## Abstract

What determines the dynamics of parasite and anaemia during acute primary malaria infections? Why do some strains of malaria reach higher densities and cause greater anaemia than others? The conventional view is that the fastest replicating parasites reach the highest densities and cause the greatest loss of red blood cells (RBCs). Other current hypotheses suggest that the maximum parasite density is achieved by strains that either elicit the weakest immune responses or infect the youngest RBCs (reticulocytes). Yet another hypothesis is a simple resource limitation model where the peak parasite density and the maximum anaemia (percentage loss of RBCs) during the acute phase of infection equal the fraction of RBCs that the malaria parasite can infect. We discriminate between these hypotheses by developing a mathematical model of acute malaria infections and confronting it with experimental data from the rodent malaria parasite *Plasmodium chabaudi*. We show that the resource limitation model can explain the initial dynamics of infection of mice with different strains of this parasite. We further test the model by showing that without modification it closely reproduces the dynamics of competing strains in mixed infections of mice with these strains of *P. chabaudi*. Our results suggest that a simple resource limitation is capable of capturing the basic features of the dynamics of both parasite and RBC loss during acute malaria infections of mice with *P. chabaudi*, suggesting that it might be worth exploring if similar results might hold for other acute malaria infections, including those of humans.

## 1. Introduction

Many aspects of malaria, and particularly the interactions between the parasite and its host which result in the disease, are still poorly understood (Miller *et al*. 2002). A malaria infection starts with the bite of an infected mosquito, which introduces a few sporozoite stages that migrate to and infect liver cells. Asexual replication in the liver cell results in the release of thousands of merozoites that initiate the blood stage of the infection. This stage is responsible for the pathology associated with malaria (anaemia, fever, cerebral malaria and coma). Merozoites infect red blood cells (RBCs) where they multiply to produce 8–32 new merozoites (depending on the malaria species), which are released by lysis of the RBCs. The released merozoites can infect new RBCs, causing a rapid increase in parasites and the infected cells. In this paper, we focus on the initial dynamics of infection of the merozoite stage of infection of naive hosts (individuals who have never had a previous malaria infection). This corresponds to the first week or the second following infection when the parasite density reaches a peak and the anaemia (loss of RBCs) may be severe.

During this initial or acute phase of infection, different parasite strains reach different densities and cause varying degrees of anaemia and other pathology (Field & Niven 1937; Field 1949; Kitchen 1949*a*; Molineaux *et al*. 2001; Mackinnon & Read 2004). While most studies agree that specific immunity does not play a major role in this initial dynamics, there is considerable controversy over which factors drive the dynamics shortly after infection (Anderson *et al*. 1989; Hellriegel 1992; Hetzel & Anderson 1996; McQueen & McKenzie 2004). Some of the hypotheses to explain the differences in the initial dynamics of malaria strains include (i) the conventional view—in both the virulence evolution and the malaria literature—that the fastest replicating parasites reach the highest densities and cause most anaemia (Antia *et al*. 1994; Mason *et al*. 1999; Chotivanich *et al*. 2000; Gandon *et al*. 2001), (ii) the hypothesis that strains infecting reticulocytes (the youngest RBCs) cause the severest anaemia (McQueen & McKenzie 2004, 2006), and (iii) the hypothesis that innate or early specific immune responses regulate the initial dynamics of infection and anaemia (Jakeman *et al*. 1999; Haydon *et al*. 2003; Dietz *et al*. 2006).

There are two reasons why it is difficult to determine the contribution of these different factors to the dynamics of acute infection. The first problem is that there is relatively limited data on the dynamics of the parasite and the loss of RBCs following infection of humans with human malaria parasites such as *Plasmodium falciparum* and *Plasmodium vivax*. One approach to overcome this limitation is to use data from well-characterized model systems such as infections of mice with species such as *Plasmodium chabaudi* or *Plasmodium yoelii*. The second problem is that the dynamics of infection may involve many interacting populations which fall into three groups: (i) the parasite and its resource, i.e. merozoites, uninfected and infected RBCs, (ii) the innate immune response, i.e. macrophages, dendritic cells, cytokines, etc., and (iii) the adaptive immune response, i.e. B and T cells and antibodies. One approach to this problem is to start with the simplest possible mathematical model of the dynamics of infection (i.e. one that consists of simply the parasite and the uninfected and infected RBCs) and by confronting the model with the experimental data to determine which features of the infection the model can explain. Importantly, identifying where and how the model fails can help us determine how the model needs to be refined in future studies. This approach is similar to that used to dissect the early dynamics of infections such as HIV and influenza (Phillips 1996; Perelson 2002; Regoes *et al*. 2004; Baccam *et al*. 2006).

