The within-host dynamics of malaria infection with immune response. Preprint Liu Z, Magal P, Ruan S (2008) Projectors on the generalized eigenspaces for functional differential equations using integrated semigroups. J Differ Equ 244:1784–1809 Liu Z, Magal

Malaria infection is one of the most serious global health problems of our time. In this article the blood-stage dynamics of malaria in an infected host are studied by incorporating red blood cells, malaria parasitemia and immune effectors into a mathematical model with nonlinear bounded Michaelis-Menten-Monod functions describing how immune cells interact with infected red blood cells and merozoites. By a theoretical analysis of this model, we show that there exists a threshold value R0, namely the basic reproduction number, for the malaria infection. The malaria-free equilibrium is global asymptotically stable if R0 < 1. If R0 > 1, there exist two kinds of infection equilibria: malaria infection equilibrium (without specific immune response) and positive equilibrium (with specific immune response). Conditions on the existence and stability of both infection equilibria are given. Moreover, it has been showed that the model can undergo Hopf bifurcation at the positive equilibrium and exhibit periodic oscillations. Numerical simulations are also provided to demonstrate these theoretical results.


1.
Introduction.Malaria is one of the three most dangerous infectious diseases worldwide (along with HIV/AIDS and tuberculosis).It is endemic in the tropical and subtropical regions of the world and caused an estimated 243 million cases led to an estimated 863,000 deaths in 2008 (WHO [42]).It is believed that half of the world's population is at risk of malaria (WHO [42]).Malaria infection in a host is caused by an inoculum of parasites from a blood-feeding female Anopheles mosquito carrying one or a combination of any of the four species of Plasmodium parasites: P. falciparum, P. malariae, P. ovale, and P. vivax.Among them, P. falciparum is 1000 YILONG LI, SHIGUI RUAN AND DONGMEI XIAO responsible for almost all of the deaths attributed to malaria (McKenzie and Bossert [25]).
The malaria parasites first penetrate liver cells of the host and then move into the blood, where they multiply and undergo replication cycles in the red blood cells (or erythrocytes): the parasites multiply in the red blood cells which cause the infected red blood cells to burst and release a mass of new parasites (called merozoites) that quickly invade other red blood cells, and the cycle is repeated.When malaria parasites evolve in the host, they can stimulate the activity of immune cells in the host which produce an immune response to fight the infection.Immune response can either prevent the re-invasion of merozoites or increase the death rate of infected red blood cells (Stevenson and Riley [38] and Good et al. [12]).
Human immune system is composed of two subdivisions, the innate (non-specific) immune system and the adaptive (specific) immune system.The innate immune system is the first line of defense against invading pathogens while the adaptive immune system acts as a second line of defense which also provides protection against re-exposure to the same pathogen.Malaria infection triggers both innate and adaptive immune responses (Augustine et al. [6], Langhorne et al. [21], Malaguarnera and Musumeci [23]).Innate immune cells such as natural killer cells and dendritic cells are involved in the clearance of circulating parasites infected red blood cells (Cuban et al. [8], Augustine et al. [6]).Adaptive immune cells such as CD4 + and CD8 + are important for protection against malaria and B cell responses are induced by Plasmodium infection (Langhorne et al. [21], Augustine et al. [6]).The immune system has both cellular and humoral components by which they carry out their protective function.Cellular immunity is that T lymphocytes secrete proteins to act directly against the pathogens and stimulate cytotoxic T-cells which protect the host cells by lysis of infected cells and reduce the production of merozoites and gametocytes.Humoral immunity is the immune protection mediated by B lymphocytes which are activated by merozoites in blood and secrete antibodies into circulation as they remove merozoites from blood (Deans and Cohen [9] and Tumwiine et al. [40]).Though antibody-mediated immunity is more effective than cell-mediated immunity (Deans and Cohen [9]), extensive numerical analysis by Anderson et al. [4] suggested that it is very difficult to eradicate the parasites from the host by antibody-mediated attack against the free merozoites alone due to their short life-expectancy outside the erythrocytes.
