Mathematical Model for the Effects of Intervention Measures on the Transmission Dynamics of Tungiasis
الموضوعات : فصلنامه ریاضیJAIROS SHINZEH 1 , Livingstone Luboobi 2
1 - The University of Dodoma.
2 - Makerere University
الکلمات المفتاحية: numerical simulation, Stability, Keywords: Tungiasis, Control strategies, Steady states, Effective reproduction number,
ملخص المقالة :
Tungiasis is a zoonosis affecting human beings and a broad range of domestic and syvatic animals caused by the penetration of an ectoparasite known as “Tunga penetrans” into the skin of its host. In this paper we derive and analyze a mathematical model of control measures and then examine the effect of the control strategies on the transmission dynamics of Tungiasis. The model effective reproduction number is determined using the next generation operator method and the analysis is performed using the stability theory of the differential equations. The analytical results show that the disease free equilibrium is locally asymptotically stable when and unstable when . Using Meltzer matrix stability theorem we found that the disease free equilibrium is globally asymptotically stable and by Lyapunov method, the endemic equilibrium is globally asymptotically stable when . From the numerical simulation it was observed that the control strategies have positive impact on the reduction of transmission of Tungiasis disease and that they work better in combination than when applied as singly. The results from simulations will help the decision makers from national health care to advise people at risk with Tungiasis to apply the control strategies based on: educational campaign, personal protection, personal treatment, environmental hygiene and insecticides application to control the flea.
Mathematical Model for the Effects of Intervention Measures on the Transmission Dynamics of Tungiasis
Abstract
Tungiasis is a zoonosis affecting human beings and a broad range of domestic and syvatic animals. It is caused by the penetration of an ectoparasite known as “Tunga penetrans” into the skin of its host. In this paper we derive and analyze a mathematical model of control measures and then examine the effect of the control strategies on the transmission dynamics of Tungiasis. The model effective reproduction number 
is determined using the next generation operator method and the analysis is performed using the stability theory of the differential equations. The analytical results showed that, the disease free equilibrium is locally asymptotically stable when 
and unstable when
. Using Meltzer matrix stability theorem we found that the disease free equilibrium is globally asymptotically stable and by Lyapunov method, the endemic equilibrium was found to be globally asymptotically stable when
. The numerical simulation was performed and it was observed that, the control strategies have the positive impact on the reduction of Tungiasis disease transmission and that, they work better in combination than when applied as singly. Therefore, the decision makers from national health care should make sure that, people at risk with Tungiasis are supported in the application of the control strategies based on: educational campaign, personal protection, personal treatment, environmental hygiene and insecticides application to control the flea.
Keywords: Tungiasis, Control strategies, Steady states, Effective reproduction number, Stability, Numerical simulation.
1 Introduction
Tungiasis is a zoonosis affecting human beings and a broad range of domestic and syvatic animals caused by the penetration of ectoparasites known as “Tunga penetrans” into the skin of its host (Eisele et al., 2003). It is a female sand flea that penetrates into the skin of the animal or human host, which results in parasite hypertrophy and egg production (Eisele et al., 2003). This ectoparasitosis occurs in many parts of Latin America, the Caribbean, and sub-Saharan Africa (Heukelbach et al., 2001). Domestic animals such as dogs, cats, pigs, cattle, goats and others have been described to be commonly infested (Heukelbach et al., 2001; Mutebi et al., 2015). Transmission of Tungiasis occurs when susceptible hosts are in contact with sandy soil in which female fleas are present, also when in contact with the infested animal reservoirs (Heukelbach et al, 2004; Pilger et al., 2008). One of the primary reasons for studying about infectious diseases is to understand its dynamics so as to improve control and ultimately to eradicate the infection from the population. Several forms of control measures exist; mostly all operate by reducing the average amount of transmission between infectious and susceptible individuals. The control measure or mixture of control measures used will depend on the disease, the hosts, and the scale of the epidemic (Keeling and Rohani, 2008).
