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A control theoretic approach to malaria immunotherapy with state jumps
Automatica 47 (2011) 1271–1277
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Brief paper
A control theoretic approach to malaria immunotherapy with state jumps✩ H. Chang a,∗ , A. Astolfi b,c , H. Shim d a
Department of Mechanical Engineering, Boston University, MA 02215, United States
b
Department of Electrical and Electronic Engineering, Imperial College London, UK
c
Dipartimento di Informatica, Sistemi e Produzione, Università di Roma Tor Vergata, Via del Politecnico 1, 00133 Roma, Italy
d
ASRI, Department of Electrical Engineering, Seoul National University, Seoul, Republic of Korea
article
info
Article history: Received 11 February 2010 Received in revised form 21 January 2011 Accepted 21 February 2011 Available online 20 April 2011
abstract We investigate a control method for malaria dynamics to boost the immune response using a model-based approach. The idea of state jump is introduced and discussed using a hybrid model. To implement the state jumps we use a physiologically feasible method for the biological system, whereby the introduction of a pathogen into the system steers it to a desirable status. It is noted that we take advantage of a pathogen to implement state jumps in the system. The study is supported by recently reported experimental results. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Malaria Immunotherapy Hybrid systems State jump Impulsive control
1. Introduction Malaria remains a lethal and prevalent infectious disease worldwide, which accounts for 300 million cases and over a million fatalities every year (World Health Organization Expert Committee on Malaria, 2003). Several recent studies have discussed modelbased approaches to malaria treatment. In Mason and McKenzie (1999) malaria dynamics have been modeled and model parameters have been estimated. The model has been studied in Gurarie, Zimmerman, and King (2006) and Mason, McKenzie, and Bossert (1999), and the equations of the immune response in the model have been employed in different malaria models (Gurarie & McKenzie, 2006; Recker et al., 2004). We investigate a control scheme by a model-based approach to enhance the immune response in malaria dynamics. In Chang and Astolfi (2008, 2009a) and Shim, Jo, Chang, and Seo (2009) the authors have studied an immune boosting mechanism for the HIV dynamic models of Wodarz (2001) and Wodarz and Nowak (1999) and this mechanism is also used for the treatment of chronic myeloid leukemia in Chang and Astolfi (2009b). In this paper we
study the malaria model in Mason and McKenzie (1999) on the basis of the immune boosting mechanism introduced in Chang and Astolfi (2009a). To this end we propose the idea of using state jumps as impulsive control1 method. In addition an implementation method for state jumps in malaria dynamics is suggested. The implementation of the state jumps is physiologically practical, as suggested in Roestenberg, McCall, and Hopman (2009). Our research on the malaria model is supported by the experimental data which have been recently published in Roestenberg et al. (2009). Note that we do not provide a formal treatment of the properties of the controlled system, which will be reported elsewhere, but focus on the methodological and conceptual aspects of the control design. The paper is organised as follows. In Section 2 we give a brief description of the experiments in Roestenberg et al. (2009). Section 3 describes the application of the proposed control ideas to the malaria model of Mason and McKenzie (1999) and discusses the robustness of the suggested control method to a range of parameter values. Finally we discuss future works and present further remarks in Section 4. 2. A summary of the experiments in Roestenberg et al. (2009)
✩ The material in this paper was presented at the 2010 American Control Conference, June 30–July 2, 2010, Baltimore, Maryland, USA. This paper was recommended for publication in revised form by Associate Editor Murat Arcak under the directions of Corresponding Guest Editors Francis J. Doyle III and Frank Allgöwer. ∗ Corresponding author. Tel.: +1 617 358 0844; fax: +1 617 353 5548. E-mail addresses: [email protected] (H. Chang), [email protected], [email protected] (A. Astolfi), [email protected] (H. Shim).
0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.03.009
In this section we briefly discuss the experiments of Roestenberg et al. (2009). Although an effective vaccine against malaria is urgently needed, a malaria vaccine is not currently available. 1 Impulsive control gives a sudden change of the state variable at discrete instants. See Yang (1999) for rigorous definition of impulsive control.
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Table 1 Malaria model parameters and their values or ranges (Mason & McKenzie, 1999). Parameter
Description
Value or range
a b g qs qn cs cn ss sn x
Replication rate of P. vivax Replication rate of P. falciparum Gametocyte conversion rate Specific immunity decay rate Non-specific immunity deactivation rate Specific immunity removal rate Non-specific immunity removal rate Specific immunity proliferation rate Non-specific immunity proliferation rate Cross-reactivity rate of P. falciparum-specific effectors Cross-reactivity rate of P. vivax-specific effectors Continuous parasite reduction ratio
1.28 1.39 0.04 0.01 0.6 0.0001–1000 0.0001–1000 0.0001–1000 0.0001–1000 0–1
y k
0–1 2.3
This is largely because immune response to Plasmodium falciparum, a species of malaria parasite, is regarded difficult to acquire. In Roestenberg et al. (2009) it is shown that immunity against a malaria challenge can be induced by the inoculation of infected mosquitoes. In the experiment of Roestenberg et al. (2009), fifteen healthy volunteers have been treated with an antimalaria drug regimen, chloroquine, when they were exposed to mosquito bites once a month for 3 months. Ten of the volunteers have been assigned to a vaccine group while the remaining 5 volunteers have been assigned to a control group. The vaccine group has been exposed to mosquitoes infected with P. falciparum. The control group has been exposed to mosquitoes not infected with the malaria parasite. After 1 month the immune response has been tested by the inoculation of five mosquitoes infected with the malaria parasite. Malaria has not developed in any of the 10 volunteers in the vaccine group. Also no serious adverse events occurred in this vaccine group. However malaria has developed in all 5 volunteers in the control group. Thus the inoculation of malaria parasite by infected mosquitoes in the presence of antimalaria drug can lead to protection against a malaria challenge. 3. State jump in malaria dynamics The clinical work of Roestenberg et al. (2009) leads to application of the mathematical method of immune enhancement, developed in HIV dynamics, to malaria dynamics. The experimental results in Roestenberg et al. (2009) can be explained by the immune system analysis in this section. We consider the malaria infection model of Mason and McKenzie (1999). The model is given by V˙ = aV − cs JV − cs xKV − cn IV − gV ,
(1)
F˙ = bF − cs KF − cs yJF − cn IF − gF , ˙I = sn (V + F ) − qn I ,
(2)
J˙ = ss V − qs J , K˙ = ss F − qs K ,
(3) (4) (5)
where the states V , F , I, J, and K describe the populations of specific cells in a unit volume of blood and therefore are meaningful only when nonnegative. Human malaria is caused by four species of Plasmodium: P. falciparum, P. malariae, P. ovale, and P. vivax (Mason et al., 1999). The model (1)–(5) includes the dynamics between P. vivax, P. falciparum and the immune system. Indeed V describes the concentration of P. vivax, F the concentration of P. falciparum, I the concentration of the effectors of the non-specific immune response, and J and K the concentrations of the effectors of the specific immune response for V and F , respectively. The remaining parameters a, b, g, qs , qn , x, y, cs , cn , ss , and sn are positive and constant. The descriptions and values of the model parameters
estimated in Mason and McKenzie (1999) are summarised in Table 1. Since the model parameters are based on clinical data, the state variables in this section represent actual data. See Mason and McKenzie (1999) for a detailed explanation of the model. We consider a single species P. falciparum case by removing the variables V and J of model (1)–(5) and taking equations (2), (3) and (5) as studied in Gurarie et al. (2006) because this is the clinical case studied in Roestenberg et al. (2009). The term of administration of antimalarials u is included by introducing a parasite killing rate k. If u = 1 the maximal drug therapy is received, while u = 0 means no medication. We set k = 2.3 consistently with published ranges of parasite reduction ratios (Mason & McKenzie, 1999). As a result the model is given by F˙ = (b − ku)F − cs KF − cn IF − gF ,
(6)
˙I = sn F − qn I , K˙ = ss F − qs K .