In the last two decades, many mathematical models have been employed to describe the within-host dynamics of malaria infection, namely the dynamics of the blood stages of the malaria parasites and their interactions with red blood cells and immune effectors.The first models were proposed by Anderson et al. [4] (see also Hetzel and Anderson [16] and Anderson [3]) which consisted of healthy red blood cells, infected red blood cells, malaria parasitemia, without or with immune effectors.These models have been generalized by many researchers for different purposes, we refer to a review by Molineaux and Dietz [29] on various such generalizations and references.
Oscillations are common in the immune system (Stark et al. [37]), in particular when the host is infected by malaria parasites.Periodic occurrence of fever is the cardinal symptom of malaria and the period has been identified with the length of the replication cycle (Rouzine and Mckenzie [34]), which is 48 hours for P. falciparum.The periodicity indicates that malaria parasite replication in the red blood cells is synchronized: parasites enter and are released from the red blood cells at approximately the same times (Rouzine and Mckenzie [34]).However, the mechanism of this synchronization is still not well-understood and quite a few models have been proposed to study the synchronization.For example, Kwiatkowsti and Nowak [20] proposed a 2-dimensional discrete model to show that the interaction between malaria parasites and red blood cells naturally tends to generate periodic fevers in the host when the replication rate is high.Rouzine and McKenzie [34] constructed an age-structured model to demonstrate that innate immune responses cause synchronization between the replication cycles of parasites in red blood cells which is reflected in periodic fevers in the host.Su et al. [39] proved the existence of Hopf bifurcation in the age-structured malaria infection model of Rouzine and McKenzie [34] by using the replication rate as the bifurcation parameter and showed numerically that synchronization with regular periodic oscillations (of period 48 h) occurs when the replication rate increases.Hoshen et al. [17] and Dong and Cui [11] introduced time delay into the basic model (without immune response) of Anderson et al. [4] to produce periodic oscillations in host-parasites.See also Mitchell and Carr [28].When immune response is included in the basic model, via numerical simulations Anderson et al. [4] and Hetzel and Anderson [16] observed that periodic oscillations occur in the model with killing of infected red blood cells or with immune response directly against merozoites and infected red blood cells.
In this article we study the blood-stage dynamics of malaria in an infected host by incorporating healthy red blood cells, infected red blood cells, malaria parasitemia and immune effectors into a mathematical model.The model is a generalization of the basic models of Anderson et al. [4] and Anderson [3] with nonlinear bounded Michaelis-Menten-Monod functions describing how immune cells interact with the infected red blood cells and merozoites.We present some local analysis of the model, namely the existence and stability of the malaria-free, malaria infection (without specific immune response), and positive (with specific immune response) equilibria, in terms of the basic reproduction number.It is shown that if the basic reproduction numbers is greater than one, then the malaria parasites can infect the host and establish a persistent infection.The model also exhibits periodic oscillations due to Hopf bifurcation at the positive equilibrium by using the proliferation rate of the immune cells induced by infected red blood cells as the bifurcation parameter, which demonstrates that synchronicity is an inherent feature of malaria infection with immune response.Thus, we provide theoretical analysis and proof of the numerical observations of Anderson et al. [4] and Hetzel and Anderson [16] on the existence of periodic oscillations in the model with immune response.We also present some numerical simulations to illustrate out results.
This paper is organized as follows.In section 2 we propose a simple mathematical model for the within-host dynamics of malaria infection based on the basic understanding of biological interactions between malaria parasites, red blood cells, and immunity effectors, and some simple assumptions about the immune system.In section 3 we analyze the equilibria and obtain the basic reproduction numbers for malaria infection.We then present some numerical simulations and give some discussion in section 4.
Red blood cells develop continuously from stem cells in the bone marrow through reticuloctyes to mature in about 7 days and live a total of about 120 days (Rapaport [33]).The population of uninfected red blood cells satisfies the equation dH dt = λ − d 1 H in the absence of any infection, which converges to a steady state λ d1 , where λ is a constant product rate from the bone marrow and d 1 is the constant death rate of uninfected red blood cells, respectively.A density of about 5 million red blood cells per µl is maintained in adult males (Rapaport [33] and McQueen and McKenzie [26]).
