- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

ISRN Biomathematics

Volume 2012 (2012), Article ID 930130, 9 pages

http://dx.doi.org/10.5402/2012/930130

## Transmission Dynamics of Lymphatic Filariasis: A Mathematical Approach

Department of Mathematics, University of Zimbabwe, Box MP 167 Mount Pleasant, Harare, Zimbabwe

Received 27 August 2012; Accepted 19 September 2012

Academic Editors: G. Bocharov and M. T. Figge

Copyright © 2012 C. P. Bhunu and S. Mushayabasa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An epidemiological model for the spread of lymphatic filariasis, a mosquito-borne infection, is developed and analysed. The epidemic thresholds known as the reproduction number and equilibria for the model are determined and stabilities analysed. Results from the analysis of the reproduction number suggest that treatment will somehow contribute to a reduction in lymphatic filariasis cases, but what it does not show is the magnitude of the reduction, a part answered by the numerical simulations. Numerical simulations show that even when all lymphatic filariasis cases displaying elephantiasis symptoms are put on treatment it will not be able to eradicate the disease. This result suggests that effective control of lymphatic filariasis may lie in treatment for those displaying symptoms as well as chemoprophylaxis for the exposed.

#### 1. Introduction

Lymphatic filariasis, a debilitating disease, is one of the most prevalent and yet one of the most neglected tropical diseases with serious economic and social consequences [1, 2]. Lymphatic filariasis affects women, men and children of all ages. It is a mosquito-borne disease caused by tissue-dwelling nematodes of * Brugia malayi, Brugia timori*, and * Wuchereria bancrofti* species [1, 3] and is estimated to affect about 120 million people worldwide [4–6].* Wuchereria bancrofti* is responsible for 90% of the cases and is found throughout the tropical and subtropical areas of the world; *Brugia malayi* is confined to southeast and eastern Asia; * Brugia timori* is found only in Timor and its adjacent islands [7]. Infection leads to lymphedema, a buildup of fluid due to impaired function of the lymph vessels, in only a small proportion people, even in areas of intense transmission [8], as most people with long-term infections are clinically asymptomatic. Recurrent bacterial infections in some lymphedema patients lead to elephantiasis [9].

Filarial parasites are a major cause of morbidity and therefore hinder socioeconomic growth in parts of Asia, Africa, and the Western Pacific [1, 10]. Despite improved knowledge of pathology of lymphatic filariasis and existence of the drugs diethylcarbamazine and albendazole necessary to treat lymphatic filariasis, it continues to be a major public health problem in tropical and subtropical countries. Lymphatic filariasis is more common in regions that have a higher incidence of poverty [11] making it a disease of the poor and serves as an indicator of underdevelopment [1]. Surveys in Ghana have indicated that * bancroftian* filariasis is present in most parts of the country with considerable variations in prevalence [12]. In countries where lymphatic filariasis is well established, the prevalence of infection continues to increase due to unplanned growth of cities and water resource development such as irrigation, which creates numerous breeding sites for the mosquitoes that transmit the disease [12].

A number of mathematical models have looked into malaria a mosquito-borne infection [13–19], to mention just a few, but only a few have looked into lymphatic filariasis [20–24]. Here the authors propose and analyse the transmission dynamics of lymphatic filariasis using differential equations to explore if treatment for those symptoms alone will be able to keep the infection under control.

#### 2. Model Description

The model subdivides the human population based on lymphatic filariasis status into the following subpopulations: susceptible humans , infected and not showing signs of elephantiasis (exposed) , infected and displaying elephantiasis symptoms . Thus, the total human population is given by The mosquito population is divided into the following subgroups: susceptible mosquitoes and infected mosquitoes , so the total mosquito population is given by The mosquitoes and human beings are recruited into their susceptible corresponding populations at rates and , respectively. Mosquitoes experience natural death rate at a rate which is proportional to the number in each mosquito class. Similarly, human beings experience natural death at a rate , which is proportional to the number in each human class.

