Abstract
Mathematical models become an important and popular tools to understand the dynamics of the disease and give an insight to reduce the impact of malaria burden within the community. Thus, this paper aims to apply a mathematical model to study global stability of malaria transmission dynamics model with logistic growth. Analysis of the model applies scaling and sensitivity analysis and sensitivity analysis of the model applied to understand the important parameters in transmission and prevalence of malaria disease. We derive the equilibrium points of the model and investigated their stabilities. The results of our analysis have shown that if , then the disease-free equilibrium is globally asymptotically stable, and the disease dies out; if , then the unique endemic equilibrium point is globally asymptotically stable and the disease persists within the population. Furthermore, numerical simulations in the application of the model showed the abrupt and periodic variations.
1. Introduction
Malaria is a mosquito-borne disease caused by Plasmodium parasite, which is transmitted through the bites of an infected mosquito. In 2017, the World Health Organization report reveals estimations of 216 million malaria cases and 445 thousand deaths due to malaria were registered worldwide in 2016. However, the most malaria cases and deaths were shared by the WHO Africa region, which account for 90% of cases and 91% deaths. The most predominant malaria parasite in the WHO Africa region is Plasmodium falciparum, accounting for 99% of malaria cases in 2016 [1].
Malaria is entirely preventable and treatable disease if the recommended interventions are properly applied. Individuals should have taken some aggressive measurements to decline malaria burden. Personal protection measures are the first line of defense against mosquito-borne diseases. Mosquito repellent is a method used for personal protection; and these are the substances used for exposed skin to prevent human-mosquito contact. Insecticide Treated Bed Nets (ITNs) are used for individuals against malaria to reduce the morbidity of childhood malaria (below five years of age) by 50% and global child mortality by % [2, 3]. When used on a large scale, ITNs are supposed to represent efficient tools for malaria vector control but there is a limitation of resistance to insecticides used for a saturated net. The resistance of the most important African malaria Anopheles gambiae to protrude is already widespread in several West African countries [4, 5].
Nowadays, mathematical models become an important and popular tools to understand the transmission dynamics of the disease and give an insight to reduce the impact of malaria burden in the society. This is because mathematical modeling can answer the following questions raised by the public health authorities and policy makers to make the correct decisions: (1) how severe will the epidemics be? (2) How long will it last? (3) How effective will an intervention be? (4) What are the effective measures to control and eliminate an endemic disease? The earliest malaria model study originated from Ross in 1911 [6] and later modification made by Macdonald [7]. Some further extensions of Ross-Macdonald models for malaria were described in [8–13]. Tumwiine et al. [13] define the reproduction number, , and show the existence and stability of the disease-free equilibrium and an endemic equilibrium.
Recently, many works on host-vector interaction models have been done in [14–24]. In [18, 20, 22, 25], global stability of equilibria has been investigated using suitable Lyapunov functions; and their results show that the disease-free and endemic equilibrium points become globally asymptotically stable if and , respectively. Application of the optimal control theory becomes an important tool for investigating the efficiency of joint control intervention strategies to minimize the impact of malaria disease and cost-effectiveness of implementing them [19, 21, 23, 24]. Their studies suggest that the optimal control strategies can effectively reduce the malaria disease.
Motivated by the above studies, we extend the model presented in [14] by taking into account a logistic model with population dependent birth rates for both human and vector populations that describes self-limiting growth of both the human host and mosquito vector populations. We consider logistic malaria model as no population can grow exponentially at all time, in general. A number of populations initially grow exponentially, but, due to competition and limited resources availability, their population size decline, after some time, to a stable size , called the maximum carrying capacity. The competition for limited resources (including food, territory, light, water, and oxygen) decreases the fertility or survival of individuals. Furthermore, this paper presents application of the model to study abrupt and periodic variations of malaria and sensitivity analysis applied to understand the important parameters in transmission and prevalence of the malaria disease. The purpose of this work is to investigate the global stability of both disease-free and endemic equilibrium points.