In the body system of an infected host, the invading parasites will infect the red blood cells of the host.We assume that malaria parasites infect the red blood cells at a rate proportional to the contact rate of their population size, αM H, where α is a positive constant which describes the rate or probability of successful infection by a malaria parasite.It has been reported that up to 500, 000 red blood cells per µl are parasitized with P. falciparum and only 25, 000 cells per µl with P. vivax, P. ovale, or P. malariae (Mandell et al. [24]).The infected cells die at rate δ R per day so that 1/δ is the life-expectancy of infected red blood cells (approximately 2 days, see Anderson et al. [4]).
Immune responses against malaria infections are complex and stage-specific.The malaria parasite induces a specific immune response which can stimulate the release of cytokines and activate the host's monocytes, neutrophils, T-cells, and natural killer cells to react to the different stage parasite (Malaguarnera and Musumeci [23]).It would be reasonable to include various innate, antibody and T-cell responses to malaria parasite in modeling the within-host dynamics (see McQueen and McKenzie [27] and Chiyaka et al. [7]).However, for the sake of simplicity and analysis, we only consider the immunity effectors E(t) as the total capacity of the immune response of the host to infected cells by parasites.
Previously, the killing of infected cells by immunity effectors has been modeled by a simple mass-action term depending only on the product of the densities of the parasite and the immune cells which is an unbounded bilinear function (see Anderson et al. [4] and Hetzel and Anderson [16]).Taking into account the fact that cell proliferation can saturate and that there is a handling time in immune responses, the more reasonable nonlinear bounded Michaelis-Menten-Monod function was firstly used by Agur et al. [2] and Antia et al. [5] and late formally derived and proposed by De Boer and Perelson [10] and Pilyugin and Antia [32] to describe the interaction between immune cells and their targets (bacteria, parasites, viruses, etc.).Though there are no clinical or experimental data to support that the interaction between immune responses and malaria parasites satisfies the Michaelis-Menten-Monod function, we follow De Boer and Perelson [10], Pilyugin and Antia [32], and Chiyaka et al. [7] to use such a function p 1 IE/(1 + βI) to describe the killing of infected cells I by the immunity effectors E, where p 1 is the rate or possibility of successful removal of infected red blood cells I by immunity effectors and 1/β is a saturation constant that simulates immune cells to grow at half their maximum rate.It is also assumed that the presence of infected cells stimulates the proliferation of immune cells at a net rate k 1 IE/(1 + βI), where k 1 is the proliferation rate of lymphocytes.Immune effectors decay at a rate d 2 .Free malaria parasites is produced from merozoites which replicate at a rate r in an infected red blood cell and die at a rate µ.Note that the replication rate r is understood as the number (r 1 ) of merozoites produced by each infected red blood cell times the rate (δ) at which the infected red blood cells burst due to infection.We also assume that antibody-mediated attack directed against the free merozoites in the blood system (Anderson et al. [4]), given by p 2 M E/(1 + γM ), and a net production rate of merozoite-specific antibodies of k 2 M E/(1 + γM ).p 2 is the rate or possibility of successful removal of free merozoites M by immunity effectors, 1/γ is a saturation constant, and k 2 is the proliferation rate of lymphocytes due to the interactions between E and M.
The mathematical model for malaria parasites infection in a host consists of four ordinary differential equations: The variables and their initial values are presented in Table 1.All parameters and their biological interpretations are given in Table 2.
Note that the terms M E/(1 + γM ) and IE/(1 + βI) describe, respectively, how the parasites and infected red cells simulate the activation of the immune effectors, they are regarded to describe the humoral and cell-mediated immunity, respectively (Anderson et al. [4], Murase et al. [30], Tumwiine et al. [40]).
We would like to make some remarks on the choice of parameter values and their units.Some parameters were adapted from other references directly, such as λ and d 2 from Anderson et al. [4].Some other parameters were obtained by conversion and calculation of that from other references.For example, the term k1IE 1+βI appeared as (k1/β)IE (1/β)+βI in Chiyaka et al. [7], so we calculated the parameter values correspondingly and changed their units accordingly.
