigma_{A}\)  +0.5073 
\(\mu_{A}\)  +0.0073 
\(\mu_{m}\)  −1.0097 
\(\mu_{h}\)  −5.5798 × 10^{−5} 
b  +0.60 
\(\beta_{m}\)  +0.50 
\(\beta_{h}\)  +0.50 
\(\gamma_{h}\)  −0.5092 
\(\rho_{h}\)  −0.0015 
5 Numerical simulations and discussion
The selection of order α is similar to the approach taken in the previous work [20]. We solved the system for a various choice of α values. Specifically, we have used the formula of \(\alpha \in\{k/100:k=1,2,\ldots,100\}\). From that, we found that a reasonable range of α is between 0.7 and 1. Note that \(\alpha=1\) gives the same solution as the integer order system.
Notice that, in Table 3, the index of the parameter b is decreased as we reduced the order α. This explained the relationship of \(R_{0}\) with the α. The threshold quantity \(R_{0}\) defined here is a memorydependent threshold quantity as \(R_{0} \propto b^{\alpha}\). As the memory of the biting rate of the mosquito increased (\(\alpha\rightarrow0\)), the basic reproduction number increased, thus, we have an increase of the transmission rate of the disease. This tells us that, in the fractional sense, the sensitivity of the biting rate parameter b of the system depends on the α value.
The parameters \(\mu_{m}\) and \(\gamma_{h}\) have a negative sensitivity index, the most negative being the mortality rate of the mosquito, \(\mu _{m}\), with \(\varUpsilon_{\mu_{m}}^{R_{0}} =  1.0097\). This tells us that if we increase the parameter value of \(\mu_{m}\) by 10%, then the basic reproduction number \(R_{0}\) will be decreasing by approximately 10%. This agreed with the numerical simulation obtained in Fig. 5.
6 Conclusions
In the present paper, a fractional order dengue epidemic model is studied. The local stability of the equilibrium points has been determined theoretically and verified numerically. The numerical experiment revealed that, for the fractional order model, the speed of convergence of the solution is slower than in the classical integer model. From the epidemiological point of view, this result is crucial, because it affects the time needed to eliminate the disease.
Also, we have performed a sensitivity analysis in order to determine the relative importance of the model parameters in the transmission of dengue. Such information allows us to determine the effective control and prevention measures, in monitoring the spread of the disease.
We believed that this work can provide an important tool to the public health practitioners in dealing with the increase of dengue cases in the reallife situation, especially in Malaysia.
Notes
Acknowledgements
The authors would like to thank the editor and reviewers for the constructive comments and useful suggestions, which improved the manuscript.
Availability of data and materials
Not applicable.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Funding
The authors acknowledge financial support by the Ministry of Education Malaysia and Universiti Teknologi MARA and also partial financial support by the Universiti Putra Malaysia providing Putra Grant GPIPS/2018/9625000.
Competing interests
The authors declare that they have no competing interests.
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