(7) (8)
The goal of the control is to enhance immunity, and this is equivalent to boosting I and K . If ˙I = sn F − qn I > 0 (or K˙ = ss F − qs K > 0), then the immunity by I (or K ) is enhanced. Now note that each inequality depends on two state variables, F and I (or F and K ), of model (6)–(8). When we consider the (I , F ) positive quadrant (or the (K , F ) positive quadrant), ˙I = 0 (or K˙ = 0) corresponds to a straight line. Thus within the region above this line ˙I > 0 (or K˙ > 0). Therefore the immune term I (or K ) increases if the state (I , F ) (or (K , F )) belongs to the region above the line. The basic control idea is to force the system state to be such that ˙I > 0 (or K˙ > 0). In this paper we consider state jumps as an impulsive control method to guarantee these inequalities. To this end the state must be moved into the set {(I , F ) : sn F − qn I > 0} (or {(K , F ) : ss F − qs K > 0}) by a ‘‘state jump’’, which is realised by mosquito inoculation in Roestenberg et al. (2009). Let Y (t ) := [F (t ), I (t ), K (t )]T and let model (6)–(8) be written as Y˙ = G(Y , u),
(9)
where u is the antimalarial drug input. In this section we consider infective mosquito bites to simulate the experiment of Roestenberg et al. (2009) in silico. One mosquito bite corresponds to a total of 300,000 malaria parasites (primary merozoites) (Gurarie & McKenzie, 2006). The experiment in Roestenberg et al. (2009) uses 36–45 infected mosquito bites so we assume 40 bites in each mosquito exposure to model (6)–(8). Note that the state variables used in Mason and McKenzie (1999) represent per-microliter densities and we assume that the total human blood volume is 4500 (ml). Thus we obtain 2.67 (=300,000 × 40/(4500 × 1000)) parasite density and we set Y ∗ = [2.67, 0, 0]T which corresponds to the amount of state jump. Also we consider the initial state Y (0) = [0, 0, 0]T , which represents the status of the subjects of the experiment in Roestenberg et al. (2009). We set the drug treatment period Td = 90 and the state jump treatment period Tt = 63 consistently with the experiment in Roestenberg et al. (2009). As in Mason and McKenzie (1999) we use cs = 0.1, cn = 0.1, ss = 0.1, and sn = 0.1. 3.1. Open-loop control In this subsection we consider open-loop control for the model (9) and the exposure of infective mosquito bites is carried out at t = 7, 35, 63, which corresponds to the case of Roestenberg et al. (2009). The system with suggested controls is given by Y˙ (t ) = G(Y (t ), 1),
if t ∈ [0, Td ] and t ̸∈ St ,
Y˙ (t ) = G(Y (t ), 0),
if t > Td ,
Y (t + ) = Y (t − ) + Y ∗ , where St = {7, 35, 63}.
if t ∈ St ,
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3
1 0.8 F
u
2 0.6 0.4
1
0.2 0
0 0
50
100
0
Time (day)
50
100
Time (day) 0.6
0.1
I
K
0.4 0.05
0.2
0
0 0
50
100
0
50
100
Time (day)
Time (day)
Fig. 1. Results of the application of the open-loop control strategy to model (9).
3
F
2 1 (I(0), F(0)) 0 0
0.02
0.04
0.06 I
0.08
0.1
0.12
3
F
2 1 (K(0), F(0)) 0 0
0.1
0.2
0.3 K
0.4
0.5
0.6
Fig. 2. The (I , F ) and (K , F ) trajectories resulting from the open-loop control strategy. The solid lines in each graph indicate the (I , F ) and (K , F ) trajectory, respectively. The dotted lines in each graph indicate the sets sn F − qn I = 0 and ss F − qs K = 0, respectively. The thick arrows on the trajectories illustrate the state jumps by impulsive control. Note that the arrows on the (I , F ) trajectory overlap.
Fig. 1 shows the results of the application of the proposed openloop control procedure. The specific immune effector K is boosted up to 0.602 at t = 66.75 (day) while the non-specific immune effector I is not. This is because of the difference in the decay rates (i.e., qs < qn ). Thus immune effectors of lower decay rates such as memory B cells (Mason & McKenzie, 1999) are enhanced by the given control method. The resulting (I , F ) and (K , F ) trajectories are displayed in Fig. 2. The thick arrows on the trajectories illustrate the state jumps by impulsive control. This shows that each trajectory enters into the immune increasing area (i.e. sn F − qn I > 0 and ss F − qs K > 0) by the action of impulsive control. In the simulation without state jumps, the state variables F , I, and K are zeros. Accordingly the state at t = 100 (day) of the simulation is [0, 0, 0]T which implies that the immune response is not enhanced, consistently with the result of Roestenberg et al. (2009).
In order to verify the robustness of the open-loop control we perform the computer simulations with Y ∗ and St being perturbed within certain bounds. Fig. 3 shows the simulation resulting from applying the control strategy to 40 cases, with parameters generated by randomly around the nominal values of Y ∗ and St : the first element of Y ∗ is perturbed up to ±50% and each inoculation timing in St is perturbed ±3.5 (days). In spite of diverse control parameters the average of the maximally boosted K during the control procedure is 0.5905 with the standard deviation 0.0904. This indicates that the suggested open-loop control has a large robust margin against control parameter uncertainty. Remark. When the (I , F ) trajectory (or (K , F ) trajectory) passes through the set sn F − qn I = 0 (or ss F − qs K = 0), we have that dI /dF = 0 (or dK /dF = 0). This implies that if we obtain a (I , F ) trajectory (or (K , F ) trajectory) with a point in which dI /dF = 0 (or dK /dF = 0) from experimental data, then we could estimate the ratio sn /qn (or ss /qs ). We now state a proposition related to the boosting level achievable with the state jump by an impulsive control technique. Proposition 1. Consider the model (9) with Y ∗ = [FM , 0, 0]T for some positive FM and assume b − k − g ≤ rm < 0 and u = 1. Also assume that the state jump is carried out if 0 ≤ F ≤ Fm for some positive Fm . Then Fm ss /qs < K (t ) < (FM + Fm )ss /qs and Fm sn / qn < I (t ) < (FM + Fm )sn /qn for t > tF for some positive tF . By Proposition 1 the minimum enhancement and the maximum enhancement of the immune response I (t ) (or K (t )) are proportional to sn /qn (or ss /qs ). This conclusion is discussed further in Section 3.2. Proof. For the case of K (t ) see Fig. 4 (the proof for I (t ) is similar). For t > tF , F (t ) ≥ Fm by the state jump and F (t ) ≤ (Fm + FM ) since F˙ < rm F < 0. Note that the state (K , F ) cannot stay close to either (Fm ss /qs , Fm ) or ((FM + Fm )ss /qs , (FM + Fm )) since F˙ < rm F < 0. Thus the state (K , F ) cannot be such that Fm ≤ F ≤ (Fm + FM ) and K ≤ Fm ss /qs because K˙ > 0 if ss F − qs K > 0. Also the state (K , F ) cannot be such that Fm ≤ F ≤ (Fm + FM ) and K ≥ (FM + Fm )ss /qs because K˙ < 0 if ss F − qs K < 0. Hence Fm ss /qs < K (t ) < (FM + Fm )ss /qs for t > tF for some positive tF .