The control of Tungiasis is based on reducing the transmission rate between susceptible individuals and the sources of infestation. The sources of infestation for human Tungiasis include sandy soils and the infested animal hosts. Since human beings and domestic animals are infested, the successful and effective control can only be achieved using a trans-disciplinary approach, i.e. human and animal health to be improved or done simultaneously. To control the epidemic, the risk factors responsible for Tungiasis transmission must be identified. The risk factors are the major targets where the control measures should be directed to, they include; domestic animals in the homes perhaps because they harbor the “Tunga penetrans” fleas (Mwangi et al., 2015; Pilger et al., 2008), poor housing with dusty and with cracks on walls and with earthen floors, provides for survival and good breeding environment for “Tunga penetrans” (Muehlen et al., 2006), lack of regular use of closed foot wear are important factors for Tungiasis (Mwangi et al., 2015; Ugbomoiko et al., 2007). Another risk factor for Tungiasis is poor environmental and personal hygiene; littered compounds attract stray dogs, cats, and rodents which are important reservoirs for sand fleas (Wafula et al., 2016). Poor personal hygiene practices such as having dirty feet and putting on dirty clothes provide a favorable environment for flea to survive and hide (Ruttoh et al., 2012). High Tungiasis transmission potential in endemic areas, results to a high demand for sustainable control strategies to be implemented. The effective control measures could be designed only if the dynamics, risk factors, and the biology of fleas causing Tungiasis in a targeted population are well understood. Standard therapy consists of removing the embedded flea under sterile condition (Heukelbach et al., 2001). Domesticated animals should be treated with available on-animal insecticides, including collars, shampoos and sprays and in addition, environmental insecticides could be used during the early and late stages of the flea to break the life cycle (Heukelbach et al., 2004). Viable prevention and intervention methods for combating the disease include paving of public areas and house floors and the use of closed shoes when feet touch the contaminated soil (Cestari et al., 2007). A more realistic and very successful preventive approach is possible by application of a plant repellent called Zanzarin, this is a derivative of coconut oil, jojoba oil, and aloe vera that has been shown to effectively prevent the infestation with sand fleas in areas with high attack rates. Studies in Brazil and Madagascar showed that, a twice-daily application of this repellent on the feet could reduce the attack rate by almost 86 percent (Feldmeier et al., 2006; Buckendahl et al., 2010; Thielecke et al., 2013).
A long-lasting reduction of incidence and of Tungiasis-associated morbidity could be achieved through an approach integrating the soil environment, animal reservoirs and the human beings using a mathematical modelling approach. There is no evidence that the mathematical modelling on the dynamics and control of Tungiasis disease has already been done, therefore in this paper the dynamical model for Tungiasis infestation with controls strategies is presented and analyzed. The rest of the paper is arranged as follows. In Section 2, we formulate the model with control measures and its basic properties. In Section 3, we carry out model analysis. In Section 4, we perform numerical simulation and discuss the results and in Section 5 is the conclusion based on the study.
2 Materials and Methods
In this study, we use an integrated approach to investigate the conditions for Tungiasis transmission and then suggest for control measures. For successful control, we integrate the dynamics between human beings, animal reservoirs and the dusty or sand environment with fleas. We formulate deferential model equations with control and perform analytical and numerical analysis for model effective reproduction number and model stability. For model building ordinary differential equations were used, for stability Meltzer matrix theory and the Lypunov function was used. We carryout numerical simulation using both primary and secondary data and MATLAB computer program was used to generate graphs which showed the behaviour of model trajectories over time. Different control measures are treated one at a time and in combination of two, three, four and all together and the behaviour of model trajectories were detected. For this paper in particular, control groups were identified and compared based of their ability to reduce the transmission of epidemic.
2.1 Formulation of a Mathematical Model With Control Measures
To control Tungiasis epidemic, we aim at reducing transmission rate between susceptible animals and the soil environment denoted by
, reducing the transmission rate between severely infested animal reservoirs and susceptible humans
, reducing transmission rate between soil environment and susceptible humans
. Spraying the animals with insecticidal dusts reduces transmission rate between infested animal reservoirs and the soil environment 
at the same time reduces the transmission rate between infested and susceptible animal reservoirs
. Also focal spraying of insecticides to the premises increases the flea death rate 
which results to the reduced attack rate. We let 
be the control effort aimed at reducing the flea population present in the soil environment by increasing their mortality rate; this can be achieved by focal application of insecticides to the premises. Let
be the control efforts aimed at reducing the transmission rate between infested animal reservoirs and the soil environment, similarly reducing the transmission rate between infested and susceptible animal reservoirs. We let 
be the control effort aimed at reducing the transmission rate between susceptible animals and flea infested soil environment; achieved by cementing the floors and environmental hygiene. We let 
be the control effort aimed at reducing the transmission rate between susceptible humans and the severely infested animal reservoirs; achieved by education campaign and lastly we let 
be the control effort aimed at reducing the transmission rate between susceptible humans and the flea infested soil environment; achieved by the use of closed footwear, cementing the floors, environmental sanitation and application of plant based repellants. The total mortality rate of sand fleas in the soil environment is due to the natural mortality rate 
and the mortality rate by insecticides applications to the premises that leads to the death of fleas at a rate
. The transmission rateof the fleas from the severely infested animal to the soil environment is reduced by the factor 
hence we have the net transmission rate
and the animal to animal effective contact rate 
is reduced at net effective contact rate
. The factor 
reduces the transmission rate between susceptible animals and the soil environment at a net effective contact rate
. The factor 
reduces the transmission rate between severely infested animal reservoirs and the susceptible humans at a net effective contact rate 
and the factor 
reduces the transmission rate between flea infested soil environment and susceptible humans at a net effective contact rate
.