The mosquito ingests microfilariae when biting a human who is infected with filariasis (elephantiasis causing nematodes) at a rate Here, is the average number of mosquito bites which cause transmission of disease from infected human to susceptible mosquito; accounts for reduced number of microfilariae in the blood stream of individuals infected but not showing elephantiasis symptoms. Upon getting infected, susceptible mosquitoes enter the infected class . Microfilariae pass through mosquito gut into hemocoel and develop into filariform juveniles. Filariform juveniles escape from mosquito’s proboscis when the insect is feeding and then penetrate wound structure of a human being at a rate where where is the average number of mosquito bites which cause transmission of disease from infectious mosquito to susceptible human per mosquito. Thus, humans are infected at a rate following a bite by a mosquito to move into the exposed class (infected and not showing signs of filariasis). Individuals in progress to the stage of showing filarasis symptoms at rate . However, some progress to the as a result of reinfection at a rate . Individuals in -class are treated using Diethycarbamzime (DEC) and albendazole drugs at a rate to move back into the susceptible class , since recovery from filarasis is not permanent. With filarasis, there is no disease-induced death. The structure of model is given in Figure 1.

Unless stated otherwise, values for the parameters in the simulations are given in Table 1 which are values used in malaria models which is also spread by the same vector as data on this neglected tropical infection is hard to come by.

Due to lack of data to calibrate the model and for parameter estimation, other parameter values are assumed within realistic ranges for illustrative purpose. Note that that “Range” refers to values used in the sensitivity analysis. Based on the given assumptions, the following system of differential equations describes the model: All parameters and state variables for model system are assumed to be nonnegative to be consistent with human and mosquito populations. All feasible solutions of model system are positive and eventually enter the invariant attracting region It is sufficient to consider solutions in . Existence, uniqueness, and continuation results for system hold in this region, and all solutions starting in remain there for all . Hence, is mathematically and epidemiologically well posed, and it is sufficient to consider the dynamics of the flow generated by the model system in .

##### 2.1. Disease-Free Equilibrium and Stability Analysis

The disease-free equilibrium of model system is given by Following van den Driessche and Watmough [26], we have which is the number secondary lymphatic filarasis cases caused by one infected individual/mosquito in a completely susceptible population. Local stability of the disease-free equilibrium is assured by Theorem 2 [26].

Theorem 1. *The disease-free equilibrium for model system is locally asymptotically stable if and unstable otherwise. *

###### 2.1.1. Analysis of the Reproduction Number

is a decreasing function of as , suggesting that mechanisms which increase mortality of mosquitoes like spraying will have a positive impact on controlling lymphatic filariasis. Now let us explore the impact of treating elephantiasis cases only by partially differentiating with respect to to obtain The fact that suggests that increase in treatment of elephantiasis cases results in a decrease of the lymphatic filariasis infections in the population.

##### 2.2. Existence and Stability Analysis of the Endemic Equilibrium

The endemic equilibrium is given by , where Substituting into the force of infection in (4), we obtain Also, substituting into the force of infection in (3), we obtain Substituting (11) into (12), we have , with corresponding to the disease-free equilibrium and corresponding to the endemic equilibrium with Clearly is positive throughout, only when and can be positive or negative depending on whether or . Solving , we obtain The analysis of (14) is Theorem 2.

Theorem 2. *Model system has *(i)*a unique endemic equilibrium if ; *(ii)*a backward bifurcation if , . *

We now employ the Centre Manifold Theory [27] to analyse the stability of this equilibrium point as described in Theorem 4.1 [28], to establish the local asymptotic stability of the endemic equilibrium. Let us make the following change of variables in order to apply the Center Manifold Theory , so that and , so that . We now use the vector notation . Then, model system can be written in the form =, such that The Jacobian matrix of system at is given by from which it can be shown that the reproduction number is Now let us consider , regardless of whether or . Taking as a bifurcation parameter and if we consider the case and solve for , we obtain Note that the linearized system of the transformed equation with the bifurcation point has a simple zero eigenvalue. Hence, the Centre Manifold Theory [27] can be used to analyze the dynamics of near . It can be shown that the Jacobian of at has a right eigenvector associated with the zero eigenvalue given by , where The left eigenvector of associated with the zero eigenvalue at is given by , where

###### 2.2.1. Computation of Bifurcation Parameters and

The sign of is associated with the following nonvanishing partial derivatives of Thus, it follows from , (20), and (21) that For system , the associated nonzero partial derivatives of : at the disease-free equilibrium associated with are given by It follows from , (20), and thatwith . It follows that or if or +.