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1
4
0.8 F
u
3 0.6
2
0.4 0.2
1
0
0 0
50 Time (day)
100
0
50 Time (day)
100
0.2 0.8 0.15
K
I
0.6 0.1 0.05
0.4 0.2
0
0 0
50
100
0
Time (day)
50 Time (day)
100
Fig. 3. Results of the application of the open-loop control strategy to model (9) for 40 different control parameters conditions.
ssF-qsK = 0 FM + Fm
F
FM
Fm Fm 0
0
Fmss/qs
K
(FM + Fm)ss/qs
Fig. 4. Proof of Proposition 1 for K (t ). The dashed line and the dotted line indicate the sets in which ss F − qs K = 0 and F = Fm , respectively. The thin solid segment represents the level of Fm while the thick solid segment represents the level of FM . As the arrows indicate, K˙ > 0 if ss F − qs K > 0 and K˙ < 0 if ss F − qs K < 0.
3.2. Sensitivity to parametric range Table 1, taken from Mason and McKenzie (1999), shows that four parameters (cs , cn , ss , and sn ) are not well-defined but can vary within a range of seven orders of magnitude. Thus we now explore the sensitivity of the control results to these parametric perturbations. Among the four parameters we especially study the implication of the variation of ss and cs since the values of the parameters are closely related to the state K which is boosted by the control method in the example of the previous subsection. Fig. 5 shows the results of the application of the open-loop control method with two values of ss , 0.05 and 0.2. The resulting (K , F ) trajectories are displayed in the lower graph of Fig. 5 where the solid line and the dotted line indicate the (K , F ) trajectories with ss = 0.05 and 0.2, respectively. The thick arrows on the trajectories illustrate the state jumps. During the control procedure the specific immune effector K is boosted up to 0.306 and 1.168, with ss = 0.05 and ss = 0.2 respectively. Compared to the maximum level 0.602 of K with
ss = 0.1 in the previous subsection the maximally enhanced level of the specific immune effector is proportional to the parameter ss . Fig. 6 shows the relation between the maximal enhancement of K and the parameter values of ss of which the total range is given in Table 1 of Mason and McKenzie (1999). From the dynamics of K in (8), the proliferation of K is described as ss F which is proportional to ss . It is consistent with the biological meaning of ss in Table 1, specific immunity (i.e. K ) proliferation rate, by which the proportional trend in Fig. 6 is explained. Accordingly we can expect that the human body with high proliferation rate of specific immunity would be induced to enhance the corresponding immunity by the suggested control scheme. Fig. 7 shows the inversely proportional relation between the maximum enhancement of K and the given parameter values of cs in Table 1 of Mason and McKenzie (1999). From (6) an increase in cs leads to a decrease in the state F and then a decrease in K is consequently induced (see (8)), by which the inversely proportional trend in Fig. 7 is explained. Thus the human body with high removal rate of specific immunity would not be expected to have strong immunity even by the suggested control scheme. Similarly there is an inverse proportional relation between the maximal enhancement of K and the parameters sn and cn . 3.3. Closed-loop control In this subsection we investigate how effectively the malaria immunity can be enhanced by a closed-loop control scheme, compared to an open-loop control scheme, i.e. the experiments of Roestenberg et al. (2009). We construct a feedback control strategy to maintain ss F − qs K ≥ 0 for 0 < t ≤ Tt so that we can enhance the immune effector K which has a lower decay rate than I. In the open-loop scheme of Section 3.1, the control action is performed at predefined time instants, i.e. t = 7, 35, 63. For the closed-loop scheme we do not have any predefined time instant for control input. Instead the occurrence of control input is decided based on the inequality ss F − qs K ≥ 0. The controlled hybrid system is given by Y˙ (t ) = G(Y (t ), 1), or t ∈ (Tt , Td ],
if t ∈ [0, Tt ] and ss F − qs K > 0,
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1 K with ss - 0.2
K with ss - 0.05
0.3
0.2
0.1
0.5
0
0 50 Time (day)
0
100
0
50 Time (day)
100
3
F
2
1
0 0
0.2
0.4
0.6
0.8
1
1.2
K with ss = 0.05 and 0.2 Fig. 5. Results of the application of the control strategy of Section 3.1 to model (9) with ss = 0.05 and 0.2. The (K , F ) trajectories with these two values of ss are presented in the lower graph where the solid line and the dotted line indicate the (K , F ) trajectories with ss = 0.05 and 0.2, respectively.
Y˙ (t ) = G(Y (t ), 0),
350
Maximum enhanced value of K
300 250 200 150 100 50 0 0
200
400
600
800
1000
Ss Fig. 6. Maximum enhancement of K with the given range of ss in Table 1 of Mason and McKenzie (1999).
1 Maximum enhanced value of K
if t > Td ,
Y (t + ) = Y (t − ) + Y ∗ ,
if t ∈ [0, Tt ] and ss F − qs K ≤ 0.
Note that, due to this system description, if one state jump cannot render ss F −qs K > 0 then multiple jump actions will be performed. For certain values of F and K , a multiple of Y ∗ guarantees ss F − qs K > 0. The result of the control action is shown in Figs. 8 and 9. The total number of mosquito inoculations is 24 in the closed-loop case, while it is 3 in the open-loop case of Section 3.1. The boosting level of K by closed-loop control is much higher than that achievable by open-loop control. (Compare the graph of K in Fig. 8 with that of K in Fig. 1.) Fig. 9 shows the (I , F ) and (K , F ) trajectories. In the lower graph of Fig. 9 the (K , F ) trajectory stays within the set ss F − qs K ≥ 0 for 0 < t ≤ Tt . Note that the (K , F ) trajectory does not stay within the region ss F − qs K ≥ 0 for the whole time interval 0 < t ≤ Tt when the open-loop control strategy of Section 3.1 is applied (see the lower graph of Fig. 2). 4. Conclusions
0.9
We have investigated a control method to enhance the immune response in malaria dynamics. The idea of state jump is used and the state jumps are implemented by the use of ‘‘pathogen’’. Thus we take advantage of a pathogen to drive a biological system to a desirable status. The immune boosting idea in this paper can be applied to various types of immune system models. Recently the authors have applied this control scheme in silico to enhance the immune response in other biological dynamic systems, such as an HIV/AIDS infection model (Chang, Astolfi, & Shim, 2010b) and a bee venom immunotherapy model (Chang, Astolfi, & Shim, 2010a).
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
200
400
600
800
1000
Cs Fig. 7. Maximum enhancement of K with the given range of cs in Table 1 of Mason and McKenzie (1999).
4.1. Future works We conclude the paper discussing future directions of this research.
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1 3 0.8 2
u
F
0.6 0.4
1
0.2 0
0 0
50 Time (day)
100
5
0.2
4
0.15
3
50 Time (day)
100
0
50 Time (day)
100
I
K
0.25
0
0.1
2
0.05
1
0
0 0
50 Time (day)
100
Fig. 8. Results of the application of the closed-loop control strategy to model (9).
