Introduction
Lymphatic filariasis has an effect on almost 120 million individuals all over the world. The disease may cause a chronic morbidity if the persons who are infected are left untreated. It is endemic in many parts of tropical countries. To prevent worldwide parasite transmission, the World Health Organization initiated the Global Programme to Eliminate Lymphatic Filariasis (GPELF) by eliminating filarial parasites from their human hosts (Molyneux & Zagaria, 2002). Various GPELF implementations are done in many participating countries. In 2004 alone there were more than thirty countries have started elimination program and this number is still rising. Various degrees of success have emerged as a result of the implementation of this program. Although it was reported that in some places the program has interrupted the transmission, in many other places the program could not stop the transmission of the disease (WHO, 2005). It has been argued that strategic choices and operational or biological factors contribute to the success or failure of the program. In general, it is difficult to evaluate the success or the failure of a health program, especially in the beginning of the program.
A mathematical model provides useful tools for planning and evaluation of control program in disease elimination (Goodman, 1994). In our earlier work (Supriatna et al., 2009) we develop a mathematical model for the transmission of Lymphatic Filariasis disease in Jati Sampurna, Indonesia. In Indonesia, the disease is already alarming. For example, the incidence of filariasis in Jati Sampurna (a district in the West Java province) is more than 1%. Within less than five years since the date of the publication confirming that Jati Sampurna is an endemic area, almost all regions nearby Jati Sampurna, and other relatively far distance areas are affected by the disease, and some of them are also categorized as endemic areas. Other cases of filarial prevalence are reported outside Java island, such as in Alor islands (the province of Nusa Tenggara Timur). On Alor islands, both B. timori and W. bancrofti are circulated, with a prevalence of up to 20% (Supali et al., 2002). Indonesia joined the GPELF since 2001 and implemented administration of a single dose regimen of diethylcarbamazine (DEC) and albendazole in endemic areas (Krentel et al., 2006). Our previous model tries to capture the effectiveness of this scenario in the attempt of controlling the spread of the disease, inspired by the transmission of the disease in Jati Sampurna.
The model assumes that acute infected humans are infectious and treatment is given to a certain number of acute infected humans found from screening process. The screening is done every time a new chronic reported. The treated acute individuals are assumed to be remains susceptible to the disease. The model is analyzed and it is found a condition for the existence and stability of the endemic equilibrium. A well known rule of thumb in epidemiological model, that is, the endemic equilibrium exists and stable if the basic reproduction number is greater than one, is established. Moreover, it is also shown that if the level of screening is sufficiently large, current medical treatment strategy will be able to reduce the long-term level of incidences. However, in practice it is not realistic for the following reasons.
One important concept in mathematical epidemiology regarding transmission of a disease is the basic reproduction number. It measures the number of new infections caused by an infective during the life time of the infective. Although our previous model is able to gain some insights on how the provision of a medical treatment can reduce the level of disease incidence, however it is worth to note that the basic reproduction number does not depend on the level of the treatment. It means that the treatment, no matter how large it is, will not be able to annihilate the endemicity of the disease. This is some what surprising and unexpected, because normally, in many epidemiological models, any medical treatment should reduce the basic reproduction number.
Our earlier work shows that the medical treatment given in the model scenario cannot eliminate the disease, in terms of reducing the basic reproduction number. Our previous model has also ignored an important factor in the transmission stage, namely the time delay. The model has assumed that once an individual infected, he/she become infectious without any delay. Nonetheless, the reproduction number can be reduced by giving additional treatments, such as reducing the biting rate and mosquito’s density. This suggests that there should be a combination of treatment to eliminate the disease. In this topic we review our earlier model of the filariasis transmission and a new model based on the earlier work is developed and analysed. The topic gives a step by step improvement of our previous model. We do not carry out a heavy mathematical analysis instead some simulations of the models are presented. Finally, some interpretations are derived from the results.
