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A mathematical model with control to analyse the dynamics of dengue disease transmission in urban Colombo

Dengue is a mosquito-borne disease. It has been an important public health problem particularly in the tropical and sub-tropical regions in the world, for which no vaccine or successful treatment has been found. Therefore, prevention and control play a vital role in minimising the risk of dengue in vulnerable populations. Various mathematical models such as the SIR model have been developed to understand the transmission dynamics of dengue. The main drawback of these dynamic models is the lack of predictability with respect to external factors such as climate, geography, demography and human behaviour due depends heavily on various external factors such as climate is a critical parameter of these models and is responsible for the local transmission of the disease. In this study, the mosquito density is modelled with respect to changing levels of climate favourable for mosquito reproduction. A climate risk index developed using fuzzy set theory is used to vary the mosquito density with respect to rainfall and temperature and this measure is included in the SIR model. Finally, two measures, u 1 and u 2 , are introduced to control adult mosquitoes and growing juveniles using two methods. u 1 and u 2 as random processes from uniform distribution. The second method involves continuous and constant control measures over time. The numerical results of the SIR model suggest that the dynamics of infections has changed and the number of


INTRODUCTION
Dengue fever (DF) and its severe form, dengue haemorrhagic fever (DHF) are the most common and widespread vector borne diseases in the world. According to the World Health Organization (WHO), dengue disease is ranked as one of the most critical infectious diseases with severe impact on public health and well-being of the society (Racloz et al., 2012;Thai, 2012). The disease has spread to almost all the tropical and sub-tropical parts in the world and it has been estimated that nearly 2.5 billion people in more than 100 countries are at risk (Bhatia et al., 2013). Globally, every year, approximately 50 million dengue infections occur; and half a million DHF cases require hospitalisation with over 20,000 deaths (Thai, 2012;Bhatia et al., 2013;Murray et al., 2013). The economic impact of DF/DHF is massive, communities. This impact varies and can include deaths, medical expenditure for hospitalisation of patients and their careful clinical management, loss in productivity of the affected workforce, strain on healthcare services due to sudden, high demand during an epidemic, considerable expenditures for large-scale emergency control actions taken by the government in an outbreak etc. (Lloyd, 2003). Journal of the National Science Foundation of Sri Lanka 46 (1) to public health and the well-being of Sri Lankan people. According to the Epidemiology unit, Ministry of Health, Sri Lanka ( half of the year 2016, there were nearly 21,500 dengue deaths. Of them, the majority have been from Colombo. Colombo is the capital and the largest city in Sri Lanka; its rapid unsystematic urbanisation and increased human movements have resulted in Colombo becoming a highly vulnerable geographic area for dengue disease (Thalagala, 2012).
for dengue disease and several researches for a vaccine are currently in progress (Parks & Lloyd, 2004;Bhatia et al., 2013). For the time being, the only mechanism for preventing and controlling DF/DHF is to ensure prompt diagnosis of cases of fever and appropriate clinical management during the hospitalisation, to reduce humanvector contact, and to control larval habitats (Parks & Lloyd, 2004).
Dengue is a vector borne disease with a complex transmission process. Dengue transmission dynamics are patterns of humans and mosquitoes, and the degree of interaction between humans versus mosquitoes. Further, both humans and mosquitoes interact heavily with the environment as well. The variation in climate factors such to dengue disease transmission. Temperature and rainfall interfere at all stages of mosquito development from the emergence and viability of eggs, to the size and longevity of adult mosquitoes, as well as their dispersal over space (Allis et al., 2011;Medeiros et al., 2011). Demographic factors such as population density, education level, human awareness and household income are also vital determinants of dengue spread. Human mobility creates a large number of human contacts and this may speed up the dengue disease transmission (Allis et al., 2011).
Exploring the complexity of infectious disease systems such as dengue disease and understanding its transmission dynamics is important to identify control measures effectively. Mathematical models of dengue transmission reveal useful information about its complex dynamic nature and the information can be applied to control the spread of the disease and minimise the disease burden.
This paper focuses mainly on the importance of controlling dengue mosquitoes at two stages of their life cycle, namely, growing juveniles and adult mosquitoes.
We consider the SIR model for dengue, which describes the transmission of dengue disease in terms of population dynamics. The human population is divided into three compartments, namely, susceptible (S), infected (I) and recovered (R) while the mosquito population is divided into two compartments, namely, susceptible and infected (Estevaa & Vargas, 1998;Pongsumpun, 2006;Ma & Xia, 2009). The mosquito density is a critical parameter in these mathematical models since it is the driving factor for the local transmission of dengue and it depends heavily on various external factors, particularly climate. We model this parameter using a discrete nonlinear dynamic model. The climate effect on mosquitoes is modelled using fuzzy set theory. The constructed fuzzy measure describes the level of favourability to dengue mosquitoes from climate. This risk measure is also included in the discrete non-linear dynamic model for mosquitoes.
We introduce two measures u 1 and u 2 to control adult mosquitoes and growing juveniles and implement control measures with several combinations of u 1 and u 2 . First we consider the two control measures as uniformly distributed random variables. Next we consider that the two measures are constants and they are continuously implemented over time. Finally the entire system is numerically solved to investigate the consequences of these control strategies. The fourth order Runge Kutta scheme is used to obtain the solutions ( The Einstein Sum is defined as (Zimmermann, 2010) ) ( . (Zimmermann, 2010