2.2 Description of Interaction
We add a distinct compartment of individuals who are treated denoted by 
to the human population sub-model on the basic Tungiasis dynamical model developed by Kahuru et al. (2017). This class is included due to surgical treatment (removal of jiggers from the human body under sterile condition). Therefore the total human population 
is sub-divided into susceptible humans 
, mildly infested humans 
, severely infested humans 
and treated humans such that: 
. The total animal population 
is sub-divided into susceptible animal reservoirs
, mildly infested animal reservoirs 
and severely infested animal reservoirs 
such that: 
. The flea population is sub-divided into larvae compartment 
and adult flea compartment
.
Susceptible human sub-populationis recruited through birth at a rate 
from the total human population. acquire infestation from severely infested animal reservoirs 
at a rate 
and move to class 
, also it may acquire infestation from flea infested soil environment at a rate
and move to class 
. Class may as well acquire infestation from flea infested environment at a rate and move to class . Individuals in classes and seek treatment at the rates 
and 
respectively and join the class 
. Treated individuals revert back to a susceptible class at a rate 
. Individuals in classes, and suffer the natural mortality at a rate 
and at class the individuals suffer the natural mortality at a rate and the disease induced death at a rate
.
The susceptible animal reservoirs are recruited through birth at a rate 
from the total animal reservoir population. acquire infestation from the severely infested animal reservoirs at a rate 
and move to class 
, also they may acquire infestation from the flea infested environment at a rate 
and move to class . Class may as well acquire infestation from flea infested soil environment at a rate and join class . Individuals in classes and suffer the natural mortality rates 
and at class the individuals suffer the natural mortality at a rate and the disease induced mortality at a rate
.
The environmental component consists of two compartments, the larvae compartment 
and adult flea compartment
. The contribution of flea eggs into the soil environment is by severely infested individuals 
and 
into larvae compartment at the rates 
and 
respectively, where, 
is a rate of deposit of flea eggs on the ground and 
is the maximal larvae carrying capacity. The larval natural mortality rate is 
and the rate of transition from larva to adult sand fleas is
. The compartment represents the adult fleas in the soil environment whose half saturation constant is 
and with total mortality caused by natural mortality rate and mortality rate due insecticides application
, so the total number of fleas leaving the soil environment due to total mortality is given by
. The severely infested animal reservoirs 
contribute the adult fleas into the compartment through shedding at a rate 
with total contribution of 
fleas into the soil environment. The term 
which leaves the compartment represents the rate of jigger fleas’ removal from the soil environments that attack the hosts. The fleas leave the soil environment to infest the human and animal hosts at the rates 
and 
with proportion of 
and 
respectively. The forces of infestation for human and animal reservoir populations are given by 
and 
respectively.
Therefore we have the forces of infestation for human and animal populations given by:
2.3 Model Assumptions
The extended model will be formulated considering the following assumptions
i. Animals are not protected neither with shoes nor with plant repellents
ii. Surgical treatment of the animal reservoirs is not considered in this model
iii. After surgical treatment an individual is not re-infested but revert back to the susceptible class
iv. The targeted population is homogeneous
v. All new-borns are susceptible to infestation and there is no vertical transmission
vi. Human to human transmission is not considered in this model
vii. Shedding of jigger eggs by mildly infested individuals is ignored because at this stage it is easy to control by removing them from the body before they shed eggs
viii. The jigger eggs are shed by the infested individuals at an equal rates
ix. The soil environment has a high concentration of jigger fleas than animal reservoirs
x. The individual in contact with infested environment will have severe infestation than an individual in contact with animal reservoir
The variables and parameters that describe the flow rates between compartments are given in Table 1 and Table 2 respectively. The possible interactions between humans, animal reservoirs and flea infested environment with control measures are presented by the model flow diagram in Figure 1 and the differential equations describing the model are also given in equations (2a) – (2i) of the model system (2).