Thus, and or depending on whether is greater or smaller than . This sign of may be expected in general for epidemic models because, in essence, using as a bifurcation parameter often ensures . Using Theorem 4.1 [28] items (i) and (iv), we establish Theorem 3.

Theorem 3. *If +, , then model system undergoes a backward bifurcation at , otherwise and a unique endemic equilibrium guaranteed by Theorem 4.1 [28] is locally asymptotically stable for , but close to 1. *

The phenomenon of backward bifurcation in disease models, where a stable endemic equilibrium coexists with a stable disease-free equilibrium when the associated reproduction number is less than unity, has important implications for disease control [29]. In such a case, having the reproduction number being less than unity becomes only a necessary but are not sufficient condition for disease elimination. Backward bifurcation in this case is caused by reinfection of the exposed individuals. In the absence of reinfection model system (7.3) will have a unique endemic equilibrium which exists whenever . Thus, for , the model exhibits a forward bifurcation. The bifurcations which occur for different signs of are shown in Figure 2 using parameters values in Table 1.

#### 3. Numerical Simulations

In this section, we make use of Matlab to analyse the effect of varying initial infected mosquito population, for different reproduction numbers of model system using model parameters in Table 1.

Figure 3 shows that whenever the reproduction number is greater than unity, then all the populations (mosquito and human) converge to their corresponding respective endemic equilibrium point regardless of the size of initial mosquito population size.

In Figure 4, the effect of varying infected mosquito population size when all human individuals displaying signs and symptoms of elephantiasis are put on treatment is illustrated. It shows that even when the reproduction number is less than unity the individuals displaying elephantiasis symptoms will not disappear in the population. This result suggests that effective control of elephantiasis may lie in treatment of both humans in the exposed and infectious classes. Reinfection of the people in the exposed class makes it difficult to eliminate lymphatic filariasis through treatment of the infectives alone.

##### 3.1. Sensitivity Analysis

In order to investigate the effects of variations in to its constituent parameters, we used Latin Hypercube Sampling and Partial Rank Correlation Coefficients (PRCCs) with 1000 simulations per run. Latin Hypercube Sampling is a statistical sampling method that allows for an efficient analysis of parameter variations across simultaneous uncertainty ranges in each parameter [30]. PRCCs illustrate the degree of the effect that each parameter has on the outcome.

Figure 5 illustrates the PRCCs using as an output variable. The parameter with the greatest effect on the outcome is the death rate of mosquitoes suggesting that mechanisms which increase death rate of mosquitoes will have a positive impact on lymphatic filariasis control. Surprisingly, the rate of progressing from the exposed class to the infectious class decreases more than treatment. This result turns to suggest that treatment of people displaying symptoms of elephantiasis symptoms alone is not enough to keep the disease under control.

Figure 6 illustrates the effect that varying six sample parameters will have on . If the mosquito death rates are sufficiently high, then the disease prevalence can be reduced , but not sufficiently enough to control it . Similarly, high treatment levels result in but are not sufficient enough to have necessary for disease control. Increases in the average number of mosquito bites () increase suggesting an increase in lymphatic filariasis cases. This further suggests that other mechanisms which reduce mosquito bites like spraying, use of mosquito nets, and wearing clothes which cover much of the body may be necessary to control lymphatic filariasis.