Mathematical model with no time delay in infection period
To formulate the model we use the assumptions that initially the human population is virgin, i.e. there is no infection, and the total population of human is constant. We assume that there is an invasion by few infective individuals of either human or mosquitoes. There is only one species of worm and one species of mosquito, and there is no vertical transmission of the disease, either in human or mosquitoes populations. The human population is divided into three subpopulations, susceptible SH, infected-carrier A and infected-chronic K, with the total number of the population given by NH . We assume that once a human individual is infected then without any delay the individual becomes infectious. However, we strictly assume that transmission to the mosquitoes is only from the acute population. All chronic individuals are isolated perfectly. This strict assumption will be relaxed in some simulation later on. The mosquitoes are divided in two subpopulations, susceptible SV and infected IV mosquitoes, with the total number NV . Related parameters in the model are the human recruitment rate RH , human death rate
, successful rate of transmission from mosquitoes to susceptible human pH , mosquitoes biting rate on human b , symptomatic rate
, mosquitoes recruitment rate RV , mosquitoes death rate
and successful rate of filarial transmission from human to susceptible mosquitoes pV . If the medical treatment is quantified by n number of people screened by the health authority, for every single chronic found, with the successful probability of the treatment p0, then the governing differential equations describing the mathematical model of the disease transmission are given by the following equations:
We can evaluate the effectiveness of the medical treatment n in managing the disease within the presumed policy, by inspecting its appearance in the endemic equilibrium and in the basic reproduction number. From the model, by assuming the host and vector populations
are constant, so that
we found the endemic and non-endemic equilibria of the model related to the basic reproduction number
We also establish a theorem saying that "if
then the endemic equilibrium of the system is locally asymptotically stable, otherwise it is unstable". The details of the derivation can be seen in Supriatna et al. (2009). In terms of controlling the disease it means that we should keep the basic reproduction number as low as possible so that it is lower than the unity by adjusting the level of the treatment n. The basic reproduction number is obtained using the next generation matrix (see Diekmann & Heesterbeek, 2000). It is worth to note that the basic reproduction number does not depend on the level of screening n, and hence, current presumed method of treatment does not annihilate the endemicity of the disease. This is partially because of the re-susceptibility of the treated population. However, our earlier work show that it indeed reduces the number of the acute population in the long-term as shown in the following section.
Numerical examples for the model with no delay time in infection period
To facilitate some interpretation regarding the results in our previous work, we present numerical examples using the parameters shown in Table 1.Powersim code for other models in the preceding section can be easily modified from this basic model. We give two examples: the first example assumes that a virgin population is invaded by acute infected human (via human immigration) and the second example assume that a virgin population is invaded by infected mosquitoes (e.g. a container un-intentionally transporting infected mosquitoes from an endemic area).
|
Parameter |
Value |
|
Parameter |
Value |
|
Rh |
2,500 |
Rv |
1,000,000 |
|
 |
1/70 |
 |
365 (1/30) |
|
 |
0.25 |
b |
250 |
|
|
Ph |
0.01 |
Pv |
0.1 |
|
|
n |
0 |
P0 |
0.75 |
Table 1. The main values of parameters used in the numerical examples
Figure 1 depicts the following scenario. Suppose that a population is initially virgin and stays at its equilibrium. We assume that it is then invaded by 10 acute infected human individual, with all the mosquitoes are also virgin. Using the parameter values given in Table 1, we obtain the value of the basic reproduction number is 3.02, which means that the disease will increase if there is no intervention. Figure 1 shows the dynamics when there is no treatment (n=0). The effect of the values of the parameters on the basic reproduction number is clear from equation (6). However its effect on the dynamics and the endemic equilibrium is not so obvious. Figure 2 shows the same dynamics as in Figure 1, with an addition that in the 25th year after the invasion of infective individuals there is a medical treatment with n=200. Figure 3 shows the same dynamics as in Figure 1, but here the treatment is carried out as early as the 5th year after the invasion with only 100 screening (n=100). These figures reveal that an early average treatment is better than a late huge treatment.
