Abstract
In this study, a mathematical model for the transmission dynamics and control of schistosomiasis is studied using a system of nonlinear ordinary differential equations. The basic reproduction number for the model is obtained, and its dependence on model parameters is discussed. Analytical results reveal that the disease-free equilibrium point is globally stable if and only if the basic reproduction number , indicating that the disease would be wiped out of the community, and the endemic equilibrium point is globally stable if and the disease would persist at the endemic steady state. Numerical simulations reveal that a combination of treatment, public health education, and chemical control intervention strategies significantly increased the number of susceptible human population and susceptible snail population while significantly decreasing the number of infected humans, miracidia, infected snails, and cercariae. The results further indicate that a combination of treatment, public health education, and chemical control intervention strategies can effectively manage the transmission of schistosomiasis in endemic areas.
1. Introduction
Schistosomiasis, commonly known as bilharzia or snail fever, is a disease caused by parasitic flukes of genus Schistosoma. It is listed among the neglected tropical diseases (NTDs) and is endemic in many countries, particularly in tropical countries of Africa, South America, and Asia. Schistosomiasis is the second most devastating tropical disease after malaria [1], and its prevalence has continued to prevail amidst efforts to curb it.
The human is the definitive host of the parasite and the snail is the intermediate host. The snail lives in fresh, shallow, and slow running water sources. The three main schistosomes that infect humans are Schistosoma japonicum, Schistosoma mansoni which cause intestinal and hepatosplenic disease, and Schistosoma haematobium that causes urogenital pathology. Each species of the schistosome is transmitted by a snail of different species; however, the species share the same pattern in the reproductive cycle [1–3].
The infection cycle begins when an infected host eliminates eggs of the parasite through urine (Schistosoma haematobium) or feaces (Schistosoma mansoni and Schistosoma japonicum) into fresh water bodies [2]. The excreted eggs can stay up to seven days and hatch into miracidia that infect fresh water snails of appropriate species [4]. The miricidia then multiply through asexual reproduction within the body of the snail [1], and within 4–6 weeks after infection, the snails release cercariae into water which can then penetrate the skin of a new definitive host (human) and take about 4–6 weeks to mature in the blood vessels of the urinary or the digestive system. The cercariae continue to develop and mature to adult size and pair in sexual reproduction [5]. The period between cercariae penetration of the human skin and release of schistosome eggs in human waste takes about five weeks [6] and subsequent egg laying continues the life cycle of the parasite. Drinking water or eating food that has been washed with untreated water can also lead to infection.
The disease is characterized by an itchy skin within the first few hours of penetration, fever, cough, headache, difficulty in breathing, loss of appetite and weight, diarrhea and blood in stool. In advanced cases, children develop thin legs, seizures and spinal cord inflammation which lead to physical and mental development problems [5, 7]. However, most of these conditions can be reversed and prevented through early and regular treatment with praziquantel [8]. Globally, 76.9 million people received treatment for schistosomiasis in 2020, and of these, 59.9 million were school aged children (55.2 million in African region) and 17 million adults (14.5 million in African region) [9].
Studies to determine the effectiveness of the control strategies on schistosomiasis have been undertaken before. Gao et al. [10] studied the effect of treatment and snail control on schistosomiasis transmission, Das et al. [11] studied the optimal control of snail population and sanitation and treatment on schistosomiasis transmission, Nur et al. [12] determined the effect of health education and snail control on schistosomiasis transmission, and Kanyi et al. [13] determined the effect of treatment and sanitation on schistosomiasis transmission. Although there have been studies that address the role of reducing the intermediate host population (snail population) in the spread and control dynamics of schistosomiasis, the combined effect of treatment, public health education, and chemical control interventions on the disease transmission dynamics has not been explicitly established. Therefore, there is need to examine the effectiveness of control measures that include environmental modifications that separate humans from contaminated water sources through public health campaigns and snail reduction with chemical control measures to limit schistosomiasis transmission. In this study, a deterministic mathematical model describing the effect of treatment, public health education, and snail chemical control strategies on schistosomiasis transmission is formulated. The model considers control of schistosomiasis through human treatment using praziquantel and chemical control intervention by use of molluscicide (niclosamide) to reduce snail population and public health education which is intended to create change in human behavior, so as to maintain improved sanitation and hence reduce schistosome worm egg disposal in water by infected individuals.