#### 4. Discussion

A mathematical model for the transmission dynamics of lymphatic filariasis with treatment for those displaying elephantiasis symptoms is presented as a system of differential equations. The reproduction number and the equilibria are computed and analysed. The model is shown to exhibit backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibrium when the reproduction number is less than unity. The presence of backward bifurcation makes the classical requirement of having the reproduction number less than unity necessary but insufficient to control the epidemic. Analysis of the reproduction number suggests that treatment of elephantiasis cases has some impact on reducing the spread of lymphatic filariasis infections. However, sensitivity analysis (Monte Carlo simulations) tends to give a better picture about the relationship between the reproduction number and the treatment factor, as they show that the treatment factor reduces the reproduction number but not to levels necessary for disease elimination. This result suggests that effective lymphatic filariasis control requires strategies beyond elephantiasis treatment only. The model presented is not exhaustive; it can be extended to incorporate chemoprophylaxis for the exposed. Given that lymphatic filariasis and malaria are both mosquito-borne infections, a coinfection model of the two would be another worth extension.

#### References

- World Health Organization, “Lymphatic filariasis: the disease and its control,”
*Fifth Report of the World Health Organization Technical Report Series*821–871, World Health Organization Expert Committee on Filariasis, 1992. - J. H. F. Remme, P. de Raadt, and T. Godal, “Tropical health. The burden of tropical diseases,”
*Medical Journal of Australia*, vol. 158, no. 7, pp. 465–469, 1993. View at Scopus - G. J. Weil and R. M. R. Ramzy, “Diagnostic tools for filariasis elimination programs,”
*Trends in Parasitology*, vol. 23, no. 2, pp. 78–82, 2007. View at Publisher · View at Google Scholar · View at Scopus - E. Michael, D. A. P. Bundy, and B. T. Grenfell, “Re-assessing the global prevalence and distribution of lymphatic filariasis,”
*Parasitology*, vol. 112, no. 4, pp. 409–428, 1996. View at Scopus - E. Michael and D. A. P. Bundy, “Global mapping of lymphatic filariasis,”
*Parasitology Today*, vol. 13, no. 12, pp. 472–476, 1997. View at Publisher · View at Google Scholar · View at Scopus - A. J. Terhell, M. Haarbrink, A. van den Biggelaar, A. Mangali, E. Sartono, and M. Yazdanbakhsh, “Long-term follow-up of treatment with diethylcarbamazine on anti-filarial IgG4: dosage, compliance, and differential patterns in adults and children,”
*American Journal of Tropical Medicine and Hygiene*, vol. 68, no. 1, pp. 33–39, 2003. View at Scopus - W. D. Melrose, “Lymphatic filariasis: new insights into an old disease,”
*International Journal for Parasitology*, vol. 32, no. 8, pp. 947–960, 2002. View at Publisher · View at Google Scholar · View at Scopus - K. T. Cuenco, M. E. Halloran, J. Louis-Charles, and P. J. Lammie, “A family study of lymphedema of the leg in a lymphatic filariasis-endemic area,”
*American Journal of Tropical Medicine and Hygiene*, vol. 70, no. 2, pp. 180–184, 2004. View at Scopus - G. Dreyer, J. Norões, J. Figueredo-Silva, and W. F. Piessens, “Pathogenesis of lymphatic disease in bancroftian filariasis: a clinical perspective,”
*Parasitology Today*, vol. 16, no. 12, pp. 544–548, 2000. View at Publisher · View at Google Scholar · View at Scopus - E. A. Ottesen and C. P. Ramachandran, “Lymphatic filariasis infection and disease: control strategies,”
*Parasitology Today*, vol. 11, no. 4, pp. 129–131, 1995. View at Publisher · View at Google Scholar · View at Scopus - J. Z. G. Tan, “The elimination of lymphatic filariasis: a strategy for poverty alleviation and sustainable development-perspectives from the Philippines,”
*Filaria Journal*, vol. 2, article 12, 2003. View at Publisher · View at Google Scholar · View at Scopus - M. A. Appawu, S. K. Dadzie, A. Baffoe-Wilmot, and M. D. Wilson, “Lymphatic filariasis in Ghana: entomological investigation of transmission dynamics and intensity in communities served by irrigation systems in the Upper East Region of Ghana,”