2. Malaria Model
We consider that the total human host population, , at a time , is divided into three disjoint compartments: susceptible , infectious , and recovered . The total vector population, , at a time , is divided into two mutually exclusive subpopulations of individuals who are susceptible, , and infectious, . The susceptible human and vector populations are recruited at the rates and , respectively, where and . The susceptible human and vector populations decrease due to natural death at a rate for humans and for vectors, and those that move to the infected classes at a rate and , respectively. The infected human population grows as a result of new infection at a rate and decline due to natural mortality, disease induced death, and recovery at a rate , , and , respectively. For details, see the schematic diagram of the model in Figure 1. The state variables and parameters for the model are described in “State Variables, Parameters, Descriptions, and Their Dimensions of Malaria Model” section.
The model has made the following assumptions: both the total sizes of human and vector populations not being constant; all variables and parameters involving the model assumed to be nonnegative; all newborns susceptible to infection; mosquitoes not dying because of infection; no recovery compartment for infected mosquitoes; and the recovered human population developing permanent immunity. From the schematics diagram of transmission of malaria between human and mosquito (see Figure 1), we have the governed differential equations which describe the dynamics of malaria,with initial conditions
At all times, and . Moreover, their differential equations are satisfyingrespectively. We may notice that the vector population equation is completely decoupled from the human equations which is physically reasonable.
If we eliminate and and add the total population equations, then we finally have
2.1. Basic Properties
Since the model system (1) involves human and mosquito populations, all its associated variables and parameters are nonnegative.
Theorem 1. Solutions of the model system (1) with positive initial data will remain nonnegative for all time
Proof. Let . Then it follows from the second equation of malaria model (1) that so that Hence, Similarly, it can be shown that , , , and for all time
This completes the proof.
2.1.1. Invariant Region
The malaria model (1) will be analyzed in biologically feasible region. Thus, the feasible solutions set for the model written by is positively invariant and then the model is biologically meaningful and mathematically well posed in the domain . The proof is omitted for simplicity.
3. Model Analysis
To analyze the malaria model in system (4), we use the normalized quantities instead of the actual populations. Since and may vary, these scales are not suitable for use in the scaling. However, the typical choice for logistic models is to use the sustainable populations and for the scales. In the present case, we shall also consider varying and . It is, therefore, convenient to write and , where and are typical sizes and where and take care of the time variations. At the moment, we just assume that . We shall scale the time with the quantity by setting . The scaling is then , , , , , , , and . The dimensionless parameters of the model become , , , , , and
The scaled equations then becomesubject to suitable initial conditions,
3.1. Stability of Disease-Free Equilibrium and Reproduction Number
Our model (9) admits two disease-free equilibria, namely, and , where and lie between 0 and 1, and a unique endemic equilibrium, . The equilibria of the system (9) are obtained by setting the right side equal to zero.
The basic reproduction number, , is the single most important parameter in epidemiological modeling. It measures the average number of the secondary infections caused by a single infective in an entirely susceptible population during its whole infectious period [26]. To derive the basic reproduction number of model (9), we use the next generation matrix approach described in [27–29]. The infected compartments of system (9) are and . Following [29], the new infection matrix and the transition matrix are given, respectively, by
Hence, the basic reproduction number, , is the dominant eigenvalue of the next generation matrix and becomesFrom (12), it is noted that the reproduction number depends on the product of the number of humans that one mosquito infects through its infectious lifetime, , and the number of mosquitoes that one human infects through its infectious lifetime, .
Theorem 2. The disease-free equilibrium point, , is locally asymptotically stable if all eigenvalues of the characteristic equation of the variational matrix lie below zero.
Proof. At the equilibrium point, , the variational matrix is given by The characteristic equation may be written as . It implies Clearly, we have Thus, this shows that the solution, , is unstable since and lie above zero.
Theorem 3. The disease-free equilibrium point, , is locally asymptotically stable if
Proof. The variational matrix at the equilibrium point, , becomes Thus, the characteristic equation of the variational matrix is given bywhereThe characteristic polynomial in (17) has roots , , , with negative real parts since . By Routh-Hurwitz criterion [28], the other roots and have negative real parts if both and lie above zero. From the second equation of (18), , and from the first equation of (18), , when . Hence, the disease-free equilibrium point, , is locally asymptotically stable.
Theorem 4. The disease-free equilibrium point, , is globally asymptotically stable in if ; otherwise it is unstable.
