- Research
- Open Access
Mathematical modeling of the effects of public health education on tungiasis—a neglected disease with many challenges in endemic communities
- Rachel A. Nyang’inja^{1, 2}Email author,
- David N. Angwenyi^{3},
- Cecilia M. Musyoka^{1} and
- Titus O. Orwa^{4}
https://doi.org/10.1186/s13662-018-1875-5
© The Author(s) 2018
- Received: 8 May 2018
- Accepted: 8 November 2018
- Published: 20 November 2018
Abstract
In this paper, we formulate and study a mathematical model for the dynamics of jigger infestation incorporating public health education using systems of ordinary differential equations and computational simulations. The basic reproduction number \(R_{E}\) is obtained and used to determine whether the disease breaks out in the population and results in an endemic equilibrium or dies out eventually corresponding to a disease-free equilibrium. We carried out an analysis of the model and established the conditions for the local and global stabilities of the disease-free and endemic equilibria points. Using the Lyapunov stability theory and LaSalle invariant principle, we found out that the disease-endemic equilibrium point is globally asymptotically stable if \(R_{E}>1\) and unstable otherwise. Numerical simulations are performed to illustrate our theoretical predictions. Both the analytical and numerical results show public health education is a very effective control measure in eradicating jigger infestation in the endemic communities at large.
Keywords
- Basic reproduction number
- Global stability
- Jigger infestation
- Lyapunov stability theory
- Public health education
1 Introduction
Tungiasis is a parasitic skin disease caused by female sand flea or jigger (tunga penetrans) that mostly lives in dry sand and soil [1]. The impregnated female parasite attacks exposed skin and burrows itself into the skin, leaving a section of its abdomen exposed through a pore in the skin to enable it breathe and carry out other metabolic functions like feeding on blood vessels [2]. The first evidence of jigger infestation is a tiny dark spot on the skin at the point of penetration. When the eggs are ready, they are released through the orifice and after about one week hatch into fleas [3]. It is important to note that one female flea may lay hundreds of eggs, meaning that if the situation is not promptly corrected it can easily result into severe health effects.
Tungiasis is endemic in many countries including Latin America, the Caribbean, and Sub-Sahara Africa [4]. Researchers agree that it is a neglected health problem in impoverished communities in the affected countries [5]. In Kenya, the neglect is evidenced by scarcity of epidemiological data [6]. It is estimated that about 2.6 million (6.5%) Kenyans suffer from tungiasis and 10 million others are at risk [7]. Tungiasis is a common problem among people who live in Mount Kenya region, western Kenya, Rift Valley, and coastal regions [2]. In 2010, the prevalence of tungiasis in Muranga South, an endemic area, was 57% among children aged 5 to 12 years [8]. Unhygienic conditions have been identified as the major causes of tungiasis [9]. The jigger menace is a common problem among the low income populations, particularly where people live with furred animals under congested and filthy conditions [10]. Dirty and warm areas are optimal breeding grounds for the flea. Severe tungiasis infestations occur in vulnerable age groups such as children and the elderly, especially those from the economically challenged communities [11]. Jiggers cause extreme local itching, pain, and sensation in the skin of their victims. Other effects include: restlessness, edginess, poor appetite, susceptibility to opportunistic infections, lymphangitis, gangrene and extreme limb amputations, sepsis, loss of toes, poor health, poor school performance and drop out, low self-esteem, permanent distortion of shape of legs and hands, and can be life threatening [12].
Treatment of tungiasis involves identification of the parasite especially through mechanical removal using a sterile, sharp pointed object such as a needle or pin followed by an antiseptic dressing. It may also be effectively treated by surgical extraction of embedded sand fleas under sterile conditions in medical facilities [13]. The fleas may also be deterred by washing the affected areas with disinfectants like potassium permanganate, topographic application of anti-parasitic agents, use of anti-inflammatory creams and use of repellents [14]. Personal hygiene and wearing of shoes are key in controlling jigger infection. Children and the elderly need to be well taken care of through provision of primary and affordable amenities and healthcare facilities within close proximity of low income populations.