Table 1: The State Variables of the Model
Variable | Description |
| Number of humans in a susceptible class at time, t |
| Number of animals in a susceptible class at time, t |
| Number of humans in mildly infested class at time, t |
| Number of animals in mildly infested class at time, t |
| Number of humans in severely infested class at time, t |
| Number of animals in severely infested class at time, t |
| Number of humans who are receiving treatments at time, t |
| The density of fleas population in the environment at time, t |
| The density of larvae population in the environment at time, t |
| Total human population at time, t |
| Total animal population at time, t |
Table 2: The Parameters of the Model
Parameter | Description |
| Maximal larval carrying capacity |
| Half saturation constant |
| Maturation (transition) rate from larvae to adult jigger fleas |
| disease induced death rates for humans and animal reservoirs respectively |
| Natural mortality rates for humans, animals, fleas and larvae respectively |
| The rate of removal of jigger fleas that leaves the soil to attack the hosts |
| Effective contact rate between environment and humans |
| Effective contact rate between environment and animal reservoirs |
| Effective contact rate between infested animals and susceptible humans |
| Effective contact rate between infested animals and susceptible animals |
| Recruitment birth rates for humans and animal reservoirs respectively |
| The rate of flea eggs deposit on the ground |
| Shedding rates for adult fleas into the environment |
| The proportions of infestation for humans and animals respectively |
| Progression rate of treated humans to susceptible class |
| Progression rates from infested human classes to the treatment class |
2.4 The Model Flow Chart
Using the above assumptions, definition of variables and parameters; the model flow diagram that depicts the dynamics of Tungiasis transmission with control measures for the humans, animal reservoirs and the flea infested soil environment is shown in Figure 1.
Figure 1: The Control Measures on the Populations of Human Beings, Animal Reservoirs and Sand Fleas.
2.5 Model Differential Equations with Control Measures
From the compartmental model flow chart in Figure 1 the following dynamical system with control measures are derived to describe the transmission dynamics of Tungiasis.
2.6 Properties of the Model
In this section, we present the mathematical analysis of the model with control measures; we investigate the invariant region and positivity of the solution. The invariant region of the extended model describes the region in which the solutions of the model system (2) are biological meaningful and the positivity describes non-negative solution of model system (2).
2.6.1 Invariant Region
To examine whether the model system (2) is epidemiologically and mathematically well posed, we investigate the bound-ness of the model solution. We use the approach by Dumont et al. (2008), and we present the system (2) in compact form as shown below:
Lemma 1: The model system (2) is well posed in the feasible region denoted by:
Proof
We write the model system (2) in the form:
and 
is the column vector:
is a Metzler matrix for all 
. Thus using the fact that
is Lipschitz continuous, therefore the model system (2) is positively invariant in
. Thus the feasible region 
for the model system is the set:
It can be verified that is positively invariant with respect to model system (2). That is, the solution remains in the feasible region if it starts in this region. Hence, it is sufficient to study the dynamics of model system (2) in .
2.6.2 Positivity of the Solution
For model system (2) to be epidemiologically meaningful and well posed, we need to prove that all state variables are non-negative for all 
.
Lemma 2: For dynamical system (2) we let the initial condition of model variables to be,
, then the solution set 
is positive for all
Proof:
From the first equation of the model system (2) we have
Integrating (8) by separating the variables
Correspondingly, by using the similar procedure it is proved that the other remaining model state variables 
are also positive. Hence the solution set 
of the model system (2) is non-negative for all
.
3 Analysis of the Model
In this section, we perform the mathematical analysis of the model system (2). According to Hirsch et al. (2012) we use stability theory of differential equations to investigate the existence and stability of the model equilibria.
3.1 Disease Free Equilibrium
The Disease Free Equilibrium (DFE) is a situation where there is no infestation for both populations of human and animal reservoir which means:
,
, 
, 
, 
and 
for model system (2). To compute the model DFE we set the equations of model system (2) equal to zero then solving for 
and 
. Thus we have DFE point given by:
where and are human and animal reservoir population sizes.
3.2 The Effective Reproduction Number
The effective reproduction number 
is the average number of infectious individuals resulting from a single infective introduced at time 
into the population, given an interventions at that time. This quantity is calculated using the next generation operator approach as described by Diekmann et al. (1990) and subsequently used by Van de Driessche and Watmough (2002). To compute we consider the compartments in which infestation is in progression and determine the Jacobian matrices 
corresponding to new infestation and the remaining transfer terms defined by:
Where;
be the rate of appearance of new infestations of individuals in compartment 
be the rate of transfer of individuals in and out of compartment
are the state variables which belong to the transmitting compartments 
At disease free equilibrium, the Jacobian matrices 
and 
respectively are given by:
We compute the inverse of the Jacobian matrix in (11) and determine the next generation matrix 
as defined by Diekmann et al. (1990 where is non-negative and 
is a singular matrix.
Using Maple software we obtain the five eigenvalues of the next generation matrix in (12) which are; 
.
It follows that 
which is the spectral radius (or largest eigenvalue) of
Therefore at DFE we have:
We have noted that the control variables 
have the positive impact on the effective reproduction number
. Increasing efforts based on insecticides spraying to the affected areas, application of insecticidal dusts 
on domestic animals and improving environmental sanitation 
lowers. This is an attempt to achieve the reduced Tungiasis transmissibility.
