Abstract
Giardiasis is among the ignored zoonotic illnesses accorded by the World Health Organization that is caused by Giardia duodenalis. The disease is ignored regardless of the harm it causes to people and other creatures. In this paper, a mathematical model for giardiasis illness transmission is formed, which considers sickness carriers and control measures such as screening, treatment, and sanitation of the environment around people. In the assessment, the basic reproduction number, , which is used for analyzing the local stability of the equilibria is determined using the state-of-the-art next-generation matrix, while the Metzler constancy speculation is used to show the overall adequacy of the global stability of the equilibrium point free from the disease. In addition, a Lyapunov function has been used to study the stability of the endemic equilibrium point. The assessment of parameters is performed to explore the limits that significantly influence the transmission components of the disease disorders using the normalizing sensitivity index method. The result revealed that the recruitment rate is the most sensitive limit to the reproduction number. The environment-human interaction parameter is the second influential factor in the transmission of giardiasis in the community. In the same manner, the outcomes recommend that carriers assume an expected part in the rate of giardiasis subsequently; disregarding them could risk endeavors to control the pestilence. Besides, the mathematical recreation of the model shows that a mix of each of the three interventions fundamentally affects the control of giardiasis. In this way, we advise implementing the strategies simultaneously in endemic areas to effectively stop the spread of the giardiasis disease in humans.
1. Introduction
Giardiasis is a digestive contamination brought about by a parasite called Giardia lamblia, Giardia gastrointestinal, or Giardia duodenalis. The main cause of diarrhea in children below the age of five is Giardia protozoan parasites, particularly in poor countries. The life cycle of Giardia normally begins with the ingestion of cysts that are located in food and water contaminated by host feces. The transmission of diarrhea increases by using unsafe water and inadequate sanitation and hygiene during food preparation [1]. It is worth noting that only one excretion of the infected person can release as much as 109 cysts, while with only 10 cysts, giardiasis can begin in the community [2, 3]. At present, there are eight groups of genes from A to H that are recognized for giardiasis, but only A and B are species that are unfavorable for humans, and other genotypes C to H mostly infect animals. Symptoms during infections might include severe diarrhea, nutrient malabsorption, cognitive and developmental defects, fever, itchy skin, weight loss, stomach cramps, greasy poop that can float, an upset stomach, or nausea, to mention a few [4, 5].
The parasite infects both developing and developed countries, with widespread in developing countries. According to the World Health Organization, this disease has been designated as an ignored Diseases Initiative in September 2004 for its high burden and association with poverty [6]. The incidence of giardiasis worldwide is estimated to be cases per year. Around 200 million people have been detected with giardiasis symptoms due to poor sanitation and access to safe drinking water in developing countries. The high incidence of disease in developing countries is due to the fact that most African countries are facing difficulties with accurately identifying, detecting, and reporting infectious diseases as a result of remote communities, poor transport, and a shortage of qualified health workers and laboratory facilities for accurate diagnosis [5, 7].
The risk of giardiasis contamination accelerates with the consumption of raw food. That is why international recommendations provide innocuousness in food preparations, like practicing appropriate hand hygiene for protection against protozoan parasites and maintaining food packed or closed. Also, it is insisted on separating raw from cooked food, using purified or boiled water, and making sure that food is cooked at a temperature of more than 70°C [8, 9].