Mathematical model for dengue transmission
We consider the mathematical model for dengue transmission described by a system of non-linear ordinary differential equations (Estevaa & Vargas, 1998;Pongsumpun, 2006). This model describes the interaction between susceptible, infected and recovered human populations together with susceptible and infected mosquito populations.
v dt . Now we assume that the mosquito population v N changes in time (is not a constant) that is However, we assume that the total human population is a constant hence hence n can be modelled as a function of time t ) (t n .
Although v N depends on time t, it should be noted that given time t. Therefore, system (3) can be reduced to

Model for mosquito density
The development of the discrete non-linear model for mosquito reproduction is discussed here.
.. (6) where s is the per capita survival rate of the adult mosquitoes and p is the per capita growing probability of juveniles.
The number of juveniles at time t, (J(t)) is regulated by the adult mosquito density in the previous time step. Thus, we can write ... (7) Using Gompertz model in equation (5) and we write ) and we write .
... (8) Now equation (6) can be expressed as . ... (9) Modelling the climate force rainfall is required to make breeding sites available for mosquitoes and the breeding sites are washed out due to the heavy rainfall, which is over 55 mm (Huang et  are defined respectively to represent respectively to represent the effect from eight weeks leading rainfall (RF), and immediate temperature (TEMP) to create an unfavourable environment for dengue mosquitoes as The trapezoidal-shaped membership functions given i ... (11) The trapezoidal-shaped membership functions given in equations (10) and (11) are illustrated in Figure 1.

Fuzzy operator
MES (x and it is given by ) ( ... (12) The behaviour of the overall effect ) (x U MES = for different membership values of RF and TEMP is represented in Figure 2. .. (13) Here, t is the time.
This fuzzy measure of climate favourability is additively included into our model and it is assumed that the adult mosquitoes in time assumed that the adult mosquitoes in time take the advantage of this climate = take the advantage of this climate favourability. Now . Finally, we can write our fuzzy discrete-time dengue mosquito density model as ).

Mathematical model with control measures
It should be noted that there is no control over the climate since it is a natural phenomenon, but the impact of climate conditions can be controlled.

March 2018
Journal of the National Science Foundation of Sri Lanka 46 (1) against adult mosquitoes, such as insecticide space sprays or residual applications (Parks & Lloyd, 2004 . This control copepods (small crustaceans) that feed on mosquito larvae] to kill or reduce larval mosquito populations in water containers and chemical methods against the mosquito aquatic stage for use in water containers (e.g. temephos sand granules) (Parks & Lloyd, 2004).
With these two control measures namely 1 ∈ u ∈ and ∈ and 2 u ∈ − ∈ , equation (15) can be expressed as

Implementing control measures
It should be noted that the scenario where It should be noted that the scenario where . We perform the simulation with four . We perform the simulation with four combinations of the two random variables as i.

Method II:
It is observed that rainfall is the driving force for dengue outbreaks in urban Colombo due to the increased level of dengue mosquitoes during the rainy seasons. The is favourable for dengue mosquitoes throughout the year. The control measures should be implemented to avoid the environmental favourability produced by this changing levels of rainfall. Thus it is expected to have a constant level of effective control activities from the public to eradicate dengue mosquitoes from the environment. Therefore, we C (as an average measure). Appropriate measures are then implemented to control the mosquito density and hence we expect to reduce the number of dengue infections in humans. We investigate the conditional distribution of infected humans given eight weeks leading rainfall .

Construction of fuzzy membership functions
according to equations (10) and (11)  , input ) (t n and solve the system (4) using Runge and solve the system (4) using Runge Kutta method of order 4. Figure 3 shows the simulated mosquito density dynamics without control measures according to equation (15) [ Figure 3(a)] and the corresponding dynamics of the infected human population [ Figure 3(b)]. The weekly average rainfall and temperature data in Colombo from year 2006 to 2011 is used for the simulation. The mosquito density shows a periodic variation in time and this is due to the varying levels of climate favourability. The infected human population is also periodic with some increasing trend.

RESULTS AND DISCUSSION
The dynamics of the mosquito density with uniformly distributed two control measures are shown in Figure 4. The corresponding infected human population dynamics is represented in Figure 5. It can be seen that the distribution of infected human population has changed after introducing  a small level of control, which varies randomly with respect to time. Figure 5 reveals theoretically that the randomly varying control measures reduce the number of infections compared to the dynamics of the infected human population in Figure 3 (right). Figure 6 represents the dynamics of mosquito density with respect to different levels of control measures. It can be seen from Figure 6 that the simulated dengue mosquito density shows a reduction as the two control measures 1 u and 2 u are increased. This simulation indicates that the effective constant continuous control strategies have a negative impact on the dengue mosquitoes. This mosquito density dynamic is used to investigate the behaviour of the infected human distribution given the rainfall variation and control measures and the results are shown in Figure 7. The results indicate that the dynamics of the infected humans have changed and the number of infections have reduced once the control strategies are implemented. The results in Figure 7 suggests that a eradicate the dengue infections in urban Colombo.

CONCLUSION
A fuzzy based mathematical model was developed to analyse mosquito density considering only the climate variation. For further improvement, other external factors favourable for mosquitoes can also be included. Optimal control techniques can be used to investigate the point in time where the control measures should be introduced to minimise the magnitude of the disease outbreak.