- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 595487, 21 pages

http://dx.doi.org/10.1155/2012/595487

## Stability Analysis and Optimal Control of a Vector-Borne Disease with Nonlinear Incidence

^{1}Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, H-12 Campus, Islamabad 44000, Pakistan^{2}Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea^{3}Department of Mathematics, Vaal University of Technology, Andries Potgieter Boulevard, Private Bag X021, Vanderbijlpark 1900, South Africa

Received 29 July 2012; Revised 17 September 2012; Accepted 17 September 2012

Academic Editor: M. De la Sen

Copyright © 2012 Muhammad Ozair et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper considers a model for the transmission dynamics of a vector-borne disease with nonlinear incidence rate. It is proved that the global dynamics of the disease are completely determined by the basic reproduction number. In order to assess the effectiveness of disease control measures, the sensitivity analysis of the basic reproductive number and the endemic proportions with respect to epidemiological and demographic parameters are provided. From the results of the sensitivity analysis, the model is modified to assess the impact of three control measures; the preventive control to minimize vector human contacts, the treatment control to the infected human, and the insecticide control to the vector. Analytically the existence of the optimal control is established by the use of an optimal control technique and numerically it is solved by an iterative method. Numerical simulations and optimal analysis of the model show that restricted and proper use of control measures might considerably decrease the number of infected humans in a viable way.

#### 1. Introduction

Vector-borne diseases are infectious diseases caused by viruses, bacteria, protozoa, or rickettsia which are primarily transmitted by disease transmitting biological agents, called vectors. Vector-borne diseases, in particular, mosquito-borne diseases such as malaria, dengue fever, and West Nile Virus that are transmitted to humans by blood-sucker mosquito, have been big problem for the public health in the world. The literature dealing with the mathematical theory and dynamics of vector-borne diseases are quite extensive. Many mathematical models concerning the emergence and reemergence of the vector-host infectious disease have been proposed and analyzed in the literature [1, 2].

Mathematical modeling became considerable important tool in the study of epidemiology because it helped us to understand the observed epidemiological patterns, disease control and provide understanding of the underlying mechanisms which influence the spread of disease and may suggest control strategies. The model formulation and its simulation with parameter estimation allow us to test for sensitivity and comparison of conjunctures. The foundations of the modern mathematical epidemiology based on the compartment models were laid in the early 20th century [3].

The incidence of a disease is the number of infection per unit time and plays an important role in the study of mathematical epidemiology. In classical epidemiological bilinear incidence rate and standard incidence rate are frequently used, where is the probability of transmission per contact, is susceptible, and is infective individuals. However, actual data and evidence observed for many diseases show that dynamics of disease transmission are not always as simple as shown in these rates. There are a number of biological mechanisms which may result in nonlinearities in the transmission rates. In 1978, Capasso and Serio [4] introduced a saturated incidence rate in an epidemic models. This is important because the number of effective contacts between infective and susceptible individuals may saturate at high infective levels due to overcrowding of infective individuals or due to protective measures endorsed by susceptible individuals. A variety of nonlinear incidence rates have been used in epidemic models [5–10]. In [10], an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors, (mosquitoes), such as malaria, yellow fever, dengue and so on.

Optimal control theory is a powerful mathematical tool to make decision involving complex dynamical systems [11]. For example, what percentage of the population should be vaccinated as time evolves in a given epidemic model to minimize both the number of infected people and the cost of implementing the vaccination strategy. The desired outcome depends on the particular situation. New drug treatments and combinations of drugs are under constant development. The optimal treatment scheme for patients remains the subject of intense debate. Further, optimal control methods have been used to study the dynamics of some diseases (see [12, 13] and the references therein).

Recently, a number of mathematical models have been proposed to study the transmission dynamics of vector-borne diseases. Cai and Li [1] describes the dynamics of a vector-borne disease considering that the infection moves from person to person directly with no environmental source and intermediate vector or host. There have been applications of optimal control methods to epidemiological models, namely, Blayneh et al. [14], Okosun and Makinde [15], Lashari and Zaman [16, 17], and so forth. Lashari and Zaman [17] used personal protection, blood screening, and vector-reduction strategies as optimal control to reduce the transmission of a vector-borne disease. Kar and Batabyal [18] analyzed a nonlinear epidemic model and used optimal control technique to reduce the disease burden with a vaccination program.