3.3 Local Stability of Disease Free Equilibrium
An equilibrium point is locally asymptotically stable if the Jacobian matrix evaluated at the disease free equilibrium point has the eigenvalues with negative real parts (Roussel, 2005). For local stability to hold the Jacobean matrix evaluated must have a negative trace and a positive determinant. Here we state and prove the following theorem.
Theorem 1: The disease free equilibrium 
whenever it exists is locally asymptotically
stable if 
and unstable otherwise.
Proof:
Model system (2) is redefined by; 
. Therefore we have:
From system (14) we construct the Jacobian matrix at the disease free equilibrium point which is defined by:
and
such that:
in this case implies:
We have already noted that from the first, second, fourth, fifth and sixth columns of the Jacobian matrix
the following five distinct negative eigenvalues
, 
, 
and 
are obtained. Removing their corresponding rows and columns, and then performing the row operation, reduces into 
Jacobian matrix 
and the other eigenvalues 
and 
are obtained. The remaining two eigenvalues are obtained from such that:
in this case implies:
, we let 
Then we have:
We have noted that from equation (16) the trace of the Jacobean matrix is negative, 
and from (18) the determinant is positive 
if and only if 
and 
. Therefore the model system (2) is locally asymptotically stable and hence we have proved Theorem 1.
3.4 Global Stability of Disease Free Equilibrium
The global stability of the disease free equilibrium (DFE) is determined using Metzler matrix stability method as stated by Castillo-Chavez et al. (2002). Under this approach the model system (2) is put in the form:
Where;
is a vector representing non disease transmitting compartments
is a vector representing disease transmitting compartments
is a vector representing non transmitting state variables at DFE
are matrices
The global stability of Disease Free Equilibrium (DFE) holds if matrix 
has real negative eigenvalues and 
is a Metzler matrix (i.e. the off-diagonal elements of are non-negative, symbolically denoted by 
,
. From model system (2) we consider the following sub-systems:
We consider the system with non-transmitting compartments of the model system (2) and put in the form as in system (19) we have:
From (22) we perform partial differentiation with respect to the non-transmitting state variables 
at disease free equilibrium (DFE) to obtain matrix as shown in equation (23).
We perform partial differentiation for system (22) with respect to the transmitting state variables; 
, we have matrix 
as shown in equation (24).
We then consider the system with transmitting compartments from model system (2) and put in the form as in system (20) as indicated in equation (25).
We perform partial differentiation for system (25) with respect to the transmitting state variables; , we obtain matrix as indicated in (26).
We have observed that matrix 
in (23) has the eigenvalues with real negative parts, and matrix 
in (26) is a Metzler matrix (Metzler matrix is one with which its off diagonal elements are positive). This shows that the systems (19) and (20) are globally asymptotically stable at Disease Free Equilibrium (DFE). Based on the above conditions our model system (2) is globally asymptotically stable which results into the following theorem:
Theorem 2: The disease-free equilibrium point 
is globally asymptotically stable if
and unstable otherwise.
3.5 The Existence of Endemic Equilibrium Point for the Model
Endemic equilibrium points are steady-state solutions whereby the disease persists in the population (Chitnis et al., 2008). It refers to the situation that shows the persistence of Tungiasis infestation in the community where 
. Let 
be an endemic equilibrium point of model system (2). The conditions for existence of endemic equilibrium point are obtained when setting the right hand side of each equation in (2) equal to zero as shown in (27a)-(27i).
To determine the endemic equilibrium point we use the approach used by Tumwiine et al. (2007) and Massawe et al. (2015) such that; for the existence of endemic equilibrium the condition 
and
must be satisfied. Adding Equations (27a)-(27i) of system (27), we have:
On simplification we have:
From equation (27h), we have:
It follows that:
We assume that all the parameters and control variables are positive and the control variable is less than one hundred percent. We have:
Based on the condition that; 
,
and 
we have:
Since 
and 
This implies that Endemic Equilibrium (EE) point of the Tungiasis disease model in human, animal reservoirs and sand fleas in the soil environment exists. Therefore, it is sufficient to determine the conditions under which, EE is stable or unstable.
3.6 Global Stability of Endemic Equilibrium Point for the Model
In this section we investigate the global stability of the Endemic Equilibrium (EE) using the Lyapunov method and LaSalle's invariance principle. The approach which has been found to be useful for compartmental epidemic models with any number of compartments (Korobeinikov and Maini, 2004). To determine the global stability of the endemic equilibrium we construct a suitable Lyapunov function of the form:
where 
are constants, 
is the population of 
compartment, and 
is the equilibrium value of . 