The transmission of the infectious diseases has been of great interest to both medics and scholars [15]. Public health programs play a very important role in defining health issues since they serve as a major source of information and influence changes in the way people behave. It is, therefore, critical to study epidemic transmission and take effective strategies to prevent, control, and contain them. Individuals response to a disease threat depends on risk perception that is gained largely through information reported by the government to the public, for example, a number of infections, hospitalizations, and deaths, as provided by the public health department [16, 17]. For instance, public health officials could be interested in epidemic measurements such as the peak number of infections, peak time, the total number of infections, and the end of epidemic, which are all directly related to important public health resources [18]. Public health programs, such as public vaccination or immunization, isolation, quarantine, and awareness creation through media, can affect disease transmission during an epidemic contagion [19].
Mathematical models can serve as powerful tools for understanding the driving mechanisms of disease transmission. A general susceptible-infected-recovered or removed model was studied in [20] by using type 2 functional response to depict the transmission rate decreased by awareness intervention. The authors established that effective awareness measures can reduce the number of infections. An effective method of reducing the spread of the disease is to make people understand the preventive measures as soon as possible. A mathematical model was presented in [21] to study the effects of public health educational campaigns as a single control strategy on sexual transmission of HIV/AIDS in the continuing absence of a preventative vaccine. Results showed that public health educational campaigns can slow down the epidemic. In [22], the authors formulated a deterministic model on HIV/AIDS transmission and studied the impact of educational programs and abstinence in Sub-Saharan Africa.
Despite the harm inflicted by tungiasis and countless challenges of the disease to the population, studies on the disease from mathematical modeling perspective are very scarce. Theoretical studies so far conducted on tungiasis have focused on the knowledge pertaining to the biology, epidemiology, and pathology of the jigger, possible causes of the disease, attitude and practices on the prevalence in different endemic communities [11, 12, 23]. Only a few works do exist in the literature that address the problem from a mathematical approach. Some of the challenges encountered in modeling of tungiasis disease and its control interventions include the limited understanding of how the disease manifests in humans, its transmission dynamics, the jigger flea, and the limited data on the disease that exist as most infested people shy away from seeking medical attention due to stigma that is associated with the disease. Recently, the authors in [24] spear-headed the application of mathematical models to tungiasis transmission dynamics. They formulated and studied a mathematical model for the dynamics of tungiasis transmission in zoonotic areas by conducting a sensitivity analysis to determine the parameters with high impact on the basic reproduction number which could be targeted with the control interventions. Their results showed that reducing the on and off-host flea population and the effective contact rate between the sand flea sources and the hosts would be effective intervention strategies to decrease the probability of a large transient of tungiasis epidemic. This modeling work was extended in [25, 26] to include stability and optimal control strategies with an aim to minimize on the number of infested humans, infested animals, and sand flea populations. In [27], the author employed a susceptible-infected-removed (SIR) model to study the dynamics of jigger infestation incorporating treatment as a control strategy against infestation. It was established that a high probability of success of treatment leads to a low jigger prevalence in a population. All of these modeling efforts have not only provided useful information for understanding the mechanics of tungiasis disease transmission, but also highlighted the need for better data to construct and validate future models of tungiasis. Disease spread is an important phenomenon that has been widely studied both in the fields of classical models [28–30] and in networks [31, 32]. However, there are currently no models that comprehensively study the dynamics of tungiasis transmission with public health education as a control strategy, and that is our motivation for this study.
In this work, therefore, we theoretically study the dynamics of jigger infestation incorporating public health education using systems of ordinary differential equations and numerical simulations. Specifically, we formulate the epidemic threshold to enable us determine whether the infection breaks out in the population and results in an endemic equilibrium or dies out eventually corresponding to a disease-free equilibrium and predict the disease prevalence. We have comprehensively analyzed the model and proved that a unique positive endemic equilibrium exists and is locally and globally asymptotically stable. Besides, we have carried out numerical simulations to verify the results and demonstrate the validity of the various assumptions made during the analysis. The paper is organized as follows: in this section, we have provided background information of jigger infestation and given a brief discussion on previous works on the disease and other findings on a general impact of public health intervention measures on epidemics; in Sect. 2, we have formulated and described the mathematical model, presented its basic properties (positivity and boundedness), and determined its equilibrium points, the basic reproduction number \(R_{E}\), and the existence and uniqueness of the disease positive endemic equilibrium. We have also established the sufficient conditions for the local and global stabilities of the disease-free and the disease-endemic equilibria and proved that they are both globally asymptotically stable. Section 3 presents numerical simulations to verify our theoretical results. Finally, we discuss and conclude this work in Sects. 4 and 5, respectively.