Model (1) generalizes several known models, including the basic models in Anderson et al. [4] and Anderson [3], the pathogen-immune interaction model developed by Nowak and Bangham [31], and some variants in Liu [22], Murase et al. [30], and Tumwiine et al. [40].When the immune response functions are unbounded bilinear functions, that is when β = γ = 0, Murase et al. [30] and Tumwiine et al. [40] studied the stability of these models.In particular, Kajiwara and Sasaki [19] proved that the models of Liu [22] and Murase et al. [30] are indeed globally stable.However, numerical simulations by Anderson et al. [4] and Hetzel and Anderson [16] indicated that periodic oscillations occur in the model with immune response.We shall study the existence and stability of the malaria-free, malaria infection, and positive equilibria and show that the model exhibits periodic oscillations via Hopf bifurcation at the positive equilibrium by using the proliferation rate of the immune cells induced by infected red blood cells as the bifurcation parameter.
3. Mathematical analysis.In the section we study the dynamics of model ( 1) which imply various outcomes of malaria parasite infection within a host.Because of the biological meaning, we consider system (1)  {(H, I, M, E) : H ≥ 0, I ≥ 0, M ≥ 0, E ≥ 0}.We can show that the first orthant R 4 + is positively invariant for flows of (1), i.e., every solution of model (1) with the initial values in R 4 + will always stay there.We first study the existence of equilibria of system (1) in R 4  + .Setting the righthand sides of system (1) to zero, we have the following equations Therefore, the existence of equilibria of system (1) in R 4 + is equivalent to that of nonnegative solutions of equations (2).It can be checked that system (1) always has one equilibrium P 0 = (λ/d 1 , 0, 0, 0) for all parameters values, which represents the state in which there is no malaria infection in the host.Hence, we call P 0 the malaria-free equilibrium.Now we find malaria infection equilibria.There are two cases for these equilibria.One case is that the host lacks immune response as malaria parasites from a blood-feeding female Anopheles mosquito invade and produce infection in a host.Thus, E = 0. We denote this equilibrium by P 1 = (H 1 , I 1 , M 1 , 0).The other case is a positive equilibrium P * = (H * , I * , M * , E * ) which implies that the host has immune response when malaria parasites invade and produce infection in a host.Let Following van den Driessche and Watmough [41] and Xiao and Bossert [43], we can see that R 0 is the basic reproduction number for the malaria infection in a host.From equations (2), we can obtain the following lemma.
Lemma 3.1.System (1) has a unique equilibrium which is the malaria-free equilibrium More precisely, (i) system (1) has only two equilibria: P 0 = (λ/d 1 , 0, 0, 0) and R0 , M 1 = d1 α (R 0 −1); (ii) system (1) has three equilibria: P 0 = (λ/d 1 , 0, 0, 0), P 1 = (H 1 , I 1 , M 1 , 0) and , where Proof.The existence of the equilibrium P 0 or P 1 can be obtained directly from (2) by setting E = 0. Thereby, we only need to seek conditions for the existence of the positive equilibrium P * = (H * , I * , M * , E * ) of system (1).Suppose that (H * , I * , M * , E * ) is a positive solution of (2).Then from the last equation of (2) we have Hence, it is necessary for the existence of the positive equilibrium P * that d 2 < k1 β + k2 γ , and On the other hand, from the second and the third equations of (2), we respectively have * by the first equation of (2).Hence, This gives the other necessary condition for the existence of the positive equilibrium P * which is R 0 > 1.Therefore, if system (1) has a positive malaria infection equilibrium P * = (H * , I * , M * , E * ), then R 0 > 1 and d 2 < k1 β + k2 γ .In the following we discuss the sufficient conditions on the existence of the positive equilibrium P * .From (2), we can obtain ( It is clear that H * > 0, I * > 0, and E * > 0 if the following conditions hold: From the last two inequalities of ( 6) and the expression of I * in (5), we obtain that This is equivalent to the following inequalities where Note that F (M * ) = 0 has a negative root and a positive root L, Note that G(M * ) = 0 also has a negative root and a positive root d1 α (R 0 − 1).Therefore, when 0 < M * < min{L, d1 α (R 0 − 1)}, we have F (M * ) < 0 and G(M * ) < 0.
We now discuss the conditions that M * should satisfy.Substituting (5) into the second equation of (2), after some calculations we obtain the equation where β and equation ( 8) has a positive solution M * with 0 < M * < min{L, d1 α (R 0 − 1)}, then (2) has a positive solution (H * , I * , M * , E * ), which implies statement (ii).We complete the proof.