The function 
is continuous and differentiable with respect to time. The global stability for endemic equilibrium point holds if 
, then the time derivative of is given by:
At endemic equilibrium, we have the following relations
Substituting the derivatives from (2) and the relations from (33) into (32) we have:
is non-positive following the approach by Korobeinikov (2002, 2004, 2007) and McCluskey, 2006). Thus 
for all 
Hence, 
in 
and is zero when
. Therefore the largest invariant set in such that 
is the singleton 
which is our endemic equilibrium point. By LaSalle’s invariant principle (LaSalle, 1976) we conclude that is globally asymptotically stable. Thus, we establish the following theorem.
Theorem 3: The endemic equilibrium point 
of model system (2) is globally
asymptotically stable in if 
and unstable otherwise.
4 Numerical Simulations and Discussion of the Results
In this section we present the simulation of the general dynamics for the extended model and for the reproduction numbers. We show that, the control strategies based on insecticides application to the premises, dusting of the animals with insecticidal powder and environmental hygiene, have the positive impact on the reduction of disease transmission. Moreover we compare the effectiveness of the control measures when applied in combination of two, three and as a single entity. We use a set of parameter values from literature and others are estimated as shown in Table 3.
Table 3: Parameter Values Used for the Simulation of Model Variables
Parameter | Value/Range | Source/References |
|
| Estimated |
|
| Estimated |
| 0.0105per day | Estimated |
| 0.011 per day | Estimated |
| 0.037 per day | Estimated |
| 0.000045 per day | UNICEF. (2015) |
|
| Gaff et al. (2007); Radostits. (2001) |
| 0.04 per day | Eisele et al. (2003), |
| 0.58 per day | Estimated |
| 0.08 per day | Estimated |
| 0.19 per day | Estimated |
| 0.48 per day | Estimated |
| 0.052 per day | Gaff et al. (2007) |
| 0.26 (0.091-0.9) per day | Allerson et al. (2013) |
| 0.00011 per day | TP, (2016). |
| 0.022 per day | Gaff et al. (2007) |
| 0.40 per day | Estimated |
| 0.12 per day | Estimated |
| 0.4 | Estimated |
| 0.6 | Estimated |
| 0.15 | Estimated |
| 0.15 | Estimated |
| 0.09 | Estimated |
| 0.3, 0.4, 0.3, 0.2, 0.3 | Estimated |
4.1 Simulation of Controlled Model Variables
In this section we perform numerical simulations to show the controlled dynamics for animal and flea populations in Figure 2(a) given the model variables 
and in Figure 2(b) is the dynamics for human population for the model variables
.
Figure 2 (a): The Controlled General Dynamics for Animal and Flea Populations
From Figure 2(a): The solid green line indicates the susceptible animal population which increases during intervention period and after about 160 days it starts decreasing and eventually attains the endemic equilibrium level. The solid magenta line indicates the mildly infested animal population which increases gradually during intervention period of 150 days, then it rapidly increases for about 50 days and after 200 days it declines to attain the endemic equilibrium level. The dotted red line indicates the severely infested animal population which decreases for sometimes and maintains a constant level, after 150 days it starts increasing and then decreases to attain the endemic equilibrium level. The dotted blue line indicates the larvae population which initially decreases, maintains the constant level for about 150 days and starts increasing and eventually declines to attain endemic equilibrium level. The dotted black line indicates the adult flea population which initially decreases, maintains the constant level for about 150 days and starts increasing and eventually declines to attain endemic equilibrium level.
Figure 2(b): The Dynamics in Human Population Variables with Control Measures Implemented
From Figure 2(b): The solid red line indicates the susceptible human population which grows and maintains constant value then after about 150 days it declines and attain endemic equilibrium level. The solid black line indicates the mildly infested human population which initially decreases and maintains constant level, then after about 150 days it starts increasing and eventually declines to attain the endemic equilibrium level. The solid cyan line indicates the severely infested human population which initially decreases for about 20 days and maintains constant level then after about 150 days it starts increasing and eventually attains the endemic equilibrium level. The solid blue line indicates the treated human population which starts from zero and increases for about 10 days then decreases to maintain a constant level and after 150 days it increases for some days and finally attains the endemic equilibrium level.
4.2 Simulation of Model Reproduction Numbers
We perform numerical simulations of the model reproduction numbers when the control measures are implemented on the Tungiasis basic dynamical model under different scenario. We show that the control measures based on insecticides application to the premises denoted by 
, dusting of the animals with insecticidal powder denoted by 
and environmental sanitation denoted by 
, have the positive impact on the reduction of the disease transmission because they lower the basic reproduction number, 
.