*Tropical Medicine and International Health*, vol. 6, no. 7, pp. 511–516, 2001. View at Publisher · View at Google Scholar · View at Scopus - R. J. Smith and S. D. Hove-Musekwa, “Determining effective spraying periods to control malaria via indoor residual spraying in Sub-Saharan Africa,”
*Journal of Applied Mathematics and Decision Sciences*, vol. 2008, Article ID 745463, 19 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus - J. L. Aron, “Mathematical modeling of immunity to malaria,”
*Mathematical Biosciences*, vol. 90, no. 1-2, pp. 385–396, 1988. View at Scopus - N. Chitnis, J. M. Cushing, and J. M. Hyman, “Bifurcation analysis of a mathematical model for malaria transmission,”
*SIAM Journal on Applied Mathematics*, vol. 67, no. 1, pp. 24–45, 2006. View at Publisher · View at Google Scholar · View at Scopus - H. M. Yang, “Malaria transmission model for different levels of acquired immunity and temperature-dependent parameters (vector),”
*Revista de Saude Publica*, vol. 34, no. 3, pp. 223–231, 2000. View at Scopus - H. M. Yang and M. U. Ferreira, “Assessing the effects of global warming and local social and economic conditions on the malaria transmission,”
*Revista de Saude Publica*, vol. 34, no. 3, pp. 214–222, 2000. View at Scopus - G. A. Ngwa, “Modelling the dynamics of endemic malaria in growing populations,”
*Discrete and Continuous Dynamical Systems B*, vol. 4, no. 4, pp. 1173–1202, 2004. View at Scopus - G. A. Ngwa and W. S. Shu, “A mathematical model for endemic malaria with variable human and mosquito populations,”
*Mathematical and Computer Modelling*, vol. 32, no. 7-8, pp. 747–763, 2000. View at Publisher · View at Google Scholar · View at Scopus - S. Swaminathan, S. P. Pani, R. Ravi, K. Krishnamoorthy, and P. K. Das, “Mathematical models for lymphatic filariasis transmission and control: challenges and prospects,”
*Parasites and Vectors*, vol. 1, article 2, 2008. View at Publisher · View at Google Scholar · View at Scopus - P. K. Das and S. Subramanian, “Modelling the epidemiology, transmission and control of lymphatic filariasis,”
*Annals of Tropical Medicine and Parasitology*, vol. 96, pp. S153–164, 2002. View at Scopus - M. S. Chan, A. Srividya, R. A. Norman et al., “EPIFIL: a dynamic model of infection and disease in lymphatic filariasis,”
*American Journal of Tropical Medicine and Hygiene*, vol. 59, no. 4, pp. 606–614, 1998. View at Scopus - R. A. Norman, M. S. Chan, A. Srividya et al., “EPIFIL: the development of an age-structured model for describing the transmission dynamics and control of lymphatic filariasis,”
*Epidemiology and Infection*, vol. 124, no. 3, pp. 529–541, 2000. View at Publisher · View at Google Scholar · View at Scopus - A. P. Plaisier, S. Subramanian, P. K. Das et al., “The LYMFASIM simulation program for modeling lymphatic filariasis and its control,”
*Methods of Information in Medicine*, vol. 37, no. 1, pp. 97–108, 1998. View at Scopus - J. Labadin, C. M. L. Kon, and S. F. S. Juan, “Deterministic malaria transmission model with acquired immunity,” Proceedings of the World Congress on Engineering and Computer Science, October 2009.
- P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,”
*Mathematical Biosciences*, vol. 180, pp. 29–48, 2002. View at Publisher · View at Google Scholar · View at Scopus - J. Carr,
*Applications Centre Manifold Theory*, Springer, New York, NY, USA, 1981. - C. Castillo-Chavez and B. Song, “Dynamical models of tuberculosis and their applications,”
*Mathematical Biosciences and Engineering*, vol. 1, no. 2, pp. 361–404, 2004. View at Publisher · View at Google Scholar - O. Sharomi and A. B. Gumel, “Curtailing smoking dynamics: a mathematical modeling approach,”
*Applied Mathematics and Computation*, vol. 195, no. 2, pp. 475–499, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. M. Blower and H. Dowlatabadi, “Sensitivity and uncertainty analysis of complex models of disease trans- mission: an HIV model, as an example,”
*International Statistical Review*, vol. 2, pp. 229–243, 1994.