**4. Conclusion**

438 Current Topics in Tropical Medicine

Suppose now that we have another scenario of treatment, that is giving prophylaxis to all the populations (set *n*=0, simply to evaluate the effectiveness of this prophylaxis). The prophylaxis works by inhibiting the growth of the worms inside human, say by delaying the recruitment into the acute population *A* from the sub acute population *Am* . Technically this

the effect of the prophylaxis application. It seems that all the graphs increase exponentially (upper figure), but in fact at the end they end up to their stable equilibrium (lower figure) with different speed and different peak. This indicates that controlling the density of worm inside the body of infective human is effective in reducing the number of filarial infection. The model assumes that the delivery of prophylaxis has a result in a constant effect over time, which doesn't reflect the reality. To increase the realism, we should consider the decrease of prophylaxis effectiveness by modifying or refining the model. Nevertheless, we still can apply the current model by only believing the short-term prediction given by the model, say only in one to two years prediction and use it as guidance in a periodic delivery

The introduction of a single exposed compartment is not without a problem. Getz and Lloyd-Smith (2006) showed that a single exposed compartment will produce an exponentially-distributed sojourn time in the exposed stage. Referring to our delay model

poor match to the real distribution of latent periods. Plant and Wilson (1986) pointed out that the drawback can be resolved by introducing a distributed delay or staging delay time approach comprising of *k* classes of sub acute or exposed individual. This approach gives a gamma-distributed total time of individuals staying in the exposed class with mean 1 /

In this part we use this approach (see also Getz and Lloyd-Smith (2006)) to our delay model by introducing multiple exposed compartments which is more appropriate to the disease like filariasis which has more than one different exposed stages. The general model is the

> 1 0 1 1 1 *<sup>m</sup> HV H <sup>m</sup> m Hm H H*

<sup>1</sup> ( ) 2,..., *mi mi*

*dA p n AA kA A A , i k dt <sup>N</sup>*

*dA pn A kA A <sup>A</sup>*

The system is much more complex since it consists of 15 differential equations compared to just 6 differential equations in the previous model. However, numerical example in Figure 15 shows that for 10 *k* (and also for any 1 *k* ), the simpler model of equations (7) to (11), qualitatively, is a good approximation of the more realistic model of the same equations but

 

0

0

*H*

*H*

2

 

*dA bp I S p n AA kA A dt N N*

 

. Note that a fixed time delay 1 /

same as equations (7) to (11) except that equations (8) and (9) are replaced by

*mi i H mi*

 

*mk H*

*dt N*

 , (13)

) in the model. Figure 14 shows the effect of delay for various sojourn time due to

(or equivalently the sub acute sojourn

while its modus is at 0 , which is a

is obtained whenever the number

, (12)

. (14)

is done by varying the values of the transition rate

of a mass drug administration program.

of delay stages *k* approaches the infinity.

(equations (8) and (9)), this distribution has mean at 1 /

time 1 /

and variance <sup>2</sup> 1/( ) *k*

In this chapter we review a mathematical model of filarial transmission in human and in mosquitoes. Some simulations are carried out to obtain some insights regarding the transmission and possible actions to control the transmission. Some refinement of the model could be done in many directions to increase the realism of the model and to obtain a more accurate prediction. New directions may include the evolutionary, sosio-economics, and climatology aspects of the disease (Levin, 2002).

In the evolutionary issues of epidemiology, some agents of diseases may develop resistance to certain drug. It is worth to explore how this affects the transmission of the diseases. In many situations, especially in developing countries, there always competing interests related to limited resources and budget. There are many other important diseases, other than filariasis, needs for attention. Choosing the right priorities are among the concerns of health managers and authorities. In the absence of sufficient health budget it is important to address questions like the long term consequences when the treatment is terminated, either purportedly, e.g. because the budget is re-allocated to a higher priority health problem (to other endemic places of the same disease or to other disease problems) or inadvertently (due to the decreasing compliance of the program implementation). This is an example of sosio-

Lymphatic Filariasis Transmission and Control: A Mathematical Modelling Approach 441

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Getz, W.M. & Lloyd-Smith, J.O. (2006). Basic Method for Modeling the Invasion and Spread

Krentel, A.; Fischer, P.; Manoempil, P.; Supali, T.; Servais, G. & Ruckert, P. (2006). Using

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Molyneux, D. & Zagaria, N. (2002). Lymphatic Filariasis Elimination: Progress in Global Programme Development. *Ann. Trop. Med. Parasitol*. 96 (Suppl 2): S15-40 Plant, R.E. & Wilson, L.T. (1986). Models for Age-Structured Population with Distributed

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Maturation Rates. *J. Math. Biol.* 23, 247-262.

aux Rate\_A = ((b\*Iv\*Sh\*ph)/Nh)-Mh\*A-((p0\*n\*A\*d\*A)/Nh)-d\*A

aux Rate\_Sh = Rh-((b\*Iv\*Sh\*ph)/Nh)-Mh\*Sh+((p0\*n\*A\*d\*A)/Nh)

aux Rate\_Iv = ((b\*Sv\*A\*pv1)/Nh)-Mv\*Iv

aux Rate\_Sv = Rv-((b\*Sv\*A\*pv1)/Nh)-Mv\*Sv

init Sv = Rv/Mv flow Sv = +dt\*Rate\_Sv

aux Rate\_K = d\*A-Mh\*K

aux Nh = Sh+A+K const b = 250 const d = 0.25 const Mh = 1/70 const Mv = 365/30

const n = 0 const p0 = 0.75 const ph = 0.01 const pv1 = 0.1 const Rh = 2500 const Rv = 1000000

**7. References** 

1862-1863

York, USA

economics issues in epidemiology (Supali *et al*., in prep.). Climate change also regarded as a factor contributes to current emerging and re-emerging infectious diseases. For example, since the global temperature is rising then suitable habitat for mosquitoes becomes wider. It is reported that many parts in the globe of previously free from mosquito is now invaded by incoming mosquitoes. To obtain a better prediction of global filarial transmission, this climatology aspect also should be considered. We believe that there are many other venues are possible for future research in mathematical aspect of filariasis transmission.