2 The model
The total human population size N at time t is subdivided into compartments: susceptible compartment \(S(t)\), in which all individuals are susceptible to jigger infestation; educated compartment \(E(t)\), in which all individuals are educated on jigger infestation prevention strategies or treatment, and infected compartment \(I(t)\), in which all individuals are infested by the disease and have infectivity. Since the incubation period is very short, we assume that the probability of survival till the infectious state for the individuals exposed to jiggers is unity and exclude the exposure stage. At each time step, each individual exists in only one of the three stated compartments. Recruitment into susceptible population takes place at the rate Λ. We assume that the mass-action incidence transmission is defined by \(\beta SI\), where β is the effective contact rate for disease transmission. When the jigger infected individuals receive treatment or other interventions, they will be cured and become susceptible again with the recovery rate γ. The rate at which public health educational strategies disseminate among susceptible individuals as per the public health intervention programs is represented by ε. Since the influence of education may not be permanent or the strategies employed may not be very effective, the knowledge of taking control measures gradually wears off, and so educated individuals will be infected at a lower rate \(\alpha\beta EI\), where \(0<\alpha<1\). The fraction α reflects the effect of reducing the infection due to public health education. In this work, \(0<\alpha<1\) because \(\alpha=0\) means education is completely effective in preventing jigger infection, and \(\alpha=1\) implies education is not useful at all. Disease mortality is assumed to take place at the rate σ, while μ is the natural death rate.
2.1 Positivity and boundedness of solutions
Theorem 1
The solutions of model system (1) are feasible for \(t>0\) if they enter the invariant region Γ.
Proof
Hence, N is bounded and all feasible solution sets of the human population of model system (1) approach, enter, or stay in the region \(\varGamma=\{(S,E,I)\in\mathbb{R}_{+}^{3}:S>0,I\geq0,E\geq 0, N\leq\frac{\varLambda}{\mu}\}\).
Therefore, the region Γ is positively invariant, that is, the solution is positive for all times \((t)\) and model system (1) is epidemiologically meaningful and mathematically well-posed in the domain Γ. Hence, it is sufficient to consider the dynamics of the flow it generates in a proper subset \(\varGamma=\{(S,E,I)\in\mathbb {R}_{+}^{3}:N\leq\frac{\varLambda}{\mu}\). □
Theorem 2
All feasible solutions of model system (1) are uniformly bounded in the subset region Γ.
Proof
Let the initial data set be \(\{S(0)>0,[I(0),E(0)\geq 0]\}\in\varGamma\). Then the solution set \(\{(S,E,I)\}(t)\) of model system (1) is nonnegative for all \(t>0\).
Consider the first equation of model system (1) at time t.
Similarly, it can be verified that the second equation of model system (1) is also positive for all time \(t>0\), since \(e^{\tau}>0\) for all \(\tau\in\mathbb{R}\). □
2.2 Equilibrium points of the model
2.3 The basic reproduction number \(R_{E}\)
The basic reproduction number \(R_{E}\) is an important epidemiological parameter which is defined as the expected number of secondary infections produced by a single infective individual in a completely susceptible population [28]. Specifically, we define the basic reproduction number \(R_{E}\) of model system (1) as the number of secondary jigger infections caused by a single jigger infected individual in the presence of public health educational interventions. When no such programs are employed, the basic reproduction number is defined by \(R_{0}\). It measures the power of a disease to invade a population under conditions that facilitate maximal growth. In general, the basic reproduction number \(R_{0}\) depends on the demographic, disease, and mobility parameters.