The rest of the study is structured as follows. In the next section, we present the schistosomiasis model. In Section 3, we find and determine sufficient conditions for the stability of the disease-free and endemic equilibria and analyze the reproduction number of the disease, while Section 4 provides numerical results, and Section 5 concludes the study.
2. Model Formulation
2.1. Variables of the Model
The model consists of the human population divided into two epidemiological classes: the susceptible human population and the infected human population such that the total human population is given by . The snail host population is categorized into two epidemiological classes, namely, the susceptible snail population and infected snail population such that the total snail population is given by . The model also includes the population of the free-living larva stages, the miracidia and cercariae .
2.2. Assumptions
The following assumptions are considered in the formulation of the model:(i)There is no vertical transmission of the disease in humans(ii)There is no immigration of infectious humans(iii)Seasonal and weather variations do not affect snail populations(iv)Susceptible host is infected only by contact with infected water(v)The public health education awareness campaign covers the entire population(vi)Treatment provides cure but does not prevent reinfection(vii)The interaction between the miracidia and the susceptible snails, and the interaction between the cercariae and the susceptible humans has negligible effect on the pathogen population
2.3. Parameters of the Model
The parameters used in the model are described as follows: : Recruitment rate into the human population : Recruitment rate into the snail population : Recovery rate due to treatment : Force of infection of the susceptible human population : Force of infection of the susceptible snail population : Public health education efficacy parameter : Number of schistosome eggs excreted by infected human : Rate at which excreted eggs hatch into miracidia : Rate at which infected snails release cercariae : Rate at which susceptible humans get infected : Rate at which susceptible snails get infected : Limitation rate of growth velocity : Saturation constant for cercariae : Saturation constant for miracidia : Disease induced mortality rates of the human and snail population, respectively : Elimination rates of snail, miracidia, and cercariae by molluscicide, respectively : Natural death rates of the human and snail population, respectively : Natural death rates of the miracidia and cercariae population, respectively
The parameters and are given by and , respectively.
2.4. Equations of the Model
From the compartmental diagram in Figure 1 and applying the assumptions, definitions of variables and parameters above, the following system of differential equations that describe the effect of integrated control strategies on schistosomiasis transmission is obtained:
Let , , , , and , then system (1) can be written as
3. Model Analysis
3.1. Positivity and Boundedness of the Solutions
Theorem 1. Let the initial population be
Then, the solution set of (2) is positive for all .
Proof. From the first equation of system (2), we getThis implies thatBy separating variables, (5) is integrated to givewhere is the constant of integration. Applying the initial condition to (6) gives .
Hence,Therefore,Again from system (2), one getsNow, separating the variables and integrating with the initial conditions givesThus, the solution set of model (2) is positive for all values of .
Theorem 2. The feasible solution set of the model system (2) is mathematically and epidemiologically well-posed in the domain as
Proof. From , it follows thatand integrating (12) with the initial condition givesand as in (13), we getFor the snail host population, we getIntegrating (15) with initial condition givesAs in (16),For the miracidia free-living larva stage subpopulation,and integrating (18) with the initial condition givesAs in (19), we getFor the cercariae free-living larva stage subpopulation,Integrating (21) with initial condition givesAs in (22), we haveThus, the region is positively invariant, and hence, the model is well posed and biologically meaningful.