A substantial contribution has been made in mathematics modeling for a better understanding of the epidemiology and dynamics of diseases, e.g., giardiasis. Saul and Nyerere [10] formulate a giardiasis mathematical model that includes humans, domestic animals, and contaminated environments in order to assess the dynamics of the disease. The results showed that the transmission from individual to individual is the most significant in the dynamics of giardiasis. The giardiasis mathematical model describing the Giardia transmission dynamics in rural Australia has also been developed by Waters et al. [16]. The study showed that the endemic infection of an animal with zoonotic protozoa can lead to epidemic infections in humans, although there is no human-to-human transmission. These findings demonstrate the importance of transmissible zoonoic species via environmental reservoirs. In addition, Li et al. [11] developed a mouse model of infection to investigate immunity against secondary infections caused by Giardia duodenalis. The study suggests that, because of the emergence of robust immunity to reinfection, an effective giardiasis vaccine can be developed in adult mouse models. Despite a number of studies looking at the dynamics of this disease, giardiasis is still affecting many people in poor countries. To achieve the eradication of this disease, current control interventions need to be assessed. Thus, if we are to eradicate or curb the disease, there is a need to assess the present control interventions. Until now, there are no previous studies that have considered screening, treatment, and sanitation interventions to study the dynamics of giardiasis. The purpose of the study is thus to gain insight into the impact of screening, treatment, and sanitation on giardiasis transmission dynamics.
The order of the manuscript is as follows: the model is formulated in Section 2, with a dynamical analysis of the mode in Section 3. The model sensitivity analysis with its interpretation has been done in Section 4. Finally, we performed numerical simulation and discussion in Section 5, which leads us to the conclusion of Section 6.
2. Model Description and Formulation
The human population is subdivided into five groups, including susceptible individuals , exposed individuals , infected individuals , carrier or asymptomatic individuals , and the removed population , as described in the basic model in Figure 1. The susceptible state is the state that involves healthy individuals who are at risk of getting giardiasis disease. The exposed population involves individuals who have been infected but have not yet developed clinical symptoms of the disease and are incapable of infecting other humans. The infectious individuals represent individuals who are actively infected and manifest all clinical symptoms of the disease. The infected individuals are capable of infecting other individuals through unhygienic interactions as well as shedding giardiasis pathogens into the environment (food and water). The infected individual under this transmission is a key player in the zoonotic aspect. Transmission of Giardia normally occurs through the ingestion of infectious cyst stages that are excreted in human or animal feces. Cysts may be present in water, food, or utensils contaminated with feces from humans and animals. The infected individual can then pollute the environment (food and water) with the Giardia through touching and making it unhygienic if proper controls like washing hands before eating and improper handling of food are not well maintained [12, 13]. The fourth state is the asymptomatic population . This population includes carrier individuals who do not show clinical signs of the disease but transmit the Giardia pathogens to other individuals through shedding pathogens into the environment in direct or indirect ways. The recovered population resents individuals who recover after gaining immune-supportive services, treatment, and/or naturally. The Giardia population is represented by the letter . The human population is recruited by birth and the loss of immunity at the rates and , respectively. A susceptible human acquires pathogens from both the environment and infected individuals with the force of infections that follows the standard mass action principle. Therefore, is the combination of three forces of infection which is defined as follows:
The parameter represents the direct transmission of the disease pathogens from the infected person to the susceptible person; represents the direct transmission of the disease pathogens from the carrier person to the susceptible person. The indirect pathway of disease pathogens from the environment to a susceptible person is represented by a parameter . Once infected, the Giardia pathogens incubate for 1–4 weeks in the human intestine and thus progress and become infectious at the incubation rate , and the remaining portion, , becomes asymptomatic to the disease. Also, a portion and recovers naturally and by treatment, respectively, while become asymptomatic or carriers of the Giardia pathogens. In addition, the infected human can die at the rate . The carrier or asymptomatic person can recover naturally at the rate and die at the rate . The general population is subjected to screening in order to identify any potential carriers; consequently, the identified carriers join the infectious population at the rate . The recovered individuals are reduced by the individuals who become susceptible after losing immunity at the waning rate . Moreover, both the infected and asymptomatic individuals can shed Giardia pathogens at the rates and , respectively. All human populations are also naturally reduced at the rate . The Giardia population grows through deposited pathogens from both the infected and asymptomatic human populations into the environment. The cysts are reduced from the environment by death at a rate of .