It is important to note that we have used the next-generation approach in estimating the basic reproduction number; however, there exist other methods, including obtaining the eigenvalues of the Jacobian matrix, the survival function technique, and the existence of the endemic equilibrium, which can be employed in models where calculations are unsuccessful [35].
2.4 Local stability of the disease-free equilibrium
The disease-free equilibrium points of the model are its steady state solutions in the absence of infection or disease. Again following Theorem 2 in Van den Driessche and Watmough [34], we present local stability property for the disease-free equilibrium in the following theorem.
Theorem 3
The disease-free equilibrium of model system (1) is locally asymptotically stable whenever \(R_{E}<1\) and unstable if \(R_{E}>1\).
Proof
We prove Theorem 3 by obtaining the eigenvalues of the Jacobian matrix and show that they have negative real parts.
\(\lambda_{1}=-(\varepsilon+\mu)\), \(\lambda_{2}=-\mu\) and \(\lambda _{3}={(\frac{\beta\mu\varLambda+\alpha\beta\varLambda\varepsilon-\mu (\varepsilon+\mu)(\gamma+\sigma+\mu)}{\mu(\varepsilon+\mu)})}=(\gamma +\sigma+\mu)(R_{E}-1)\). If \(R_{E}<1\), then \(\lambda_{1}\), \(\lambda _{2}\), and \(\lambda_{3}\) are all negative. The eigenvalues have negative real parts. It follows that the trace of \(J(D^{0})\) is negative and the determinant is positive. Therefore the disease-free equilibrium (DFE) is locally asymptotically stable in the region Γ if and only if \(R_{E} < 1\). Thus, Theorem 3 is proved. □
2.5 Global stability of the disease-free equilibrium
Theorem 4
The disease-free equilibrium \((D^{0})\) of model system (1) is globally asymptotically stable whenever \(R_{E}<1\) and unstable if \(R_{E}>1\)
Proof
We use geometry to prove the global stability of the disease-free equilibrium (DFE) of model system (1). From Theorem 3, we know that if \(0< R_{E}<0\), the equilibrium \(D^{0}\) is locally asymptotically stable and there are no endemic equilibria in the region Γ when \(0< R_{E}<1\). According to Perko [36], any solution of model system (1) starting in Γ must approach either an equilibrium or a closed orbit in Γ. With reference to Kelley and Peterson [37], if the solution path approaches a closed orbit, then this closed orbit must enclose an equilibrium. Nevertheless, the equilibrium existing in Γ is \(D^{0}\) when \(0< R_{E}<1\) and it is located in the boundary of Γ, therefore there is no closed orbit in Γ. Hence, any solution of system (1) with initial condition in Γ must approach the point \(D^{0}\) as \(t\rightarrow\infty\). Therefore, the disease-free equilibrium \(D^{0}\) is globally asymptotically stable in Γ when \(0< R_{E}<1\). When \(R_{E}>1\), then \(\lambda_{3}>0\), that is, the root will be positive. Therefore, the equilibrium \(D^{0}\) will be unstable. Thus, we have proved Theorem 4. □
2.6 Existence of a unique positive endemic equilibrium \(\varphi _{*}(S_{*},E_{*},I_{*})\)
Theorem 5
An endemic equilibrium \(\varphi_{*}(S_{*},E_{*},I_{*})\) exists provided that \(R_{E}>1\).
Proof
To calculate endemic equilibrium (EE), we set \(\varphi _{*}(S_{*},E_{*},I_{*})\neq0\).
- (a)
If \(B<0\) and \(C=0\) or \(\sqrt{B^{2}-4AC}=0\), then equation (26) has a unique disease-endemic equilibrium point (one positive root) and there is no possibility of backward bifurcation.
- (b)
If \(C>0\) and \(B>0\) and \(B^{2}-4AC>0\), then equation (26) has two endemic equilibria (two positive roots), and thus there is possibility of backward bifurcation to occur.
- (c)
Otherwise, there is none.
Theorem 6
- (i)
Precisely one unique endemic equilibrium if \(C<0 \Leftrightarrow R_{E}>0\).