F
3 2 1 0 0
0.05
0.1
0.15
0.2
0.25
3
4
5
I
F
3 2 1 0 0
1
2
3. When we consider the proposed control method to apply to most people living in malaria-infested areas to boost up their immunity against malaria, we cannot expect precise control of the timing of mosquito bites and the amount of pathogen in a bite to the practical point of view. Thus, as the next step of this research, we should regard the problem of how to achieve a predefined target level of the specific immunity with a condition of the duration of antimalaria drug treatment and with considerable tolerance of the timing and the pathogen amount of a mosquito bite. To this end we should also study sensitivity analysis on the variations of inoculation time of mosquito bite and the dose of malaria pathogen from the open-loop control method while such a simple analysis is preliminarily performed in Fig. 3. This approach will be potentially useful particularly in the perspective of medical application.
K Fig. 9. The (I , F ) and (K , F ) trajectories resulting from the closed-loop control strategy. The solid lines in each graph indicate the (I , F ) and (K , F ) trajectories, respectively. The dotted lines in each graph indicate the sets sn F − qn I = 0 and ss F − qs K = 0, respectively.
1. The experimental results in Roestenberg et al. (2009) related to single species malaria infection could give evidence to support the theoretical studies of open-loop control of the malaria dynamic model in this paper. At the moment the malaria model used in the paper can provide only a crude cartoon of the complex dynamics between participating components and reactions related to intruding parasite or pathogen (Gurarie et al., 2006). Further research with more complex models will be undertaken. With such a modeling we could evaluate whether a boosted value of immunity would be enough to be protective in a malaria patient, and also we could interpret the range of the immunity level in terms of biological implication. 2. Although the closed-loop control scheme is capable of boosting the specific immunity in silico its implementation has some clinical difficulties. In the suggested control method we implicitly assume that the states F and K are continuously measurable, and that malaria mosquito bites can be used in so far as we want. However these assumptions are currently not practical.
Acknowledgments The authors are grateful to the anonymous reviewers, whose valuable comments have been helpful to improve the quality of the paper. References Chang, H., & Astolfi, A. (2008). Control of HIV infection dynamics: approximating high-order dynamics by adapting reduced-order model parameters. IEEE Control Systems Magazine, 28, 28–39. Chang, H., & Astolfi, A. (2009a). Activation of immune response in disease dynamics via controlled drug scheduling. IEEE Transactions on Automation Science and Engineering, 6, 248–255. Chang, H., & Astolfi, A. (2009b). Enhancement of the immune response to chronic myeloid leukaemia via controlled treatment scheduling. In Proc. of 31st annual EMBS international conference (pp. 3889–3892). Chang, H., Astolfi, A., & Shim, H. (2010a). A control theoretic approach to venom immunotherapy with state jumps. In Proc. of 32nd annual EMBS international conference (pp. 742–745). Chang, H., Astolfi, A., & Shim, H. (2010b). Immunotherapy for HIV and malaria: a control theoretic approach with state jumps. In Proc. of American control conference (pp. 474–479). Gurarie, D., & McKenzie, F. E. (2006). Dynamics of immune response and drug resistance in malaria infection. Malaria Journal, 5(1), 86.
H. Chang et al. / Automatica 47 (2011) 1271–1277 Gurarie, D., Zimmerman, P. A., & King, C. H. (2006). Dynamic regulation of singleand mixed-species malaria infection: insights to specific and non-specific mechanisms of control. Journal of Theoretical Biology, 240(2), 185–199. Mason, D. P., & McKenzie, F. E. (1999). Blood-stage dynamics and clinical implications of mixed Plasmodium vivax–Plasmodium falciparum infections. American Journal of Tropical Medicine and Hygiene, 61(3), 367–374. Mason, D. P., McKenzie, F. E., & Bossert, W. H. (1999). The blood-stage dynamics of mixed Plasmodium malariae–Plasmodium falciparum infections. Journal of Theoretical Biology, 198(4), 549–566. Recker, M., Nee, S., Bull, P. C., Kinyanjui, S., Marsh, K., Newbold, C., et al. (2004). Transient cross-reactive immune responses can orchestrate antigenic variation in malaria. Nature, 429, 555–558. Roestenberg, M., McCall, M., Hopman, J., et al. (2009). Protection against a malaria challenge by sporozoite inoculation. New England Journal of Medicine, 361(5), 468–477. Shim, H., Jo, N. H., Chang, H., & Seo, J. H. (2009). A system theoretic study on a treatment of AIDS patient by achieving long-term non-progressor. Automatica, 45, 611–622. Wodarz, D. (2001). Helper-dependent vs. helper-independent CTL responses in HIV infection: implications for drug therapy and resistance. Journal of Theoretical Biology, 213, 447–459. Wodarz, D., & Nowak, M. A. (1999). Specific therapy regimes could lead to long-term immunological control of HIV. Proceedings of National Academy Science, 96(25), 14464–14469. World Health Organization Expert Committee on Malaria. (2003). 20th Report. WHO Regional Office for Africa. Yang, T. (1999). Impulsive control. IEEE Transactions on Automatic Control, 44(5), 1081–1083.
H. Chang received the B.S. and the M.S. degrees from the School of Electrical Engineering of Seoul National University, in 1998 and 2004m and the Ph.D. degree from the Electrical and Electronic Engineering Department of Imperial College London in 2009. He is currently working at Boston University as Post-doctoral researcher. His research interests include nonlinear control theory, nonlinear systems analysis, and theoretical biology. Email: [email protected].
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A. Astolfi was born in Rome, Italy, in 1967. He graduated in electrical engineering from the University of Rome in 1991. In 1992 he joined ETH-Zurich where he obtained a M.Sc. in Information Theory in 1995 and the Ph.D. degree with Medal of Honour in 1995 with a thesis on discontinuous stabilisation of nonholonomic systems. In 1996 he was awarded a Ph.D. from the University of Rome ‘‘La Sapienza’’ for his work on nonlinear robust control. Since 1996 he is with the Electrical and Electronic Engineering Department of Imperial College, London (UK), where he is currently Professor in Nonlinear Control Theory. From 1998 to 2003 he was also an Associate Professor at the Dept. of Electronics and Information of the Politecnico of Milano. Since 2005 he is also Professor at Dipartimento di Informatica, Sistemi e Produzione, University of Rome Tor Vergata. He has been visiting lecturer in ‘‘Nonlinear Control’’ in several universities, including ETH-Zurich (1995–1996); Terza University of Rome (1996); Rice University, Houston (1999); Kepler University, Linz (2000); SUPELEC, Paris (2001). His research interests are focused on mathematical control theory and control applications, with special emphasis for the problems of discontinuous stabilisation, robust stabilisation, robust and adaptive control, model reduction. He is author of more than 100 journal papers, of 20 book chapters and of over 200 papers in refereed conference proceedings. He is author (with D. Karagiannis and R. Ortega) of the monograph ‘‘Nonlinear and Adaptive Control with Applications’’ (Springer Verlag). He is Associate Editor of Systems and Control Letters, Automatica, the International Journal of Control, the European Journal of Control, the Journal of the Franklin Institute, and the International Journal of Adaptive Control and Signal Processing. He has also served in the IPC of various international conferences. He is the Chair of the IEEE CSS Conference Editorial Board. Email: [email protected].
H. Shim received his B.S., M.S. and Ph.D. degrees from Seoul National University, Korea in 1993, 1995 and 2000, respectively. From 2000 to 2002 he had a position as a post-doctoral fellow for the Center for Control Engineering and Computation at University of California, Santa Barbara. In March of 2002, he joined the faculty of the Division of Electrical and Computer Engineering, Hanyang University, Seoul, Korea. Since September of 2003, he has been with the School of Electrical Engineering at Seoul National University, Korea, where he is currently an associate professor. Email: [email protected].