From the descriptions of the parameters in Table 1 and the model in Figure 1, a system of the nonlinear ordinary differential equations is formulated as follows: together with nonnegative initial conditions
2.1. The Basic Characteristics of the Giardiasis Model
2.1.1. Positivity of the Solution
The positivity and boundedness of the solutions of the model system (2) are tested to see if the model is well-posed and biologically meaningful [17, 18]. Here, it suffices to see that all variables of the model are positive and well-posed in the invariant region.
, where is the total human population, and, hence the following Lemma:
Lemma 1. Given , , , , , and , the region of the model (1) positively invariant .
Proof. The model is biologically meaningful if state variables are nonnegative . Now, by using the first equation of the model (2) we then have integrating equation (5) using separation of variables techniques results in Therefore, using equation (6) as , . Similarly, other model variables can be shown and verified nonnegative. This concludes that all equations of the model system (2) have nonnegative solutions such that for , we have , , , , and .
2.1.2. Invariant Region
According to proposition 4.1 of [18, 19], it immediately follows that is positively invariant for the dynamical system (2) which means that any trajectory starting from the region remains inside for the time .
Proof. Using the dynamical system (2) and initial conditions , the sum of human population is as follows:
Using the positivity of the solution, at disease-free, . Hence,
As a result, the general formula for equation (8) is given as follows in applying the variable separation technique:
As , then that is . Hence, the initial conditions are bounded in time .
Also, by considering the Giardia pathogens population, we have
Hence, the solution . This shows that all solutions (, , , , , and ) of the model system (2) are attracted in the invariant region . Therefore, model system (2) is biologically meaningful.
3. Dynamical Analysis of the Model
3.1. Equilibrium Point Free from Disease
The model system (2) has a unique equilibrium point free from disease which is obtained by putting all of the infected classes equal to zero, that is, at an equilibrium point free from disease and . Thus, solving the system is given by
3.2. Basic Reproductive Number
The basic reproductive number referred to the number of secondary cases caused by one infectious person during the entire [20] in a totally susceptible population. Using the next-generation operator method, the basic reproduction number is calculated. It shall be obtained by taking a dominant eigenvalue from the matrix. It is achieved by taking a lead eigenvalue from the matrix where is the rate of appearance of new infection in compartment , is the transfer rate of individuals from one compartment to another, with denoting the rate of transfer of individuals out of the compartment , is the rate of transfer of individuals into compartment , is the infected classes (, , , and ) of model system (2), and is the disease-free equilibrium point. For the model system (2), the infectious classes are as follows: where
From the system (13), we obtain
Then, the partial derivative of and with respect to , , , and evaluated at gives
It follows that the reproduction number is the spectral radius of the next-generation matrix given as given that where are partial reproduction numbers induced by being susceptible to infectious transmission, susceptible to asymptomatic carrier transmission, and susceptible to environmental transmission, respectively.
3.3. Local Stability of an Equilibrium Point Free from Disease
Local stability of an equilibrium point free from the disease is determined by first finding the Jacobian matrix of the model system (2) concerning each state variable (i.e., , , , , , and ). Based on the sign of the real parts of the eigenvalues evaluated at of the Jacobian matrix, the stability of the model system (2) will be evaluated. The partial differentiation of the model system (2) with respect to , , , , , and at gives the Jacobian matrix as
where , , , , , , and have the same meaning as in (14).
From matrix (20), the first columns have diagonal entries. Therefore, the diagonal is the first eigenvalue of the Jacobian matrix (20). Thus, excluding that column and row containing , the rest eigenvalues are calculated. Then, the reduced matrix from (20) becomes
Again, it is easily seen that one eigenvalue of matrix is . After omitting the diagonal entry of matrix , the matrix (20) is reduced to matrix and becomes and the characteristic polynomial for the remaining matrix (22) is
The corresponding Routh-Hurwitz matrix of the polynomial (23) is where
The equilibrium point free from the disease is locally asymptotically stable if the principal leading minors of are all positive for Thus,
Then, the remaining four eigenvalues of the Jacobian matrix (22) have negative real parts if they satisfy the Routh-Hurwitz criteria [21, 22], that is, , . It can be noted that since , if , if, and if and .