- (ii)
Precisely one unique endemic equilibrium if \(B<0\) and \(C=0\) or \(B^{2}-4AC=0\).
- (iii)
Precisely two unique endemic equilibria if \(C>0\), and \(B<0\) or \(B^{2}-4AC>0\).
- (iv)
None, otherwise.
From (29) it is observed that there is only one positive root of equation (27). That is, \(I_{*}>0\). We can therefore conclude that there exists one unique positive disease-endemic equilibrium.
2.7 Local stability of the endemic equilibrium
Theorem 7
If \(R_{E}>1\), then the disease-endemic equilibrium of model system (1) is locally asymptotically stable.
Proof
Therefore, we can conclude that model system (1) has an asymptotically stable endemic equilibrium whenever \(R_{E}>1\). Thus, Theorem 7 is proved. □
2.8 Global stability of the endemic equilibrium
We study the global asymptotic stability of the endemic equilibrium using the Lyapunov direct method and LaSalle’s invariance principle [38, 39].
Theorem 8
If \(R_{E}>1\), then the endemic equilibrium of model system (1) is globally asymptotically stable.
Proof
3 Numerical simulations
Parameter | Symbol | Value | Source |
---|---|---|---|
Recruitment rate of susceptible class | Λ | 4.4 × 10^{−3} day^{−1} | [41] |
Effective contact rate | β | 1.4989 × 10^{−2} day^{−1} | [42] |
Dissemination rate | ε | 1.431 × 10^{−2} day^{−1} | Assumed |
Scaling factor | α | 0.2–0.990 | Assumed |
Disease-induced mortality rate | σ | 5.0 × 10^{−2} day^{−1} | [42] |
Recovery rate | γ | 4.27 × 10^{−1} day^{−1} | [42] |
Per capita natural death rate | μ | 1.6 × 10^{−2} day^{−1} | [41] |
4 Discussion
5 Conclusion
In this paper, the jigger infestation mathematical model with public health education as an intervention strategy was formulated and comprehensively analyzed. Using the theory of differential equations, the invariant set in which the solutions of the model are biologically meaningful was derived. Boundedness of solutions was also proved. Analysis of the model showed that there exist two possible solutions, namely the disease-free equilibrium point and the positive endemic equilibrium point. Further analysis showed that both the disease-free and the endemic equilibria points are locally and globally asymptotically stable. It means that the jigger menace will disappear if \(R_{E}<1\), otherwise, it will be prevalent if \(R_{E}>1\).
Numerical simulations have been performed to support the analytical results. At the same time, we have discussed the influence of different values of α on the total density of the jigger-infested individuals, establishing that public health education is an effective measure of controlling the jigger menace as it reduces the spreading threshold. From this work, we recommend that control measures be put in place especially in resource-poor communities where the disease is endemic. Even though tungiasis is a public health hazard, reliable data on the disease occurrence are very scarce. The disease is in many ways a hidden disease that is generally neglected by healthcare providers and political decision makers.
This work provides a basic dynamical model that can be used to understand the impact of public health education on jigger infestation. Finally, the model has mass-action transmission, which assumes homogeneous mixing; however, in reality, the contact process of population cannot be uniform collision, different people contact person may be entirely different per unit of time. For future work, we propose that complex networks [31, 32, 43–46], which consider the mechanisms of epidemic transmission in large-scale social networks with distinct heterogeneities, be incorporated into the model.
Declarations
Acknowledgements
We are very grateful to the Journal editors and the anonymous reviewers for their careful reading, valuable comments, and recommendations that have greatly improved the presentation of this manuscript. The first author acknowledges the support received from the Department of Mathematics of Shanghai University and Taita Taveta University.
Funding
This work was partially supported by China Scholarship Council under CSC No. 2015404006 and Taita Taveta University.
Authors’ contributions
RAN designed the study, performed the mathematical analysis of the model and discussions, and also wrote and typeset the manuscript. DNA and TOO performed numerical simulations. TOO also edited the manuscript. CMM proofread the final manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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