Proof. Consider the Lyapunov function where The time derivative of the function along the solutions of (9) becomes Thus if and the equality holds if and only if . Therefore, the largest compact invariant set in is the singleton , where is the disease-free equilibrium. LaSalle’s Invariant Principle [30] implies that is globally asymptotically stable in .
3.2. Stability of Endemic Equilibrium
To find the endemic equilibrium , we shall keep the assumptions about and from the singular perturbation equations (see the fourth and fifth equations of system (9)) and then focus on the first three equations of system (9). We consider the equilibrium solutions, using as the basic quantity. From the third equation of (9), we have an expression for :Addition of the second and third equations of system (9) givesNow, let us consider the first equation of (9) which connects and . That is,Equation (24) has only real and positive solutions for if , since the maximum value of is . If , then (24) has two solutions:for each value of .
One might wonder what happens when the inequality is violated and . The solution of (24) will then be majorized by the equation which has only one unstable equilibrium point at and otherwise tends to 0. Moreover, it is clear that needs to be larger or equal to . Let us, therefore, consider and in the light of this restriction. The various situations are best described in terms of the graph shown in Figure 2.
The parabola is the locus of the solutions for and when . The line, , defines the limit of the region for solutions fulfilling . In the graph, the upper solution (upper red dot) is acceptable since , whereas the lower solution is outside the region and hence unacceptable. Further inspection of the graph shows the following.(1)If , both solutions, and , are larger than (consider the line ).(2)If , only is larger than when is positive (consider the line ).(3)For , one or both of the solutions are acceptable.
The main conclusion is that there are acceptable solutions with respect to size for all where .
Assume that , where only is acceptable with respect to size. Then substituting expressions (22) and (25) into the equation in (23) and simplifying lead to the following quadratic equation:where
From (27), it can easily be seen that . Further, if , then . Thus, the number of possible positive real roots of (27) can depend on the signs of . This can be analyzed using the Descartes rule of signs on the quadratic polynomial (27). The different possibilities for the roots of (27) are tabulated in Table 1.
Thus, the malaria model has a unique endemic equilibrium if and whenever cases and are satisfied. Hence, the endemic equilibrium then becomesand is the unique positive root of (27).
Theorem 5. If , then the endemic equilibrium of the malaria model (9) is globally asymptotically stable in .
Proof. We shall propose the Lyapunov functionwhereThe Lyapunov function is continuous for all . The time derivative of the function along the solutions of system (9) becomesFrom the equilibrium point of the malaria model (9), we have the following relations:By adding and subtracting and using (31) and (33) in (32), after intensive simplification, we haveSince the arithmetic mean is greater than or equal to the geometric mean, then we haveAlso,ifFurthermore, if Hence, it follows from (34), (35), (36), and (38) that in . Thus, the equality holds only when , , , , and . Therefore, the largest compact invariant set in is the singleton , where is the endemic equilibrium. From the LaSalle’s invariant principle [30], the unique equilibrium of system (9) is globally asymptomatically stable for .
3.3. Sensitivity Analysis
To understand the relative importance of parameters which are responsible for transmission and prevalence of malaria disease, described in the model (9), we perform a sensitivity analysis. Sensitivity indices help us to measure the relative change in a state variable while a parameter changes. The normalized sensitivity index of a variable to a parameter is the ratio of the relative change in the variable to the relative change in the parameter [27]. We calculate the sensitivity indices of to assess which parameter has a great impact on and hence the greatest effect in determining whether the disease dies out or persists with population.
Let be the generic parameter of model (9). We, now only, derive the normalized sensitivity index of to each of the parameters involved in , defined by the ratio of the relative change in to the relative change in the parameter ; that is, This index shows how sensitive is to a change in the parameter . We notice that This indicates that does not depend on any parameter value. Similarly, for the other parameters, we have
We evaluate the above sensitivity indices, in Table 3, using the parameter values in Table 2. The basic reproduction number, , is most sensitive to the contact rates of human to vector and vector to human, with and as it can be seen in Table 3. This shows that any increase (decrease) by in or will increase (decrease) by in . The other parameters with highest sensitivity indices are , with , and , with . Increasing (decreasing) by 10% will decrease (increase) in by and increasing (decreasing) by will decrease (increase) in by or vice versa. The rest of parameters, and , have less significant effect in .