In this work, we consider a vector host epidemic model with nonlinear incidence rate. Our aim is to carry out qualitative behavior and present a rigorous analysis of the resulting model to investigate the parameters to show how they affect the vector-borne disease transmission. We perform sensitivity analysis of the basic reproductive number and the endemic equilibrium with respect to epidemiological and demographic parameters. From the sensitivity analysis, we find that the reproductive number is most sensitive to the biting and mortality rates of mosquito. Further, the treatment rate of infectious humans is also a sensitive parameter for equilibrium proportion of infectious humans. These suggest us to develop strategies that target the mosquito biting rate, mosquito death rate, and treatment of infectious individuals in controlling the disease. Based on sensitivity analysis, we formulate an optimal control problem to minimize the number of infected human using three main efforts as control measures. Unfortunately, there is no vaccine nor specific treatment against vector-borne disease is available; that is why the main measures to limit the impact of such epidemic have to be considered. Therefore, we look at time-dependent prevention, treatment efforts and breeding sites destruction, for which optimal control theory is applied.

This paper is organized as follows. The model is developed in Section 2. The analysis of global stability of the equilibria of the model is investigated in Section 3. Section 4 focuses on the sensitivity analysis. Section 5 describes the extended model with three control measures and numerical simulations are presented in Section 6. Finally, conclusions are summarized in Section 7.

#### 2. Model Formulation

The total human population, denoted by , is split into susceptible individuals () and infected individuals () so that . Whereas, the total vector population, denoted by , is subdivided into susceptible vectors () and infectious vectors (). Thus .

The dynamics of the disease are described by the following system of differential equations:

Susceptible humans are recruited at a rate , whereas susceptible vectors are generated by . We assume that the number of bites per vector per host per unit time is , the proportion of infected bites that gives rise to the infection is , and the ratio of vector numbers to host numbers is . Let , let be the transmission rate from vector to human, and let be the transmission rate from human to vector. is the transmission probability from human to human. is natural death rate of human, is death rate of vectors, respectively. We assume that infectious individuals do not acquire permanent immunity and become susceptible again by the rate . Further we assume that incidence terms for human population and vector population that transmit disease are saturation interactions and are given by , , and , where , , and determine the level at which the force of infection saturates.

Obviously, is positively invariant, system (2.1) is dissipative, and the global attractor is contained in .

The total dynamics of vector population are . Thus we can assume without loss of generality that for all provided that . On , . Therefore, we attack system (2.1) by studying the subsystem

From biological considerations, we study system (2.2) in the closed set , where denotes the nonnegative cone of including its lower dimensional faces. It can be easily verified that is positively invariant with respect to (2.2).

#### 3. Mathematical Analysis of the Model

The dynamics of the disease are described by the basic reproduction number . The threshold quantity is called the reproduction number, which is defined as the average number of secondary infections produced by an infected individual in a completely susceptible population. The basic reproduction number of model (2.2) is given by the expression Direct calculation shows that system (2.2) has two equilibrium states. For , the only equilibrium is disease-free equilibrium . For , there is an additional equilibrium which is called endemic equilibrium, where and is the root of the following quadratic equation. with From (3.3), we see that if and only if . Since , (3.3) has a unique positive root in feasible region. If , then . Also, it can be easily seen that for . Thus, by considering the shape of the graph of (3.3) (and noting that ), we have that there will be zero (positive) endemic equilibrium in this case. Therefore, we can conclude that if , (3.3) has no positive root in the feasible region. If, , (3.3) has a unique positive root in the feasible region. This result is summarized below.

Theorem 3.1. *System (2.2) always has the infection-free equilibrium . If , system (2.2) has a unique endemic equilibrium defined by (3.2) and (3.3).*

##### 3.1. Global Stability of Disease-Free Equilibrium

In this subsection, we analyze the global behavior of the equilibria for system (2.2). The following theorem provides the global property of the disease-free equilibrium of the system.