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
A control theoretic approach to malaria immunotherapy with state jumps✩ H. Chang a,∗ , A. Astolfi b,c , H. Shim d a
Department of Mechanical Engineering, Boston University, MA 02215, United States
b
Department of Electrical and Electronic Engineering, Imperial College London, UK
c
Dipartimento di Informatica, Sistemi e Produzione, Università di Roma Tor Vergata, Via del Politecnico 1, 00133 Roma, Italy
d
ASRI, Department of Electrical Engineering, Seoul National University, Seoul, Republic of Korea
article
info
Article history: Received 11 February 2010 Received in revised form 21 January 2011 Accepted 21 February 2011 Available online 20 April 2011
abstract We investigate a control method for malaria dynamics to boost the immune response using a model-based approach. The idea of state jump is introduced and discussed using a hybrid model. To implement the state jumps we use a physiologically feasible method for the biological system, whereby the introduction of a pathogen into the system steers it to a desirable status. It is noted that we take advantage of a pathogen to implement state jumps in the system. The study is supported by recently reported experimental results. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Malaria Immunotherapy Hybrid systems State jump Impulsive control
1. Introduction Malaria remains a lethal and prevalent infectious disease worldwide, which accounts for 300 million cases and over a million fatalities every year (World Health Organization Expert Committee on Malaria, 2003). Several recent studies have discussed modelbased approaches to malaria treatment. In Mason and McKenzie (1999) malaria dynamics have been modeled and model parameters have been estimated. The model has been studied in Gurarie, Zimmerman, and King (2006) and Mason, McKenzie, and Bossert (1999), and the equations of the immune response in the model have been employed in different malaria models (Gurarie & McKenzie, 2006; Recker et al., 2004). We investigate a control scheme by a model-based approach to enhance the immune response in malaria dynamics. In Chang and Astolfi (2008, 2009a) and Shim, Jo, Chang, and Seo (2009) the authors have studied an immune boosting mechanism for the HIV dynamic models of Wodarz (2001) and Wodarz and Nowak (1999) and this mechanism is also used for the treatment of chronic myeloid leukemia in Chang and Astolfi (2009b). In this paper we
study the malaria model in Mason and McKenzie (1999) on the basis of the immune boosting mechanism introduced in Chang and Astolfi (2009a). To this end we propose the idea of using state jumps as impulsive control1 method. In addition an implementation method for state jumps in malaria dynamics is suggested. The implementation of the state jumps is physiologically practical, as suggested in Roestenberg, McCall, and Hopman (2009). Our research on the malaria model is supported by the experimental data which have been recently published in Roestenberg et al. (2009). Note that we do not provide a formal treatment of the properties of the controlled system, which will be reported elsewhere, but focus on the methodological and conceptual aspects of the control design. The paper is organised as follows. In Section 2 we give a brief description of the experiments in Roestenberg et al. (2009). Section 3 describes the application of the proposed control ideas to the malaria model of Mason and McKenzie (1999) and discusses the robustness of the suggested control method to a range of parameter values. Finally we discuss future works and present further remarks in Section 4. 2. A summary of the experiments in Roestenberg et al. (2009)
✩ The material in this paper was presented at the 2010 American Control Conference, June 30–July 2, 2010, Baltimore, Maryland, USA. This paper was recommended for publication in revised form by Associate Editor Murat Arcak under the directions of Corresponding Guest Editors Francis J. Doyle III and Frank Allgöwer. ∗ Corresponding author. Tel.: +1 617 358 0844; fax: +1 617 353 5548. E-mail addresses: [email protected] (H. Chang), [email protected], [email protected] (A. Astolfi), [email protected] (H. Shim).
0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.03.009
In this section we briefly discuss the experiments of Roestenberg et al. (2009). Although an effective vaccine against malaria is urgently needed, a malaria vaccine is not currently available. 1 Impulsive control gives a sudden change of the state variable at discrete instants. See Yang (1999) for rigorous definition of impulsive control.
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Table 1 Malaria model parameters and their values or ranges (Mason & McKenzie, 1999). Parameter
Description
Value or range
a b g qs qn cs cn ss sn x
Replication rate of P. vivax Replication rate of P. falciparum Gametocyte conversion rate Specific immunity decay rate Non-specific immunity deactivation rate Specific immunity removal rate Non-specific immunity removal rate Specific immunity proliferation rate Non-specific immunity proliferation rate Cross-reactivity rate of P. falciparum-specific effectors Cross-reactivity rate of P. vivax-specific effectors Continuous parasite reduction ratio
1.28 1.39 0.04 0.01 0.6 0.0001–1000 0.0001–1000 0.0001–1000 0.0001–1000 0–1
y k
0–1 2.3
This is largely because immune response to Plasmodium falciparum, a species of malaria parasite, is regarded difficult to acquire. In Roestenberg et al. (2009) it is shown that immunity against a malaria challenge can be induced by the inoculation of infected mosquitoes. In the experiment of Roestenberg et al. (2009), fifteen healthy volunteers have been treated with an antimalaria drug regimen, chloroquine, when they were exposed to mosquito bites once a month for 3 months. Ten of the volunteers have been assigned to a vaccine group while the remaining 5 volunteers have been assigned to a control group. The vaccine group has been exposed to mosquitoes infected with P. falciparum. The control group has been exposed to mosquitoes not infected with the malaria parasite. After 1 month the immune response has been tested by the inoculation of five mosquitoes infected with the malaria parasite. Malaria has not developed in any of the 10 volunteers in the vaccine group. Also no serious adverse events occurred in this vaccine group. However malaria has developed in all 5 volunteers in the control group. Thus the inoculation of malaria parasite by infected mosquitoes in the presence of antimalaria drug can lead to protection against a malaria challenge. 3. State jump in malaria dynamics The clinical work of Roestenberg et al. (2009) leads to application of the mathematical method of immune enhancement, developed in HIV dynamics, to malaria dynamics. The experimental results in Roestenberg et al. (2009) can be explained by the immune system analysis in this section. We consider the malaria infection model of Mason and McKenzie (1999). The model is given by V˙ = aV − cs JV − cs xKV − cn IV − gV ,
(1)
F˙ = bF − cs KF − cs yJF − cn IF − gF , ˙I = sn (V + F ) − qn I ,
(2)
J˙ = ss V − qs J , K˙ = ss F − qs K ,
(3) (4) (5)
where the states V , F , I, J, and K describe the populations of specific cells in a unit volume of blood and therefore are meaningful only when nonnegative. Human malaria is caused by four species of Plasmodium: P. falciparum, P. malariae, P. ovale, and P. vivax (Mason et al., 1999). The model (1)–(5) includes the dynamics between P. vivax, P. falciparum and the immune system. Indeed V describes the concentration of P. vivax, F the concentration of P. falciparum, I the concentration of the effectors of the non-specific immune response, and J and K the concentrations of the effectors of the specific immune response for V and F , respectively. The remaining parameters a, b, g, qs , qn , x, y, cs , cn , ss , and sn are positive and constant. The descriptions and values of the model parameters
estimated in Mason and McKenzie (1999) are summarised in Table 1. Since the model parameters are based on clinical data, the state variables in this section represent actual data. See Mason and McKenzie (1999) for a detailed explanation of the model. We consider a single species P. falciparum case by removing the variables V and J of model (1)–(5) and taking equations (2), (3) and (5) as studied in Gurarie et al. (2006) because this is the clinical case studied in Roestenberg et al. (2009). The term of administration of antimalarials u is included by introducing a parasite killing rate k. If u = 1 the maximal drug therapy is received, while u = 0 means no medication. We set k = 2.3 consistently with published ranges of parasite reduction ratios (Mason & McKenzie, 1999). As a result the model is given by F˙ = (b − ku)F − cs KF − cn IF − gF ,
(6)
˙I = sn F − qn I , K˙ = ss F − qs K .