3.2. The Disease-Free Equilibrium Point
The disease-free equilibrium point is obtained by setting the right-hand side of (2) to zero and is defined at the steady state where there is no disease. Therefore, in absence of the disease, we have . Hence, the disease-free equilibrium point is given as
3.3. Basic Reproduction Number
The basic reproduction number is the average number of secondary infections caused by an infectious individual in a fully susceptible population. It is a threshold for predicting the disease as well as evaluating the effectiveness of the intervention strategies.
The value of is obtained using the next generation operator approach described in [14, 15]. Let be the matrix of the rates of the appearance of new infections in compartment and be matrix of the rates of the transfer of individuals into and out of compartment . Thus, from (2), we have
Let and be the Jacobian matrices of and , respectively, evaluated at the disease-free equilibrium point (24). Then, one gets
Evaluating , we get
The eigenvalues of matrix are obtained fromwhere is a 4 by 4 identity matrix. This givesand the roots of the characteristic (29) are and .
The basic reproduction number is the dominant eigenvalue of the matrix and is thus given aswhere
Thus, the basic reproduction number, , can be written in terms of and , where describes the average number of secondary infections by an infected snail in a completely susceptible population (the average number of susceptible humans infected by the infectious snail) and describes the average number of secondary infections by an infected human in a completely susceptible population (the average number of susceptible snails infected by the infectious human).
3.4. The Endemic Equilibrium Point
The endemic equilibrium point determines the persistence of the disease in the community and this is evaluated when .
Theorem 3. System (2) has a unique endemic equilibrium point when .
Proof. Setting the right-hand side of system (2) to zero, we haveAdding the first two equations of (15) and rearranging givesFrom the third equation of (15), . Substituting this into the fourth equation givesPutting (33) and (34) into the fifth equation of (32) givesFrom the sixth equation of (32), we haveConsequently, (35) and (36) yieldNow, using (34) and (36) in the second equation of (32) and then simplifying giveswhereFrom (38), corresponds to the disease-free equilibrium and indicates endemicity of the disease. Thus, system (2) has a unique endemic equilibrium point when , where
3.5. Local and Global Stability of the Disease-Free Equilibrium Point
Theorem 4. The disease-free equilibrium point of system (2) is locally asymptotically stable whenever and unstable whenever .
Proof. The Jacobian matrix of system (2) evaluated at the disease-free equilibrium point is given byThe disease-free equilibrium point is locally asymptotically stable if the real parts of the eigenvalues of are negative; otherwise, it is unstable.
The eigenvalues of are obtained by setting the characteristic polynomial.
equals to zero. Thus, we haveFrom (42), it is clear that , . The other remaining eigenvalues are obtained by considering the matrixwhose characteristic equation iswhere .
According to [16], the Routh–Hurwitz conditions for the stability of the polynomial (44) are as follows:(i)(ii)(iii)(iv)Clearly (i) is satisfied and (iv) holds true whenever . It remains to check conditions (ii) and (iii):
andandThus, the Routh–Hurwitz conditions hold when all the eigenvalues have negative real parts making locally asymptotically stable whenever .
To study the global asymptotic stability of the disease-free equilibrium, the method described by Castillo-Chavez et al. [17] and in [18] has been employed. The model system (2) is expressed in the form:where comprises of the uninfected subpopulation and comprises of the infected subpopulation. Let be the disease-free equilibrium point of system (47). Then, is globally asymptotically stable provided that and that the following conditions hold true: : For is globally asymptotically stable : for , where is an M-matrix (the off-diagonal elements are non-negative) and is the region where the model makes biological sense
Theorem 5. The disease-free equilibrium point for system (2) is globally asymptotically stable provided and conditions and hold true.
Proof. For system (2), we haveAs , we observe that , that is, and . Thus, holds true:andFrom (49), clearly is an M-matrix (all the off-diagonal elements are non-negative), and from (50), since and . Thus, holds true, and the proof is complete.
3.6. Local Stability of the Endemic Equilibrium
The center manifold theory by Castillo-Chavez [19] is used to establish the local asymptotic stability of the endemic equilibrium point.