We now start to study the stability of these equilibria of system (1).We compute the Jacobian matrix of system (1) at point P = (H, I, M, E), denoted by J(P ).Then Local and global stability of the malaria-free equilibrium P 0 .At the malaria-free equilibrium P 0 = (λ/d 1 , 0, 0, 0), we have the Jacobian matrix and its characteristic equation is From ( 9), it can be seen that all eigenvalues are negative if R 0 < 1 and one of the eigenvalues is positive if R 0 > 1.Therefore, we have the following lemma.
According to Lemma 3.1, we know that system (1) has a unique equilibrium P 0 if R 0 ≤ 1.We will show that the malaria-free equilibrium is globally stable in R 4 Theorem 3.3.The malaria-free equilibrium P 0 = (λ/d 1 , 0, 0, 0) is globally asymptotically stable in R 4  + if R 0 ≤ 1.
Proof.We first note the fact that dH dt < 0 in R 4 + if H(t) ≥ λ d1 , and system (1) has a unique equilibrium P 0 in R 4 + since R 0 ≤ 1.Thus, we only need to consider the stability of P 0 in the region Choosing the Liapunov function V = rI + δM in the region D, we calculate the derivative of V along the solutions of system (1) as follows: Also equation dV dt | (1) = 0 has a unique solution P 0 of system (1) in D. By LaSalle's Invariance Principle we know that the malaria-free equilibrium P 0 is globally asymp- This result indicates that malaria infection cannot be established within a host if R 0 ≤ 1 (see Figure 1).

3.2.
Local stability of the malaria infection equilibrium P 1 (without specific immune response).If R 0 > 1, then system (1) has a malaria infection equilibrium P 1 = (H 1 , I 1 , M 1 , 0) with The Jacobian matrix at P 1 is .

3.3.
Local stability of the positive equilibrium P * (with specific immune response).From Lemma 3.1, we know that the coordinates of the two malaria infection equilibria P 1 = (H 1 , I 1 , M 1 , 0) and P * = (H * , I * , M * , E * ) have the following relationships: These imply that Thus, if the positive equilibrium P * exists, then P 0 and P 1 are always unstable.Now we study the stability of the positive equilibrium P * .The local stability of P * is established from the Jacobian matrix at P * given by where This result indicates that when the host is infected by malaria parasites, a persistent malaria infection with specific immune response can be established.Oscillations in the quantities of H, I, M and E in the host can be observed.We would like to mention that though the positive equilibrium P * (H * , I * , M * , E * ) is not given explicitly in terms of parameters due to the complexity of the model, we have given some sufficient conditions symbolically for the stability of P * and the existence of Hopf bifurcation in Theorems 3.5 and 3.6, respectively.In next section, numerical simulations will show validity of these theoretical results, that is, the positive equilibrium of system ( 1) is stable for some parameter values, and it will become unstable and a family of periodic solutions will bifurcate from the positive equilibrium via Hopf bifurcation when k 1 passes through a critical value (see Figures 3  and 4).
Remark 1.Notice that in the bifurcation analysis we selected k 1 , the proliferation rate of the immune cells induced by infected red blood cells, as the bifurcation parameter.Similarly, we may choose k 2 , the proliferation rate of immune cells due to the interactions between the immune response and merozoites, as the bifurcation parameter and obtain analogous results under certain conditions (such as R 0 > 1 and k2 γ < d 2 < k1 β )(see Figures 5 and 6).These agree with the numerical simulations by Anderson et al. [4] and Hetzel and Anderson [16] that periodic oscillations occur in the model with killing of infected red blood cells or with immune response directly against merozoites and infected red blood cells.When R 0 = 0.0025 < 1, the disease-free equilibrium P 0 = (5 × 10 6 , 0, 0, 0) is globally asymptotically stable.Here the parameter values are given in Table 2 and H(0) = 5 × 10 5 .4. Numerical simulations and discussion.In this section we provide some numerical simulations to illustrate the dynamics of model (1).First, with parameter values giving in Table 2, we can verify that R 0 = 0.0025 < 1.Thus, Theorem 3.3 implies that the malaria-free equilibrium P 0 = (5 × 10 6 , 0, 0, 0) is globally stable (see Figure 1).