The following theorem is therefore established.
Theorem 2. If the leading minors of are all positive, then the equilibrium point free from the disease of the model system (2) is locally asymptotically stable when .
Biologically, it indicates that giardiasis can be eliminated from the population provided that the size of the populations of the model (2) is on the basis of the attraction of an equilibrium point free from the disease when , that is to say, an outbreak does not arise when infectious human beings are introduced into a community of susceptible individuals. By contrast, if it is surely that giardiasis in the population, then it means that this disease is persisting.
3.4. Global Stability of the Equilibrium Point Free from the Disease
The approach of Castillo-Chavez is applied to analyze the global stability of the disease-free equilibrium solution of the model system (2) (Castillo et al. [23]). Using this approach, the model system (2) can be written as follows: where stands for classes of at-risk individuals such as . stands for carrier and infected individuals, that is, , where stands for transpose of and while is at equilibrium point free from the disease . The matrices and are obtained by differentiating the nontransmitting equations of the model system (2) to nontransmitting and transmitting state variables, respectively. The equilibrium point free from the disease is globally asymptotically stable if the real part of the eigenvalues of is negative and is a Metzler matrix (that is, the off-diagonal elements of are nonnegative). Thus, from the model system (2)
The eigenvalues of are and . Furthermore, the matrix is obtained by differentiating the transmitting equations of the model system (2) with respect to transmitting variables at the equilibrium point free from the disease, and it is given as follows:
It can be noted, that the eigenvalues of are all negative and real. In addition, the matrix is a Metzler matrix since all of its off-diagonal elements are positive. This shows that at the equilibrium point free from the disease, is globally asymptotically stable.
3.5. The Endemic Equilibrium Point
The giardiasis present equilibrium point is a point that found when the model system (2) is set to zero and solved simultaneously [24] in conditions that , , and , where
3.6. Stability of Endemic Equilibrium Point
Theorem 3. The endemic equilibrium point of the model system (2) is globally asymptotically stable if .
Proof. Consider the following Lyapunov function as used in [18, 24] with state variables and nonnegative Lyapunov constants , then the first derivative of function is By using the model system (2), equation (32) is expressed as follows:
At the equilibrium point, we have the following constants:
By inserting the constants (34) into equation (33) we then have
The function (35) became less than or equal to zero when , , , , , and , following the methodology employed in [18, 24, 25]. As a result, the endemic equilibrium point, , is the singleton with the largest compact invariant set in the domain for which . We demonstrate that the endemic equilibrium point is globally asymptotically stable in the invariant set for as noted in the work of Mukandavire et al. [25] and LaSalle [26]. Numerically, Figure 2 shows the phase portraits that are asymptotically converging to the endemic point.
4. Sensitivity Analysis
In this section, we study the robustness of the model parameters through sensitivity analysis using the basic reproduction number . The analysis helps identify which parameters require greater attention to successfully manage the disease at the appropriate moment. The sensitivity index of each parameter of the model (2) is analytically calculated through the normalized forward sensitivity index [24, 27–29]. For example, the sensitivity index of for is obtained using the formula . Sensitivity indices for other parameters of can be computed using a similar method. It can be noticed from Table 2 that the parameters , , , , , , and have positive sensitivity indices stipulating that the increasing of these parameters leads to the increase of and thus brings a chance of disease outbreak. On the other hand, the parameters , , , , , , , , , , and have negative indices. This indicates that increasing any parameter in this group while maintaining other parameters constant lowers the basic reproduction number, therefore lowering the burden of disease in the population. The decreases with an increase in the parameters catering for control measures (, , and ). This suggests that a suitable combination of sanitation, screening for asymptomatic carriers, and treatment of symptomatic patients as an optimal control measure is required to address the problem of giardiasis in the population.