Theorem 3.2. *If , then the infection-free equilibrium is globally asymptotically stable in the interior of .*

*Proof. *To establish the global stability of the disease-free equilibrium, we construct the following Lyapunov function:
Calculating the time derivative of along the solutions of system (2.2), we obtain
Thus is negative if . When , the derivative if and only if , while in the case , the derivative if and only if or . Consequently, the largest compact invariant set in , when , is the singelton . Hence, LaSalle’s invariance principle [19] implies that is globally asymptotically stable in . This completes the proof.

##### 3.2. Global Stability of the Endemic Equilibrium

Here, we use the geometrical approach of Li and Muldowney to investigate the global stability of the endemic equilibrium in the feasible region . We have omitted the detailed introduction of this approach and we refer the interested readers to see [20]. We summarize this approach below.

Consider a map from an open set to such that each solution to the differential equation is uniquely determined by the initial value . We have following assumptions: is simply connected; there exists a compact absorbing set ; Equation (3.7) has unique equilibrium in .

Let be a nonsingular matrix-valued function which is in and a vector norm on , where .

Let be the Lozinskiĭ measure with respect to the . Define a quantity as where , the matrix is obtained by replacing each entry of by its derivative in the direction of , , and is the second additive compound matrix of the Jacobian matrix of (3.7). The following result has been established in Li and Muldowney [20].

Theorem 3.3. *Suppose that , and hold, the unique endemic equilibrium is globally stable in if .*

Obviously is simply connected and is unique endemic equilibrium for in . To apply the result of the above theorem for global stability of endemic equilibrium , we first state and prove the following result.

Lemma 3.4. *If , then the system (2.2) is uniformly persistent; that is, there exists (independent of initial conditions), such that , , and *

*Proof. *Let be semidynamical system (2.2) in , let be a locally compact metric space, and let . is a compact subset of and is positively invariant set of system (2.2). Let be defined by and set , where is sufficiently small so that
Assume that there is a solution such that for each . Let us consider
where is sufficiently small so that

By direct calculation we have
where

This implies that as . However is bounded on . According to [21, Theorem 1] the proof is completed.

The boundedness of and the above lemma imply that (2.2) has a compact absorbing set [22]. Now we shall prove that the quantity . We choose a suitable vector norm in and a matrix valued function Obviously is and nonsingular in the interior of . Linearizing system (2.2) about an endemic equilibrium gives the following Jacobian matrix. The second additive compound matrix of is given by where The matrix can be written in block form as with where Consider the norm in as where denotes the vector in . The Lozinskiĭ measure with respect to this norm is defined as , where From system (2.2) we can write Since is a scalar, its Lozinskiĭ measure with respect to any vector norm in will be equal to . Thus and will become Also , and are the operator norms of and which are mapping from to and from to , respectively, and is endowed with the norm. is the Lozinskiĭ measure of matrix with respect to norm in . Hence Thus, Since (2.2) is uniformly persistent when , so for such that implies , and for all . Thus for all , which further implies that . Therefore all the conditions of Theorem 3.3 are satisfied. Hence unique endemic equilibrium is globally stable in .

#### 4. Sensitivity Analysis

We would like to know different factors that are responsible for the disease transmission and prevalence. In this way we can try to reduce human mortality and morbidity due to disease. Initial disease transmission depends upon the reproductive number whereas disease prevalence is directly related to the endemic equilibrium point. The class of infectious humans is the most important class because it represents the persons who may be clinically ill and is directly related to the disease induced deaths. We will calculate the sensitivity indices of the reproductive number, , and the endemic equilibrium point with respect to the parameters given in Table 1 for the model. By the analysis of these indices we could determine which parameter is more crucial for disease transmission and prevalence.

*Definition 4.1. *The normalized forward sensitivity index of a variable, , that depends differentiably on a parameter, , is defined as .

Table 2 represents sensitivity indices of model parameters to .