(7) (8)
The goal of the control is to enhance immunity, and this is equivalent to boosting I and K . If ˙I = sn F − qn I > 0 (or K˙ = ss F − qs K > 0), then the immunity by I (or K ) is enhanced. Now note that each inequality depends on two state variables, F and I (or F and K ), of model (6)–(8). When we consider the (I , F ) positive quadrant (or the (K , F ) positive quadrant), ˙I = 0 (or K˙ = 0) corresponds to a straight line. Thus within the region above this line ˙I > 0 (or K˙ > 0). Therefore the immune term I (or K ) increases if the state (I , F ) (or (K , F )) belongs to the region above the line. The basic control idea is to force the system state to be such that ˙I > 0 (or K˙ > 0). In this paper we consider state jumps as an impulsive control method to guarantee these inequalities. To this end the state must be moved into the set {(I , F ) : sn F − qn I > 0} (or {(K , F ) : ss F − qs K > 0}) by a ‘‘state jump’’, which is realised by mosquito inoculation in Roestenberg et al. (2009). Let Y (t ) := [F (t ), I (t ), K (t )]T and let model (6)–(8) be written as Y˙ = G(Y , u),
(9)
where u is the antimalarial drug input. In this section we consider infective mosquito bites to simulate the experiment of Roestenberg et al. (2009) in silico. One mosquito bite corresponds to a total of 300,000 malaria parasites (primary merozoites) (Gurarie & McKenzie, 2006). The experiment in Roestenberg et al. (2009) uses 36–45 infected mosquito bites so we assume 40 bites in each mosquito exposure to model (6)–(8). Note that the state variables used in Mason and McKenzie (1999) represent per-microliter densities and we assume that the total human blood volume is 4500 (ml). Thus we obtain 2.67 (=300,000 × 40/(4500 × 1000)) parasite density and we set Y ∗ = [2.67, 0, 0]T which corresponds to the amount of state jump. Also we consider the initial state Y (0) = [0, 0, 0]T , which represents the status of the subjects of the experiment in Roestenberg et al. (2009). We set the drug treatment period Td = 90 and the state jump treatment period Tt = 63 consistently with the experiment in Roestenberg et al. (2009). As in Mason and McKenzie (1999) we use cs = 0.1, cn = 0.1, ss = 0.1, and sn = 0.1. 3.1. Open-loop control In this subsection we consider open-loop control for the model (9) and the exposure of infective mosquito bites is carried out at t = 7, 35, 63, which corresponds to the case of Roestenberg et al. (2009). The system with suggested controls is given by Y˙ (t ) = G(Y (t ), 1),
if t ∈ [0, Td ] and t ̸∈ St ,
Y˙ (t ) = G(Y (t ), 0),
if t > Td ,
Y (t + ) = Y (t − ) + Y ∗ , where St = {7, 35, 63}.
if t ∈ St ,
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3
1 0.8 F
u
2 0.6 0.4
1
0.2 0
0 0
50
100
0
Time (day)
50
100
Time (day) 0.6
0.1
I
K
0.4 0.05
0.2
0
0 0
50
100
0
50
100
Time (day)
Time (day)
Fig. 1. Results of the application of the open-loop control strategy to model (9).
3
F
2 1 (I(0), F(0)) 0 0
0.02
0.04
0.06 I
0.08
0.1
0.12
3
F
2 1 (K(0), F(0)) 0 0
0.1
0.2
0.3 K
0.4
0.5
0.6
Fig. 2. The (I , F ) and (K , F ) trajectories resulting from the open-loop control strategy. The solid lines in each graph indicate the (I , F ) and (K , F ) trajectory, respectively. The dotted lines in each graph indicate the sets sn F − qn I = 0 and ss F − qs K = 0, respectively. The thick arrows on the trajectories illustrate the state jumps by impulsive control. Note that the arrows on the (I , F ) trajectory overlap.
Fig. 1 shows the results of the application of the proposed openloop control procedure. The specific immune effector K is boosted up to 0.602 at t = 66.75 (day) while the non-specific immune effector I is not. This is because of the difference in the decay rates (i.e., qs < qn ). Thus immune effectors of lower decay rates such as memory B cells (Mason & McKenzie, 1999) are enhanced by the given control method. The resulting (I , F ) and (K , F ) trajectories are displayed in Fig. 2. The thick arrows on the trajectories illustrate the state jumps by impulsive control. This shows that each trajectory enters into the immune increasing area (i.e. sn F − qn I > 0 and ss F − qs K > 0) by the action of impulsive control. In the simulation without state jumps, the state variables F , I, and K are zeros. Accordingly the state at t = 100 (day) of the simulation is [0, 0, 0]T which implies that the immune response is not enhanced, consistently with the result of Roestenberg et al. (2009).
In order to verify the robustness of the open-loop control we perform the computer simulations with Y ∗ and St being perturbed within certain bounds. Fig. 3 shows the simulation resulting from applying the control strategy to 40 cases, with parameters generated by randomly around the nominal values of Y ∗ and St : the first element of Y ∗ is perturbed up to ±50% and each inoculation timing in St is perturbed ±3.5 (days). In spite of diverse control parameters the average of the maximally boosted K during the control procedure is 0.5905 with the standard deviation 0.0904. This indicates that the suggested open-loop control has a large robust margin against control parameter uncertainty. Remark. When the (I , F ) trajectory (or (K , F ) trajectory) passes through the set sn F − qn I = 0 (or ss F − qs K = 0), we have that dI /dF = 0 (or dK /dF = 0). This implies that if we obtain a (I , F ) trajectory (or (K , F ) trajectory) with a point in which dI /dF = 0 (or dK /dF = 0) from experimental data, then we could estimate the ratio sn /qn (or ss /qs ). We now state a proposition related to the boosting level achievable with the state jump by an impulsive control technique. Proposition 1. Consider the model (9) with Y ∗ = [FM , 0, 0]T for some positive FM and assume b − k − g ≤ rm < 0 and u = 1. Also assume that the state jump is carried out if 0 ≤ F ≤ Fm for some positive Fm . Then Fm ss /qs < K (t ) < (FM + Fm )ss /qs and Fm sn / qn < I (t ) < (FM + Fm )sn /qn for t > tF for some positive tF . By Proposition 1 the minimum enhancement and the maximum enhancement of the immune response I (t ) (or K (t )) are proportional to sn /qn (or ss /qs ). This conclusion is discussed further in Section 3.2. Proof. For the case of K (t ) see Fig. 4 (the proof for I (t ) is similar). For t > tF , F (t ) ≥ Fm by the state jump and F (t ) ≤ (Fm + FM ) since F˙ < rm F < 0. Note that the state (K , F ) cannot stay close to either (Fm ss /qs , Fm ) or ((FM + Fm )ss /qs , (FM + Fm )) since F˙ < rm F < 0. Thus the state (K , F ) cannot be such that Fm ≤ F ≤ (Fm + FM ) and K ≤ Fm ss /qs because K˙ > 0 if ss F − qs K > 0. Also the state (K , F ) cannot be such that Fm ≤ F ≤ (Fm + FM ) and K ≥ (FM + Fm )ss /qs because K˙ < 0 if ss F − qs K < 0. Hence Fm ss /qs < K (t ) < (FM + Fm )ss /qs for t > tF for some positive tF .