5. Numerical Simulation and Discussion
In this section, the numerical simulations of the model system (2) were carried out using the MATLAB ODE45 solver. In order to illustrate the specific behavior of the model, the initial conditions of the state variables have been arbitrarily chosen and are as follows: , , , , , and . To support some of the analytical results previously presented, various graphical representations are presented and discussed. Since many of the parameters were not readily available, we used those we found in the literature review while making assumptions about others for the sake of illustration. Parameter values in Table 3 were used to perform the simulations.
5.1. Effect of Treatment Intervention
Figures 3(a)–3(c) demonstrate that with an increased treatment rate, , there is a corresponding decrease in exposed individuals, . It can also be witnessed from Figure 3(b) that an increase in treatment rates tends to significantly lower the infection related to this disease in infectious individuals, . On the other hand, it can be observed that adequate treatment indicates that more individuals can recover; see Figure 3(c).
(a)
(b)
(c)
5.2. Effects of Sanitation Intervention
Figures 4(a) and 4(b) show that as the sanitation rate increases from 1% to 60%, there is a significant decline in the number of both infectious and carrier individuals. Also, it can be noticed that as the sanitation rate increases, the number of giardiasis pathogens in the surroundings decreases, as depicted in Figure 4(c). The results suggest the significance of undertaking sanitation if we are to end this epidemic.
(a)
(b)
(c)
5.3. Effect of Screening Intervention
Here, we go over how screening interventions can help prevent the giardiasis disease. Figure 5(a) demonstrates that an increase in screen rate results in a drop in carriers, as those who are being screened eventually enroll in the infectious (see Figure 5(b)) class and receive treatment. This finding suggests that if giardiasis carriers are not detected and treated, the disease will spread more quickly. Similarly, it can be observed from Figure 5(c) that, as a result of more screening, giardiasis is greatly minimized as screened individuals receive medical attention, which also reduces environmental contamination.
(a)
(b)
(c)
5.3.1. Effect of Combined Interventions
The impact on the transmission dynamics of giardiasis disease of using sanitation and screening as complementary to treatments is shown in Figure 6. The data showed that there was a larger reduction in the number of isolated individuals when all three interventions were used at the same time, as opposed to applying two interventions simultaneously.
6. Conclusion
A deterministic mathematical model for the transmission dynamics of giardiasis has been developed and analyzed in the present work. The model, which considers both direct and indirect transmission, consists of five compartments: susceptible human, exposed human, infectious human, recovered human, and the Giardia population. The model is a system of ordinary differential equations that include multiple interventions in the presence of carriers. We calculated the reproduction number using the next-generation, , and utilized it to ascertain the stability of the DFE point. It was determined that the DFE point was unstable when and locally asymptotically stable for . The DFE point was globally asymptotically stable when , according to the Metzler stability theory. The developed model includes intervention strategies such as sanitation, treatment, and carrier screening. Our results demonstrate that including these control measures in the model provides more major benefits for the eradication of the Giardia epidemic. The analysis of the model parameters reveals that hygiene, screening, and treatment control strategies have a negative value as their increase reduces the risk of transmission of disease in the community. As control interventions increase, model simulations demonstrate that the number of infectious individuals decreases more rapidly over time. In addition, numerical simulations demonstrate that when all three interventions are combined (treatment, screening, and sanitation), they significantly reduce the prevalence of disease in the population faster than with just two interventions. Since the model has shown that treatment, screening, and sanitation facilities have a major influence in reducing the number of giardiasis cases in the community, we thus recommend that these areas be enhanced in endemic regions. Since the model presented in this work is not all-inclusive, its underlying assumptions can be modified to take into consideration additional treatments, including public health education, which is essential in increasing illness awareness. In addition, the assumptions might be loosened to accommodate the cost-effectiveness tactics of the control.
Data Availability
The authors certify that the data used to support the conclusion of the study are included in the manuscript and its references.
Conflicts of Interest
There are no competing interests among the writers.
Acknowledgments
The writers express their gratitude to the College of Business Education and Dar es Salaam University College of Education for providing the necessary time and space to enable them to successfully complete the writing of the manuscript.