In conclusion, the vector death rate, the human induced death rate, and the contact rates are important parameters in the model which have a significant impact on prevalence and transmission of the malaria disease; these parameters are able to control so that an intensive effort/work has to be done to eradicate the malaria disease from the population. Furthermore, one can understand from the sensitivity indices that vector control is the most effective control strategy.
4. Application of the Model
In this section, we present more simulations illustrating the abrupt and periodic variations of the model. We fix a reasonable parameter values of the model for numerical simulations.
We allow the mosquito sustainable level, , noted in Section 3, to vary with respect to time. However, we first keep fixed in order to investigate the impact of fast variation in on the human populations. Periodic variations in are shown in these plots for different periods. Plots of the abrupt changes in the sustainable populations, (see in Section 3) and , are located in Figures 3–7, whereas plots of periodic variations are shown in Figures 8 and 9.
Abrupt changes in the human and mosquito populations may, for example, be due to intensive spraying of the mosquitoes some massive emigration (refugee camps) or immigration for the humans. In Figures 3 and 4, steps down and up in vector sustainable population about 50% are plotted. These plots show that the transition occurs very fast and the system adjusts quickly to the new equilibrium. In Figures 5–7 increases about 50% and 100% and decrease about 50% in are shown. Step change in humans needs caution since it may lead to unphysical solutions. In Figure 5, after a transient, the solution converges to the new equilibrium. An increase in in Figure 6 may lead to an increase in the fraction of the human susceptible population for small time intervals, but not much dramatic change is shown. However, the slow change in human population after transient is shown in Figure 7. In the plots of periodic variation, one observes that the fast variation is quickly adapted by the vector population. This is because of the fast time scale for the mosquito population. Most of the figures show that the humans follow a slow variations relative to the vectors. Therefore, it is possible to say that fast variations in do not imply large variation in human population. In general, for periods shorter than , the human population do not in practice show the variations in the vector populations, but for long periods they do, but apparently weaker.
5. Conclusions
In this work, we developed and analyzed a logistic malaria model to study the global stability of both disease-free and endemic equilibrium points. Mathematically, we formulated a five-dimensional system of deterministic ordinary differential equations and defined the domain where the model is epidemiologically feasible and mathematically well-posed. The model used the next generation matrix approach to obtain an explicit formula for a reproduction number, , which is the expected number of secondary cases produced by a single infectious individual during its entire period of infectiousness in a fully susceptible populations.
Qualitative analysis of the model determines stability analysis of the equilibrium points. Accordingly, we obtained two diseases-free equilibrium points and . The equilibrium point, , is unstable and unphysical, while the equilibrium point, , becomes both locally and globally stable whenever and , respectively. We also have shown that the endemic equilibrium point, , is globally asymptotically stable if . Furthermore, sensitivity analysis of the model shows that the human induced death rate, the contact rates (human to mosquito or vice versa), and mosquito death rate have a significant effect on transmission and prevalence of the malaria disease. Moreover, numerical simulations are carried out in the application of the model to investigate how variations in the sustainable level of the vectors affect the human population. One can see from these simulations that fast variations in do not lead to large variations in the human population.
State Variables, Parameters, Descriptions, and Their Dimensions of Malaria Model
| : | Number of susceptible humans at a time |
| : | Number of infected humans at a time |
| : | Number of recovered humans at a time |
| : | Number of susceptible mosquitoes at a time |
| : | Number of infected mosquitoes at a time |
| : | The total human population at a time |
| : | The total mosquito population at a time |
| : | Sustainable level of human population at a time |
| : | Sustainable level of mosquito population at a time |
| : | Per capita birth rate of human population. Dimension: |
| : | Per capita natural death rate for humans. Dimension: |
| : | Per capita disease-induced death rate for humans. Dimension: |
| : | The human contact rate. Dimension: |
| : | Per capita recovery rate for humans. |
| : | Per capita birth rate of mosquitoes. Dimension: |
| : | Per capita natural death rate of mosquitoes. Dimension: |
| : | The mosquito contact rate. Dimension: . |
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The author gratefully acknowledges Harald E. Krogstad for his special support. Also, the author would like to acknowledge Haramaya University for providing access to the completion of this work.