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1
4
0.8 F
u
3 0.6
2
0.4 0.2
1
0
0 0
50 Time (day)
100
0
50 Time (day)
100
0.2 0.8 0.15
K
I
0.6 0.1 0.05
0.4 0.2
0
0 0
50
100
0
Time (day)
50 Time (day)
100
Fig. 3. Results of the application of the open-loop control strategy to model (9) for 40 different control parameters conditions.
ssF-qsK = 0 FM + Fm
F
FM
Fm Fm 0
0
Fmss/qs
K
(FM + Fm)ss/qs
Fig. 4. Proof of Proposition 1 for K (t ). The dashed line and the dotted line indicate the sets in which ss F − qs K = 0 and F = Fm , respectively. The thin solid segment represents the level of Fm while the thick solid segment represents the level of FM . As the arrows indicate, K˙ > 0 if ss F − qs K > 0 and K˙ < 0 if ss F − qs K < 0.
3.2. Sensitivity to parametric range Table 1, taken from Mason and McKenzie (1999), shows that four parameters (cs , cn , ss , and sn ) are not well-defined but can vary within a range of seven orders of magnitude. Thus we now explore the sensitivity of the control results to these parametric perturbations. Among the four parameters we especially study the implication of the variation of ss and cs since the values of the parameters are closely related to the state K which is boosted by the control method in the example of the previous subsection. Fig. 5 shows the results of the application of the open-loop control method with two values of ss , 0.05 and 0.2. The resulting (K , F ) trajectories are displayed in the lower graph of Fig. 5 where the solid line and the dotted line indicate the (K , F ) trajectories with ss = 0.05 and 0.2, respectively. The thick arrows on the trajectories illustrate the state jumps. During the control procedure the specific immune effector K is boosted up to 0.306 and 1.168, with ss = 0.05 and ss = 0.2 respectively. Compared to the maximum level 0.602 of K with
ss = 0.1 in the previous subsection the maximally enhanced level of the specific immune effector is proportional to the parameter ss . Fig. 6 shows the relation between the maximal enhancement of K and the parameter values of ss of which the total range is given in Table 1 of Mason and McKenzie (1999). From the dynamics of K in (8), the proliferation of K is described as ss F which is proportional to ss . It is consistent with the biological meaning of ss in Table 1, specific immunity (i.e. K ) proliferation rate, by which the proportional trend in Fig. 6 is explained. Accordingly we can expect that the human body with high proliferation rate of specific immunity would be induced to enhance the corresponding immunity by the suggested control scheme. Fig. 7 shows the inversely proportional relation between the maximum enhancement of K and the given parameter values of cs in Table 1 of Mason and McKenzie (1999). From (6) an increase in cs leads to a decrease in the state F and then a decrease in K is consequently induced (see (8)), by which the inversely proportional trend in Fig. 7 is explained. Thus the human body with high removal rate of specific immunity would not be expected to have strong immunity even by the suggested control scheme. Similarly there is an inverse proportional relation between the maximal enhancement of K and the parameters sn and cn . 3.3. Closed-loop control In this subsection we investigate how effectively the malaria immunity can be enhanced by a closed-loop control scheme, compared to an open-loop control scheme, i.e. the experiments of Roestenberg et al. (2009). We construct a feedback control strategy to maintain ss F − qs K ≥ 0 for 0 < t ≤ Tt so that we can enhance the immune effector K which has a lower decay rate than I. In the open-loop scheme of Section 3.1, the control action is performed at predefined time instants, i.e. t = 7, 35, 63. For the closed-loop scheme we do not have any predefined time instant for control input. Instead the occurrence of control input is decided based on the inequality ss F − qs K ≥ 0. The controlled hybrid system is given by Y˙ (t ) = G(Y (t ), 1), or t ∈ (Tt , Td ],
if t ∈ [0, Tt ] and ss F − qs K > 0,
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1 K with ss - 0.2
K with ss - 0.05
0.3
0.2
0.1
0.5
0
0 50 Time (day)
0
100
0
50 Time (day)
100
3
F
2
1
0 0
0.2
0.4
0.6
0.8
1
1.2
K with ss = 0.05 and 0.2 Fig. 5. Results of the application of the control strategy of Section 3.1 to model (9) with ss = 0.05 and 0.2. The (K , F ) trajectories with these two values of ss are presented in the lower graph where the solid line and the dotted line indicate the (K , F ) trajectories with ss = 0.05 and 0.2, respectively.
Y˙ (t ) = G(Y (t ), 0),
350
Maximum enhanced value of K
300 250 200 150 100 50 0 0
200
400
600
800
1000
Ss Fig. 6. Maximum enhancement of K with the given range of ss in Table 1 of Mason and McKenzie (1999).
1 Maximum enhanced value of K
if t > Td ,
Y (t + ) = Y (t − ) + Y ∗ ,
if t ∈ [0, Tt ] and ss F − qs K ≤ 0.
Note that, due to this system description, if one state jump cannot render ss F −qs K > 0 then multiple jump actions will be performed. For certain values of F and K , a multiple of Y ∗ guarantees ss F − qs K > 0. The result of the control action is shown in Figs. 8 and 9. The total number of mosquito inoculations is 24 in the closed-loop case, while it is 3 in the open-loop case of Section 3.1. The boosting level of K by closed-loop control is much higher than that achievable by open-loop control. (Compare the graph of K in Fig. 8 with that of K in Fig. 1.) Fig. 9 shows the (I , F ) and (K , F ) trajectories. In the lower graph of Fig. 9 the (K , F ) trajectory stays within the set ss F − qs K ≥ 0 for 0 < t ≤ Tt . Note that the (K , F ) trajectory does not stay within the region ss F − qs K ≥ 0 for the whole time interval 0 < t ≤ Tt when the open-loop control strategy of Section 3.1 is applied (see the lower graph of Fig. 2). 4. Conclusions
0.9
We have investigated a control method to enhance the immune response in malaria dynamics. The idea of state jump is used and the state jumps are implemented by the use of ‘‘pathogen’’. Thus we take advantage of a pathogen to drive a biological system to a desirable status. The immune boosting idea in this paper can be applied to various types of immune system models. Recently the authors have applied this control scheme in silico to enhance the immune response in other biological dynamic systems, such as an HIV/AIDS infection model (Chang, Astolfi, & Shim, 2010b) and a bee venom immunotherapy model (Chang, Astolfi, & Shim, 2010a).
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
200
400
600
800
1000
Cs Fig. 7. Maximum enhancement of K with the given range of cs in Table 1 of Mason and McKenzie (1999).
4.1. Future works We conclude the paper discussing future directions of this research.
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1 3 0.8 2
u
F
0.6 0.4
1
0.2 0
0 0
50 Time (day)
100
5
0.2
4
0.15
3
50 Time (day)
100
0
50 Time (day)
100
I
K
0.25
0
0.1
2
0.05
1
0
0 0
50 Time (day)
100
Fig. 8. Results of the application of the closed-loop control strategy to model (9).