The scenario in Figures 1 to 3 assumes that the medical treatment given to the infected persons does not affect the transmission parameters given in Table 1 other than the screening parameter n. The screening parameter n does not appear in the basic reproduction number formula (6). Hence, this treatment does not affect the endemic status of the disease. In reality, there are some treatments that could alter the values of the disease transmission parameters. For example, if we assume that some portion of the population is treated by giving them some insect repellent, then the biting rate b could be altered. Let us assume that an effective insect repellent could decrease the biting rate to 50% of its current level. Figure 4 shows the dynamic when there exist this effective insect repellent, and used from the 5th year in the absence of the medical treatment (n=0) and Figure 5 shows the same scenario as in the previous figure but in the presence of the medical treatment with n=100 given by the same time as the insect repellent provision. Compared to the case when there is no insect repellent (Figure 1), the introduction of the insect repellent is significantly reduces the level of the disease outbreak (Figure 4) and in the same time reduces the endemic level of the disease (changing the value of the basic reproduction number from 3.02 to 1.51). Meanwhile, if we also apply the medical treatment with only average treatment (n=100), then the level of the outbreak is relatively the same, but apparently with a shorter period of the outbreak (Figure 5).
Fig. 1. The dynamics of infected population when there is no medical treatment after the invasion of 10 infected human.
Fig. 2. The dynamics of infected population when there is a medical treatment in the 25th year with n=200.
Fig. 3. The dynamics of infected population when there is a medical treatment in the 5th year with n=100.
Fig. 4. The dynamics of infected population when there is an effective insect repellent which changes the biting rate to its 50% of the current level with no medical treatment in the 5th year after the disease invasion (n=0).
Other scenarios could also be considered. Some are already known to be ineffective if only applied solely, such as fogging (Soewono & Supriatna, 2002) and other still unexplored, such as newly developed method for shortening mosquitoes life expectancy (Turley et al., 2009). Supposed that with some ways we can reduce the mosquito life expectancy down to 50 % of the existing level (from 30 days as in Table 1 to 15 days). Figure 6 shows its dynamics which is the same as the dynamics in Figure 4.
Fig. 5. The dynamics of infected population when there is an effective insect repellent which changes the biting rate to its 50% of the existing level combining with average medical treatment in the 5th year after the disease invasion (n=100).
This is not surprising considering the form of the basic reproduction number (equation (6)), in which the decrease of biting rate acts the same as the decrease of the mosquitoes life expectancy (equivalently the increase of the mosquitoes mortality rate μν ). If we decrease both values, i.e. the values of the biting rate and the life expectancy, then their effect in reducing the basic reproduction number doubled, such as shown by Figure 7, resulting in the value of the basic reproduction number to be less than one (only 0.755), which means the disappearance of the disease is guaranteed. Even in the absence of medical treatment, Figure 8 shows that if we do this strategy before one year has elapsed then the disease does not have any chance to grow. This suggests that preventive action is better than curative action.
In the previous example we assume that invasion is done by infected human. Next in the following example we assume that invasion is done by infected mosquitoes from an endemic area. Considering the short distance of the mosquito flight, we can assume that this invasion happens un-deliberately, for example via container and other transportation modes. However, considering the stability theorem of the endemic equilibrium point in our previous work (Supriatna et al., 2009), we expect that the long term behaviour of the disease transmission dynamics would be the same as in the first example. In other words, there is an independence of initial values, such as illustrated by Figure 9, in which we assume that there are 100 infected mosquitoes invades the virgin population as described in the first example (Figure 1).
Fig. 6. The dynamics of infected population when there is an intervention which changes the mosquitoes life expectancy to its 50% of the current level with no medical treatment in the 5th year after the disease invasion (n=0).
Fig. 7. The dynamics of infected population when there is an intervention which changes both the mosquitoes life expectancy and the biting rate to their 50% level with no medical treatment in the 5th year after the disease invasion (n=0).
Fig. 8. The dynamics of infected population when there is an intervention which changes both the mosquitoes life expectancy and the biting rate to their 50% level done before one year after the disease invasion has elapsed (n=0).
Fig. 9. The dynamics of infected population when there is no medical treatment after the invasion of 100 infected mosquitoes into a totally virgin population. The figure is similar to Figure 1 in which invasion is done by 10 infected human.