By analyzing sensitivity indices we observe that the most sensitive parameters are biting rate of mosquitoes and death rate of mosquitoes . The reproductive number () is directly related to the biting rate of mosquitoes and inversely related to the death rate of mosquitoes. We can say that an increase (or decrease) in biting rate by 10% increases (or decreases) by 20%. Similarly increase (or decrease) in death rate of mosquitoes by 10% decreases (or increases) by 20%. This suggests that strategies that can be applied in controlling the disease are to target the mosquito biting rate and death rate such as the use of insecticide-treated bed nets and indoor residual spray.

##### 4.1. Sensitivity Indices of Endemic Equilibrium

We have numerically calculated the sensitivity indices at the parameter values given in Table 1. The most sensitive parameter for is mosquito biting rate. Change in mosquito biting rate is directly related to change in and inversely related to change in . This suggests that personal protection and human treatment strategies can lead to marvelous decrease in . The most sensitive parameter for is mosquito death rate , followed by mosquito biting rate. We observe that can be reduced by personal protection, larvcide adulticide, and so forth.

The analysis of the sensitivity indices of , , and suggests us that three controls, personal protection, larvacide, and adulticide and treatment of infectious humans, can play an effective role to control the disease. The sensitivity indices for , , and with respect to all parameters are given in Table 3.

#### 5. Analysis of Optimal Control

In this section, model (2.1) is extended to assess the impact of some control measures, namely, prevention, treatment, and spray of insecticide against vector. In the human population, the associated force of infection is reduced by a factor of and the reproduction rate of the mosquito population is reduced by a factor of . It is assumed that under the successful control efforts the mortality rate of mosquito population increases at a rate proportional to , where is a rate constant. The per capita recovery rate is proportional to , where is a rate constant. One has The control variable represents the use of drugs or vaccine which are preventive measures to minimize vector human contacts. The control function represents the treatment supplied to the infected humans. The control function represents the level of larvacide and adulticide used for vector control applied at those places at which vector breeding occurs.

To investigate the optimal level of efforts that would be needed to control the disease, we give the objective functional , which is to minimize the number of infected human and the cost of applying the control . One has where , , and are positive weights. We choose a quadratic cost on the controls and this is similar with what is in other literature on epidemic controls [15]. With the given objective function , our goal is to minimize the number of infected humans, while minimizing the cost of control , , and . We seek an optimal control such that where such that measurable with is the control set. The necessary conditions that an optimal must satisfy come from the Pontryagin’s Maximum Principle [24]. This principle converts (5.1) and (5.2) into a problem of minimizing pointwise a Hamiltonian , with respect to , , and where for are adjoint variables. In the following we will state and prove the existence of the optimal control by using the result by Fleming and Rishel [25].

Theorem 5.1. *There exists an optimal control that minimize over subject to the control system (5.1). Further, for the system (5.1), there exists adjoint variables satisfying
**
with transversality conditions . The optimal controls are given by
*

*Proof. *The integrand of the objective functional given by (5.2) is a convex function of and the state system satisfies the Lipschitz property with respect to the state variables since state solutions are bounded. The existence of an optimal control follows [25]. The equations governing the adjoint variables are obtained by differentiation of the Hamiltonian function with respect to , , , and , respectively, evaluated at the optimal control. To get the characterization of the optimal control given by (5.6), solving the equations
on the interior of the control set and using the property of the control space , we can derive the desired characterization (5.6).

#### 6. Numerical Results and Discussions

Here, we investigate numerically the effect of the optimal control strategies on the spread of the disease in a population.

##### 6.1. Use of Preventive Measures and Treatment

With this strategy, only the preventive control on the vector biting rate and treatment is used to optimize the objective function , while the control on reducing the vector population is set to zero. Figure 1 shows no significant difference in the number of infected mosquitoes between the case with control and the case without while there is a significant difference in the number of infected humans with control and the case without control. The control profile is shown in Figure 1(d); the control rise to the upper bound after (days), while control rise to the upper bound after (days). This shows that an effective and optimal use of preventive measure in the population without the use of larvacide against the vector will not be beneficial if total elimination of the disease is desirable in the community.