Figure 3: Variation of Reproduction Numbers with Contact Rate
From Figure 3: Given that 
is the effective reproduction number when all control measures are implemented, 
is the effective reproduction number when a control effort based on insecticides spraying on the soil environment is implemented, 
is the effective reproduction number when a control effort based on environmental sanitation is implemented, 
is the effective reproduction number when a control effort based on dusting the domestic animals with ant-flea compound is implemented. 
is the effective reproduction number when a controls effort and are combined and implemented, 
is the effective reproduction number when the control efforts and are combined and implemented, 
is the effective reproduction number when the control efforts and are combined and implemented and 
is the model basic reproduction number. The model effective reproduction numbers are less than the basic reproduction number such that; 
. We have also noted that the control measures implemented two at a time based on 
and
yield better results than
. This implies that any combination which associates the control measure based on insecticides application to the premises yield the better results than other combinations without it. Moreover we have observed that multiple controls are not necessarily better than a single control strategy due to the fact that
. However, the best results are achieved when all the three control measures are implemented.
5 Conclusion
In this paper, a mathematical model of control strategies for Tungiasis disease in endemic settings has been established and both qualitative and numerical analysis of the model was done. The disease free and endemic equilibrium points were obtained and their stabilities investigated. The model showed that the disease free equilibrium is locally and globally asymptotically stable at the threshold parameter less than unity
and unstable at the threshold parameter greater than unity. Using Lyapunov method, endemic equilibrium is globally asymptotically stable when. The numerical simulation was conducted to shows how the model trajectories behaves over time when the control measures are implemented, to investigate the effects of control strategies on the spread of the disease and to evaluate the effectiveness of the control strategies when the control measures are implemented one at a time, two at a time and three at a time. It was observed that the control measures implemented on the basic dynamical model have the positive impact on the reduction of transmission of Tungiasis epidemic and that they work better in combination than when applied as a single entity. Further we have observed that; any combination involving insecticides application yield better results than any other combination, so insecticides spraying control strategy is more effective followed by on animal insecticidal powder spray control strategy. Since reduced transmission leads to lower prevalence of Tungiasis in the long-term, the decision markers from national health care should therefore seek to ensure that all people at risk with Tungiasis have access to and are supported in the use of control strategies based on; educational campaign, personal protection such as; wearing of solid closed shoes, surgical extraction of the embedded fleas from the human body and application of plant based repellents especially on the human feet, environmental sanitation, on-animal insecticide powder application on animal furs and insecticides spraying on the premises regardless of their social and economic status.
References
1. Allerson, M., Deen, J., Detmer, S. E., Gramer, M. R., Joo, H. S., Romagosa, A., and Torremorell, M. (2013). The impact of maternally derived immunity on influenza A virus transmission in neonatal pig populations. Vaccine, 31(3), 500-505.
2. Buckendahl, J., Heukelbach, J., Ariza, L., Kehr, J. D., Seidenschwang, M., and Feldmeier, H. (2010). Control of tungiasis through intermittent application of a plant-based repellent: An intervention study in a resource-poor community in Brazil. PLoS Negl Trop Dis, 4(11), e879.
3. Castillo-Chávez, C., Feng, Z., and Huang, W. (2002). On the computation of basic reproduction number and its role in global stability. Mathematical approaches for emerging and reemerging infectious diseases: an introduction, 1, 229.
4. Cestari, T. F., Pessato, S., and Ramos-e-Silva, M. (2007). Tungiasis and myiasis. Clinics in dermatology, 25(2), 158-164.
5. Diekmann, O., Heesterbeek, J. A. P., and Metz, J. A. (1990). On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of mathematical biology, 28(4), 365-382.
6. Dumont, Y., Chiroleu, F., and Domerg, C. (2008). On a temporal model for the Chikungunya disease: Modeling, theory and numerics. Mathematical biosciences, 213(1), 80-91.
7. Eisele, M., Heukelbach, J., Van Marck, E., Mehlhorn, H., Meckes, O., Franck, S., and Feldmeier, H. (2003). Investigations on the biology, epidemiology, pathology and control of Tunga penetrans in Brazil: I. Natural history of tungiasis in man. Parasitology research, 90(2), 87-99.
8. Feldmeier, H., Kehr, J. D., and Heukelbach, J. (2006). A plant-based repellent protects against Tunga penetrans infestation and sand flea disease. Acta tropica, 99(2), 126-136.
9. Gaff, H. D., Hartley, D. M., and Leahy, N. P. (2007). An epidemiological model of Rift Valley fever. Electronic Journal of Differential Equations, 2007(115), 1-12.
10. Heukelbach, J., De Oliveira, F. A. S., Hesse, G., a Feldmeier, H. (2001). Tungiasis: a neglected health problem of poor communities. Tropical Medicine and International Health, 6(4), 267-272.
11. Heukelbach J., Costa AML, Wilcke T, Mencke N, Feldmeier H (2004) The animal reservoir of Tunga penetrans in severely affected communities of north-east Brazil. Med Vet Entomol 18:329-335
12. Hirsch, M. W., Smale, S., and Devaney, R. L. (2012). Differential equations, dynamical systems, and an introduction to chaos. Academic press.