Theorem 6. Consider the following general system of ordinary differential equations with a parameter :and without loss of generality, it is assumed that 0 is an equilibrium point of the system (51) for all values of the parameter , that is,
Assume the following points: . is the linearization matrix of system (51) around the equilibrium 0 with evaluated at 0. Zero is a simple eigenvalue of A, and other eigenvalues have negative real parts. . The matrix has a non-negative right eigenvector and a left eigenvector each corresponding to the zero eigenvalue.
Let be the component of , and then, we get
Then, the local dynamics of the system around 0 are completely determined by the signs of and that can be given as follows:(i). When with , 0 is locally asymptotically stable and there exists a positive unstable equilibrium; when , 0 is unstable, and there exists a negative and locally asymptotically stable equilibrium.(ii). When with , 0 is unstable; when , 0 is locally asymptotically stable, and there exists a positive unstable equilibrium.(iii). When with , 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when , 0 is stable, and a positive unstable equilibrium appears.(iv). When changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable.
Theorem 7. The unique endemic equilibrium point, of system (2) is locally asymptotically stable if but close to 1.
Proof. We apply the change of variable on system (2) as follows. Let . The model system (2) can be written as , where and :Taking as the bifurcation parameter while considering the case givesThe linearization around the disease-free equilibrium point evaluated at givesMatrix has zero as an eigenvalue, and this can be deduced from (21)–(23) with , which yieldsThe right eigenvector corresponding to the zero eigenvalue is computed from , which yieldsThe left eigenvector is computed from which yieldsLetting , it can be seen from (58) that any other right eigenvector of matrix corresponding to the zero eigenvalue is a constant multiple of (the same is true for the left eigenvector in (59) by letting ). Hence, zero is a simple eigenvalue.
In the computation of and , only nonzero partial derivatives of system (54) are used. Since , the nonzero partial derivatives computed at the disease-free equilibrium are from , and . These are obtained as and . Thus, it follows thatSince and , then by condition of Theorem 5, system (54) exhibits a forward bifurcation at , implying that the unique endemic equilibrium is locally asymptotically stable when and close to 1.
3.7. Global Stability of the Endemic Equilibrium Point
Theorem 8. If , the endemic equilibrium of system (2) is globally asymptotically stable.
Proof. To analyze the global stability of endemic equilibrium, we consider the candidate Lyapunov function given byClearly, , and , otherwise. Moreover, is radially unbounded in . So, it remains to determine the sign of . Differentiating (62) with respect to givesSubstituting for values of , , , , , and in (63) yieldsRearranging positive and negative terms in (64) giveswhereIf , then noting that if and only if , , , , , and . Therefore, the largest compact invariant set in is the singleton , where is the endemic equilibrium point of system (2). Thus, by Lasalle’s invariance principle [20], it implies that is globally asymptotically stable in if ; which holds if and only if .
3.8. Sensitivity Analysis
In this subsection, sensitivity analysis of the reproduction number on different parameters is carried out in order to determine their relative importance to the disease transmission in the community. The approach by Chitnis et al. [21] has been employed to obtain the relative change in a state variable when the parameter changes. An expression for the sensitivity of with respect to is given bywhere is any sensitive parameter to be considered in relation to .
From (67), expressions for the sensitivity indices are as follows:
The parameter values used are , and .
Table 1 shows the sensitivity indices of with respect to different parameters from the most sensitive to the least sensitive. It is noted that the value of increases when parameter values , , , , , , and increase, while the other parameters values are kept constant. Since these parameters have positive indices, the endemicity of the disease is increased. When parameter , , , , , , , and are increased while the other parameters values are kept constant, the value of decreases implying that the endemicity of the disease is decreased.
4. Numerical Simulations
In this section, we numerically simulate the combined effect of treatment with praziquantel, public health education and chemical control strategies on schistosomiasis transmission. The model was simulated using ODE solvers coded in MATLAB computer software. The parameter values used are shown in Table 2 with the initial conditions such as , , , , , and .