##### 6.2. Use of Preventive Measures and Larvacide Only

With this strategy, we set the control to zero and use only preventive control and larvacide to optimize the objective function . In Figures 2(a) and 2(b) show a significant difference in the infected humans and infected mosquitoes , respectively, with control compared to the situation where there is no control. More specifically, we observe a decrease in and while an increase was observed in the uncontrolled cases. The control profile is shown in Figure 2(d), where we see that the control is at the upper bound for (days).

##### 6.3. Use of Treatment and Larvacide

With this strategy, we set control to zero, while the control and the control are both used to optimize the objective function . In Figures 3(a) and 3(b), we observed that the control strategies resulted in a decrease in the number of infected humans (), infected mosquitoes (), and susceptible humans () while an increase is observed in the number of infected humans () and infected mosquitoes () in strategy without control. The control profile is shown in Figure 3(d), where we see that the control is at the upper bound for (days) and the control is at the upper bound for days and then dropped to the lower bound at the final time.

##### 6.4. Use of Preventive Measures , Treatment , and Larvacide

With this strategy, the control , control , and the control are all used to optimize the objective function . In Figures 4(a) and 4(b), we observed that the control strategies resulted in a decrease in the number of infected humans (), infected mosquitoes, () and susceptible humans () while an increase is observed in the number of infected humans () and infected mosquitoes () in strategy without control. The control profile is shown in Figure 4(d), where we see that the control is at the upper bound for (days) before dropping gradually to the lower bound, the control is at the upper bound for (days) and then dropped to the lower bound at the final time, and the control is maintained at the upper bound for (days).

#### 7. Conclusion

In this paper, we have studied a vector host epidemic model with saturated incidence rate. The global stability of the disease-free steady state is established by direct Lyapunov method and a geometric approach is used for the global stability of the endemic steady state. The model has a globally asymptotically stable disease-free solution whenever the basic reproduction number is less than or equal unity and has a unique positive globally asymptotically stable endemic equilibrium whenever exceeds unity. We found also from the sensitivity indices analysis that the most sensitive parameters are mosquito biting and death rates. The paper was also extended to asses the impact of some control measures. By the application of optimal control theory, we derived and analyzed the conditions for optimal control of the disease with personal protection, treatment and spray of insecticides. The optimal control has a very desirable effect for reducing the number of infected individuals and comparison between optimal control and without control is shown in figures. From our numerical results we found that an effective and optimal use of preventive measure in the population without the use of larvacide against the vector will not be beneficial if total elimination of the disease is desirable in the community. Control programs that follow these strategies can effectively reduce the spread of a vector-borne disease in the community.

#### Acknowledgments

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (2012-000599).