The conditions for the existence of Hopf bifurcation can be stated as follows: R 0 = rαλ d1µδ > 1, k2 γ < d 2 < k1 β , there exists a k * 1 > 0 such that (13) hold and ψ (k * 1 ) = 0. Rewrite the first condition as r µ λ d1 α δ > 1. Recall the biological meaning of these parameters, we know that λ d1 is the initial density of red blood cells (RBCs), r µ represents the successful invasion of the malaria parasites during their life time, and α δ describes the successful infection of RBCs in that process.Thus, R 0 > 1 means that, before encountering immune response, with given initial density of RBCs, when there are enough numbers of malaria parasites that cause successful infection of RBCs, then the host is infected with malaria.The second condition indicates that the decay rate d 2 is somehow balanced between the proliferations induced by the malaria parasites k2 γ and infected RBCs k1 β .The remaining conditions are mainly on the proliferation rate of the immune cells induced by infected red blood cells k 1 , which roughly means that there is a critical value of k 1 , once it is reached periodic oscillations in all components will occur.This demonstrates that synchronicity is an inherent feature of malaria infection with immune response.These are helpful for us to better understand how immune response defends against the malaria parasite infection and possibly causes the periodic fevers in the host.
We would like to make some remarks about our model (1).The model is a generalization of the basic model of Anderson et al. [4] and Anderson [3] with a nonlinear bounded Michaelis-Menten-Monod function describing the interaction between healthy red blood cells, infected red blood cells, malaria parasitemia and immune effectors.Since the replication rate of merozoites is described by the number of merozoites produced by each infected red blood cell times the rate at which the infected red blood cells burst due to infection, Soul [36] pointed out the possible unrealistic large growth of parasites in the absence of immunity by the model in Anderson et al. [4] considering the parasite growth cycle (which is 48 hours for P. falciparum).To address this problem, Gravenor and Lloyd [13] (see also Gravenor et al. [14,15]) proposed to estimate the dynamics of malaria parasites by using multiple stages for the infected red blood cells.The overall parasite life-span is now described by a sum of n exponential distributions and the modified multiple stage model is a system of n + 2 ordinary differential equations.Interestingly, Gravenor and Lloyd [13] found that the basic model of Anderson et al. [4] leads to equilibrium solutions that are identical to those obtained from the multiple stage model.Adda et al. [1] and Iggidr et al. [18] preformed global stability analysis of the multiple stage model and showed the existence and global stability of a unique endemic equilibrium which rules out the existence of possible oscillations via bifurcations.Our model predicts periodic oscillations in all components that are induced by Hopf bifurcation at the positive equilibrium by using the proliferation rate of the immune cells induced by infected red blood cells as the bifurcation parameter.Notice that immune response was not included in the multiple stage model of Gravenor and Lloyd [13].It will be interesting to see if the immune system can also induce Hopf bifurcation in their model.Another option is, as did in Hoshen et al. [17], Dong and Cui [11], and Mitchell and Carr [28], to introduce a time delay into the basic model of Anderson et al. [4] to describe the parasite growth cycle which will produce the observed periodic oscillations in host-parasite dynamics.
Another remark we would like to make is that in model ( 1) we used only one component E(t) to represent the total capacity of the immune response of the host to infected cells by parasites for the sake of simplicity and analysis.Immune responses against malaria infections are complex and stage-specific.The malaria parasite induces a specific immune response which can stimulate the release of cytokines and activate the host's monocytes, neutrophils, T-cells, and natural killer cells to react to the different stage parasites (Malaguarnera and Musumeci [23]).It would be more reasonable to model cellular and humoral immune responses separately by including various innate, antibody and T-cell responses to malaria parasites in modeling the within-host dynamics (see McQueen and McKenzie [27] and Chiyaka et al. [7]).However, that would increase the number of equations in the model and make the analysis much more difficult if it is not impossible.Our work focuses on studying the nonlinear dynamics of a basic simple model including the essential parameters of within-host malaria by a single compartment of parasites.This study provides an example of how basic mathematical frameworks may be used to explore the mechanisms of complex parasite dynamics within their hosts.As pointed out by Augustine et al. [6], many coinfections that have profound effects on the immune system, such as infection with human immunodeficiency virus (HIV) and Mycobacterium tuberculosis (TB), are common in people living malaria endemic regions.It will be interesting to study the effect of immune response to the coinfection of malaria and HIV (Xiao and Bossert [43]) or TB.We leave this for future consideration.