13. Kahuru, J., Luboobi, L., and Nkansah-Gyekye, Y. (2017). Modelling the dynamics of Tungiasis transmission in zoonotic areas. Journal of Mathematical and Computational Science, 7(2), 375-399.
14. Keeling, M. J., and Rohani, P. (2008). Modeling infectious diseases in humans and animals. Princeton University Press.
15. Korobeinikov, A., and Wake, G. C. (2002). Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models. Applied Mathematics Letters, 15(8), 955-960.
16. Korobeinikov, A and Maini, P. (2004). A Lyapunov function and global properties for SIR and SEIR epidemi-olgical models with nonlinear incidence, Math. Biosci. Eng., 1:57–60.
17. Korobeinikov, A. (2007). Global properties of infectious disease models with nonlinear incidence. Bulletin of Mathematical Biology, 69(6), 1871-1886.
18. LaSalle, J.P. (1976). The stability of dynamical systems, CBMS-NSF regional conference series in applied mathematics 25. SIAM, Philadelphia
19. Massawe, L.N., Massawe, E.S. and Makinde, O.D. (2015) Temporal Model for Dengue Disease with Treatment. Advances in Infectious Diseases, 5, 21. http://dx.doi.org/10.4236/aid.2015.51003
20. McCluskey, C. C. (2006). Lyapunov functions for tuberculosis models with fast and slow progression. Math. Biosci. Eng. 3, 603-614.
21. Muehlen, M., Feldmeier, H., Wilcke, T., Winter, B., and Heukelbach, J. (2006). Identifying risk factors for tungiasis and heavy infestation in a resource-poor community in northeast Brazil. Transactions of the Royal Society of Tropical Medicine and Hygiene, 100(4), 371-380.
22. Mutebi, F., Krücken, J., Feldmeier, H., Waiswa, C., Mencke, N., Sentongo, E., and von Samson-Himmelstjerna, G. (2015). Animal reservoirs of zoonotic tungiasis in endemic rural villages of Uganda. PLoS Negl Trop Dis, 9(10), e0004126.
23. Mwangi, J. N., Ozwara, H. S., and Gicheru, M. M. (2015). Epidemiology of tunga penetrans infestation in selected areas in Kiharu constituency, Murang’a County, Kenya. Tropical Diseases, Travel Medicine and Vaccines, 1(1), 13. doi:10.1186/s40794-015-0015-4
24. Pilger, D., Schwalfenberg, S., Heukelbach, J., Witt, L., Mehlhorn, H., Mencke, N., and Feldmeier, H. (2008). Investigations on the biology, epidemiology, pathology, and control of Tunga penetrans in Brazil: VII. The importance of animal reservoirs for human infestation. Parasitology research, 102(5), 875-880.
25. Radostits, O. M., Leslie, K. E., and Fetrow, J. (1994). Herd health: food animal production medicine (No. Ed. 2). WB Saunders Company.
26. Roussel, M. R. (2005). Stability analysis for odes. Nonlinear Dynamics, lecture notes. University Hall, Canada.
27. Ruttoh, S. K., Ochieng'Omondi, D., and Wanyama, N. I. (2012). Tunga penetransnA Silent Setback to Development in Kenya. Journal of Environmental Science and Engineering. B, 1(4), 527–34.
28. Tanzania population 2016. http://worldpopulationreview.com/countries/tanzania-population/. Retrieved on 3rd August, 2019.
29. Thielecke, M., Raharisolo, C., Ramarokoto, C. H., Rogier, C., Randriamanantena, H., and Stauss-Grabo, M. (2013). Prevention of tungiasis and tungiasis-associated morbidity: a randomized, controlled field study in rural Madagascar. PLoS Negl Trop Dis, 7, e2426.
30. Tumwiine, J., Mugisha, J. Y. T., and Luboobi, L. S. (2007). A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity. Applied Mathematics and Computation, 189(2), 1953-1965.
31. Ugbomoiko, U.S., (2007). Tungiasis: High prevalence, parasite load, and morbidity in a rural community in Lagos State, Nigeria. International Journal of Dermatology, 46(5), pp. 475-481.
32. UNICEF. (2015). State of The World’s Children 2015 Country Statistical Information.
33. Van den Driessche, P., and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1), 29-48.
34. Wafula, S. T., Ssemugabo, C., Namuhani, N., Musoke, D., Ssempebwa, J., and Halage, A. A. (2016). Prevalence and risk factors associated with tungiasis in Mayuge district, Eastern Uganda. The Pan African Medical Journal, 24, 77. http://doi.org/10.11604/pamj.2016.24.77.8916.