#### References

- L. Cai and X. Li, “Analysis of a simple vector-host epidemic model with direct transmission,”
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 679613, 12 pages, 2010. View at Publisher · View at Google Scholar - L. Cai, S. Guo, X. Li, and M. Ghosh, “Global dynamics of a dengue epidemic mathematical model,”
*Chaos, Solitons and Fractals*, vol. 42, no. 4, pp. 2297–2304, 2009. View at Publisher · View at Google Scholar - S. Liu, Y. Pei, C. Li, and L. Chen, “Three kinds of TVS in a SIR epidemic model with saturated infectious force and vertical transmission,”
*Applied Mathematical Modelling*, vol. 33, no. 4, pp. 1923–1932, 2009. View at Publisher · View at Google Scholar - V. Capasso and G. Serio, “A generalization of the Kermack-McKendrick deterministic epidemic model,”
*Mathematical Biosciences*, vol. 42, no. 1-2, pp. 43–61, 1978. View at Publisher · View at Google Scholar - X. Zhang and X. Liu, “Backward bifurcation of an epidemic model with saturated treatment function,”
*Journal of Mathematical Analysis and Applications*, vol. 348, no. 1, pp. 433–443, 2008. View at Publisher · View at Google Scholar - D. Xiao and S. Ruan, “Global analysis of an epidemic model with nonmonotone incidence rate,”
*Mathematical Biosciences*, vol. 208, no. 2, pp. 419–429, 2007. View at Publisher · View at Google Scholar - W. Wang, “Epidemic models with nonlinear infection forces,”
*Mathematical Biosciences and Engineering*, vol. 3, no. 1, pp. 267–279, 2006. View at Publisher · View at Google Scholar - S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,”
*Journal of Differential Equations*, vol. 188, no. 1, pp. 135–163, 2003. View at Publisher · View at Google Scholar - W. M. Liu, H. W. Hethcote, and S. A. Levin, “Dynamical behavior of epidemiological models with nonlinear incidence rates,”
*Journal of Mathematical Biology*, vol. 25, no. 4, pp. 359–380, 1987. View at Publisher · View at Google Scholar - L.-M. Cai and X.-Z. Li, “Global analysis of a vector-host epidemic model with nonlinear incidences,”
*Applied Mathematics and Computation*, vol. 217, no. 7, pp. 3531–3541, 2010. View at Publisher · View at Google Scholar - S. Lenhart and J. T. Workman,
*Optimal Control Applied to Biological Models*, Mathematical and Computational Biology Series, Chapman and Hall/CRC, London, UK, 2007. - I. H. Jung, Y. H. Kang, and G. Zaman, “Optimal treatment of an SIR epidemic model with timedelay,”
*Biosystems*, vol. 98, no. 1, pp. 43–50, 2009. View at Publisher · View at Google Scholar - A. A. Lashari, S. Aly, K. Hattaf, G. Zaman, I. H. Jung, and X. Z. Li, “Presentation of
malaria epidemics using multiple optimal controls,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 946504, 17 pages, 2012. View at Publisher · View at Google Scholar - K. Blayneh, Y. Cao, and H.-D. Kwon, “Optimal control of vector-borne diseases: treatment and prevention,”
*Discrete and Continuous Dynamical Systems A*, vol. 11, no. 3, pp. 587–611, 2009. View at Publisher · View at Google Scholar - K. O. Okosun and O. D. Makinde, “Impact of chemo-therapy on optimal control of malaria disease
with infected immigrants,”
*BioSystems*, vol. 104, no. 1, pp. 32–41, 2011. - A. A. Lashari and G. Zaman, “Global dynamics of vector-borne diseases with horizontal transmission in host population,”
*Computers & Mathematics with Applications*, vol. 61, no. 4, pp. 745–754, 2011. View at Publisher · View at Google Scholar - A. A. Lashari and G. Zaman, “Optimal control of a vector borne disease with horizontal transmission,”
*Nonlinear Analysis. Real World Applications*, vol. 13, no. 1, pp. 203–212, 2012. View at Publisher · View at Google Scholar - T. K. Kar and A. Batabyal, “Stability analysis and optimal control of an SIR epidemic model with
vaccination,”
*BioSystems*, vol. 104, no. 2-3, pp. 127–135, 2011. View at Publisher · View at Google Scholar - J. P. LaSalle,
*The Stability of Dynamical Systems*, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1976. - M. Y. Li and J. S. Muldowney, “A geometric approach to global-stability problems,”
*SIAM Journal on Mathematical Analysis*, vol. 27, no. 4, pp. 1070–1083, 1996. View at Publisher · View at Google Scholar - A. Fonda, “Uniformly persistent semidynamical systems,”
*Proceedings of the American Mathematical Society*, vol. 104, no. 1, pp. 111–116, 1988. View at Publisher · View at Google Scholar - G. Butler, H. I. Freedman, and P. Waltman, “Uniformly persistent systems,”
*Proceedings of the American Mathematical Society*, vol. 96, no. 3, pp. 425–430, 1986. View at Publisher · View at Google Scholar - N. Chitnis, J. M. Hyman, and J. M. Cushing, “Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model,”
*Bulletin of Mathematical Biology*, vol. 70, no. 5, pp. 1272–1296, 2008. View at Publisher · View at Google Scholar - L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko,
*The Mathematical Theory of Optimal Processes*, vol. 4, Gordon & Breach Science Publishers, New York, NY, USA, 1986. - W. H. Fleming and R. W. Rishel,
*Deterministic and Stochastic Optimal Control*, Springer-Verlag, New York, NY, USA, 1975.