Figures 2 and 3 show numerical simulation for a mathematical model of schistosomiasis with the treatment strategy only, where , , , , and .
The results indicate that with the treatment strategy only, the subpopulations of infected humans, , and miracidia, , decrease in the early stages of treatment and begin to increase again. The subpopulations of susceptible humans and susceptible snails decrease, while the infected snails and cercariae increase. It is noted that with treatment intervention only, schistosomiasis persists within the population.
Figure 4 shows the numerical simulation of schistosomiasis model with treatment and public health education control strategy when , 0.5, and 0.9. The results reveal that the susceptible humans and snail subpopulations increase with increase in public health awareness from to . The subpopulations of infected humans, infected snails, miracidia, and cercariae decrease as increases. This is because of increased awareness among the human population to maintain proper sanitation as well as reduced contact with contaminated water.
Figure 5 shows numerical simulation of schistosomiasis model with the treatment and chemical control intervention strategy only with parameter values , , , , and . From the results, it is noted that the subpopulations of susceptible humans and susceptible snails increase relatively higher than that illustrated in Figures 2 and 3 which considers treatment only. The subpopulations of infected humans, infected snails, miracidia, and cercariae also decrease further as compared to those in Figures 2 and 3.
Figure 6 shows numerical simulation of schistosomiasis model with treatment, public health education, and chemical control intervention strategies with parameter values , , , , and . The results indicate that the subpopulations of infected humans, miracidia, infected snails, and cercariae decrease very significantly compared to other strategies illustrated in Figures 2–5, whereas the subpopulations of susceptible humans and susceptible snail increase significantly.
5. Discussion and Conclusion
In this study, a deterministic mathematical model describing the effect of treatment, public health education, and chemical control intervention strategies on schistosomiasis transmission was formulated and analyzed. The model was found to be well posed and biologically meaningful.
Analysis of the model showed existence of two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. The basic reproductive number of the model was determined, and its biological interpretation explained. was further decomposed into individual reproductive numbers of the transmission agents of schistosomiasis; that is, is the average number of secondary infections by an infected snail in a completely susceptible population and is the average number of secondary infections by an infected human in a completely susceptible population. The disease-free steady state was found to be locally stable whenever . The disease-free equilibrium point was also globally stable implying that all initial conditions would converge to it over time; hence, treatment can decrease the disease prevalence. The endemic equilibrium point of the model was found to be locally stable if and only if . This meant that interventions by policy makers are necessary to eradicate schistosomiasis over time. In order to make endemic equilibrium point unstable, control measures such as mass drug administration in endemic areas are necessary.
Sensitivity analysis identifies the rate at which susceptible humans get infected, and rate at which susceptible snails get infected, as the main parameters fueling the spread, while the and are the elimination rates of miracidia and snails by molluscicide (chemical control) as inhibitors of the disease. Thus, from the results, it is noted that a combination of treatment, public health education, and chemical control intervention strategies can effectively manage the transmission of schistosomiasis in endemic areas.
Numerical simulations reveal that a combination of treatment, public health education, and chemical control intervention strategies significantly increase the number of susceptible human and susceptible snail populations while significantly decreasing the number of infected humans, miracidia, infected snails, and cercariae as shown in Figure 6. The study shows that treatment only in Figures 2 and 3 is less effective and the disease persists within the population, which agrees with the results of Gurarie et al. The study also reveals that a combination of both treatment and chemical control intervention, illustrated in Figure 5 has more impact compared to that of treatment only in Figures 2 and 3 which agrees with the results of Gao et al. [10].
In conclusion, in order to effectively manage the prevalence of schistosomiasis transmission, there should be a combined control intervention of treatment, public health education, and chemical control intervention strategies.
Data Availability
The data to support this study were obtained from the literature.
Conflicts of Interest
The authors declare that there are no conflicts of interest.