F
3 2 1 0 0
0.05
0.1
0.15
0.2
0.25
3
4
5
I
F
3 2 1 0 0
1
2
3. When we consider the proposed control method to apply to most people living in malaria-infested areas to boost up their immunity against malaria, we cannot expect precise control of the timing of mosquito bites and the amount of pathogen in a bite to the practical point of view. Thus, as the next step of this research, we should regard the problem of how to achieve a predefined target level of the specific immunity with a condition of the duration of antimalaria drug treatment and with considerable tolerance of the timing and the pathogen amount of a mosquito bite. To this end we should also study sensitivity analysis on the variations of inoculation time of mosquito bite and the dose of malaria pathogen from the open-loop control method while such a simple analysis is preliminarily performed in Fig. 3. This approach will be potentially useful particularly in the perspective of medical application.
K Fig. 9. The (I , F ) and (K , F ) trajectories resulting from the closed-loop control strategy. The solid lines in each graph indicate the (I , F ) and (K , F ) trajectories, respectively. The dotted lines in each graph indicate the sets sn F − qn I = 0 and ss F − qs K = 0, respectively.
1. The experimental results in Roestenberg et al. (2009) related to single species malaria infection could give evidence to support the theoretical studies of open-loop control of the malaria dynamic model in this paper. At the moment the malaria model used in the paper can provide only a crude cartoon of the complex dynamics between participating components and reactions related to intruding parasite or pathogen (Gurarie et al., 2006). Further research with more complex models will be undertaken. With such a modeling we could evaluate whether a boosted value of immunity would be enough to be protective in a malaria patient, and also we could interpret the range of the immunity level in terms of biological implication. 2. Although the closed-loop control scheme is capable of boosting the specific immunity in silico its implementation has some clinical difficulties. In the suggested control method we implicitly assume that the states F and K are continuously measurable, and that malaria mosquito bites can be used in so far as we want. However these assumptions are currently not practical.
Acknowledgments The authors are grateful to the anonymous reviewers, whose valuable comments have been helpful to improve the quality of the paper. References Chang, H., & Astolfi, A. (2008). Control of HIV infection dynamics: approximating high-order dynamics by adapting reduced-order model parameters. IEEE Control Systems Magazine, 28, 28–39. Chang, H., & Astolfi, A. (2009a). Activation of immune response in disease dynamics via controlled drug scheduling. IEEE Transactions on Automation Science and Engineering, 6, 248–255. Chang, H., & Astolfi, A. (2009b). Enhancement of the immune response to chronic myeloid leukaemia via controlled treatment scheduling. In Proc. of 31st annual EMBS international conference (pp. 3889–3892). Chang, H., Astolfi, A., & Shim, H. (2010a). A control theoretic approach to venom immunotherapy with state jumps. In Proc. of 32nd annual EMBS international conference (pp. 742–745). Chang, H., Astolfi, A., & Shim, H. (2010b). Immunotherapy for HIV and malaria: a control theoretic approach with state jumps. In Proc. of American control conference (pp. 474–479). Gurarie, D., & McKenzie, F. E. (2006). Dynamics of immune response and drug resistance in malaria infection. Malaria Journal, 5(1), 86.
H. Chang et al. / Automatica 47 (2011) 1271–1277 Gurarie, D., Zimmerman, P. A., & King, C. H. (2006). Dynamic regulation of singleand mixed-species malaria infection: insights to specific and non-specific mechanisms of control. Journal of Theoretical Biology, 240(2), 185–199. Mason, D. P., & McKenzie, F. E. (1999). Blood-stage dynamics and clinical implications of mixed Plasmodium vivax–Plasmodium falciparum infections. American Journal of Tropical Medicine and Hygiene, 61(3), 367–374. Mason, D. P., McKenzie, F. E., & Bossert, W. H. (1999). The blood-stage dynamics of mixed Plasmodium malariae–Plasmodium falciparum infections. Journal of Theoretical Biology, 198(4), 549–566. Recker, M., Nee, S., Bull, P. C., Kinyanjui, S., Marsh, K., Newbold, C., et al. (2004). Transient cross-reactive immune responses can orchestrate antigenic variation in malaria. Nature, 429, 555–558. Roestenberg, M., McCall, M., Hopman, J., et al. (2009). Protection against a malaria challenge by sporozoite inoculation. New England Journal of Medicine, 361(5), 468–477. Shim, H., Jo, N. H., Chang, H., & Seo, J. H. (2009). A system theoretic study on a treatment of AIDS patient by achieving long-term non-progressor. Automatica, 45, 611–622. Wodarz, D. (2001). Helper-dependent vs. helper-independent CTL responses in HIV infection: implications for drug therapy and resistance. Journal of Theoretical Biology, 213, 447–459. Wodarz, D., & Nowak, M. A. (1999). Specific therapy regimes could lead to long-term immunological control of HIV. Proceedings of National Academy Science, 96(25), 14464–14469. World Health Organization Expert Committee on Malaria. (2003). 20th Report. WHO Regional Office for Africa. Yang, T. (1999). Impulsive control. IEEE Transactions on Automatic Control, 44(5), 1081–1083.
H. Chang received the B.S. and the M.S. degrees from the School of Electrical Engineering of Seoul National University, in 1998 and 2004m and the Ph.D. degree from the Electrical and Electronic Engineering Department of Imperial College London in 2009. He is currently working at Boston University as Post-doctoral researcher. His research interests include nonlinear control theory, nonlinear systems analysis, and theoretical biology. Email: [email protected].
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A. Astolfi was born in Rome, Italy, in 1967. He graduated in electrical engineering from the University of Rome in 1991. In 1992 he joined ETH-Zurich where he obtained a M.Sc. in Information Theory in 1995 and the Ph.D. degree with Medal of Honour in 1995 with a thesis on discontinuous stabilisation of nonholonomic systems. In 1996 he was awarded a Ph.D. from the University of Rome ‘‘La Sapienza’’ for his work on nonlinear robust control. Since 1996 he is with the Electrical and Electronic Engineering Department of Imperial College, London (UK), where he is currently Professor in Nonlinear Control Theory. From 1998 to 2003 he was also an Associate Professor at the Dept. of Electronics and Information of the Politecnico of Milano. Since 2005 he is also Professor at Dipartimento di Informatica, Sistemi e Produzione, University of Rome Tor Vergata. He has been visiting lecturer in ‘‘Nonlinear Control’’ in several universities, including ETH-Zurich (1995–1996); Terza University of Rome (1996); Rice University, Houston (1999); Kepler University, Linz (2000); SUPELEC, Paris (2001). His research interests are focused on mathematical control theory and control applications, with special emphasis for the problems of discontinuous stabilisation, robust stabilisation, robust and adaptive control, model reduction. He is author of more than 100 journal papers, of 20 book chapters and of over 200 papers in refereed conference proceedings. He is author (with D. Karagiannis and R. Ortega) of the monograph ‘‘Nonlinear and Adaptive Control with Applications’’ (Springer Verlag). He is Associate Editor of Systems and Control Letters, Automatica, the International Journal of Control, the European Journal of Control, the Journal of the Franklin Institute, and the International Journal of Adaptive Control and Signal Processing. He has also served in the IPC of various international conferences. He is the Chair of the IEEE CSS Conference Editorial Board. Email: [email protected].
H. Shim received his B.S., M.S. and Ph.D. degrees from Seoul National University, Korea in 1993, 1995 and 2000, respectively. From 2000 to 2002 he had a position as a post-doctoral fellow for the Center for Control Engineering and Computation at University of California, Santa Barbara. In March of 2002, he joined the faculty of the Division of Electrical and Computer Engineering, Hanyang University, Seoul, Korea. Since September of 2003, he has been with the School of Electrical Engineering at Seoul National University, Korea, where he is currently an associate professor. Email: [email protected].