In this study, we bring mathematical models of the early stages of malaria infection into contact with detailed experimental data on the time-course of both the parasite density and the loss of RBCs following infection. We used data from infections of C57BL/6 mice with two strains of *P. chabaudi* (De Roode *et al*. 2004, 2005*a*,*b*) as shown in figure 2. The two parasite strains reach different maximum densities and cause different levels of anaemia—with strain AJ reaching higher densities and causing greater anaemia than strain AS. An examination of these data suggests immediately that the differences in the dynamics of infection cannot be accounted for by differences in the growth rates (defined as the initial exponential rate of growth) as the growth rates of the two strains are not statistically different (*p*>0.05). The next step is to develop a simple ‘resource-limitation’ model to describe the dynamics of infection of mice with the AS and AJ parasite strains and to identify the properties of these strains which account for the difference in their dynamics. We then test our conclusions by determining whether our model with parameters from the single-strain infections can describe the dynamics of co-infection of mice with both the strains.

## 2. Results

We use an age-structured model of the dynamics of parasites and RBCs (figure 1 and the appendix). We combine the age-structured model for RBCs described by Mackey (1996) with the model of the infection process along the lines of McQueen & McKenzie (2004). A key component in the model is that merozoites may preferentially infect RBCs of different ages as described by an age-dependent force of infection *β*(*τ*). The infected RBCs lyse after *t*_{m} days to produce *m* merozoites/infected RBC.

We have independent estimates of all of the parameters of the model except *β*(*τ*) from the literature (table 1). The terms and parameters for RBC homeostasis were obtained from Mary *et al*. (1980) and Mackey (1996), and for the life cycle of infected RBCs from Cox (1988) and de Roode (2005).

We briefly outline how we infer the age preference of merozoites for RBCs, *β*(*τ*), from the data on dynamics of single infections; the details are given in the appendix. As a first approximation, we take *β*(*τ*) to be constant within some range and zero outside it, as shown in figure 1. We thus need to estimate three parameters, namely *β*_{max} and the youngest and oldest ages of RBCs that are susceptible. Given the maturation time of infected RBCs, *t*_{m}, and the number of merozoites produced per RBC, *m*, for the model to explain the rapid exponential growth, the infection process must be highly efficient (i.e. *β*_{max} must be sufficiently large that at the beginning of the infection almost all free merozoites infect susceptible RBCs before they are cleared). This provides a lower bound for *β*_{max} (and we show that our results are not very sensitive to its precise value provided it is above this bound). The next step is to estimate the age range of susceptible RBCs. Given that *β*_{max} is high, almost all susceptible RBCs are infected and killed, and the maximum level of anaemia equals the proportion of the RBC population that is susceptible to infection, *σ*. We estimated the fraction of RBCs that can be infected by the AJ and AS strains, *σ*_{AJ}≃0.8 and *σ*_{AS}≃0.5. The beginning of the age range can be estimated as follows: the time between maximum anaemia (approx. day 9) and the beginning of the second peak of parasitaemia (approx. day 24 for AS and approx. day 20 for AJ from (de Roode *et al*. 2005*a*) figure 3 single infections) is the time required for new target cells to appear. This is roughly the sum of the time lag associated with increased RBC production (2.5 days) and the time taken for these newly produced RBCs to enter the susceptible age range (*τ*_{min}). We thus obtain the age ranges for AJ as being 0.2–1.0 and AS being 0.4–0.9. These figures are in units of the average lifetime of RBCs, which is 40 days in mice. In other words, AJ is able to infect a wider range of RBCs than AS, and the age preference of AS is contained entirely within that of AJ, i.e. AJ can infect both younger and older cells than AS.

In figure 2, we show that the model with these parameter estimates captures the key features of the experimental data of the dynamics of single-strain infections over the first 10–12 days of infection; the two strains have indistinguishable growth rates, but reach different peak levels and induce different degrees of anaemia. The simulations also reproduce the observed delays between the peak in parasite density and the time of maximum anaemia. Following the end of the second week of the infection, the results of the simulations depart from the data. These observations can be explained using knowledge of the underlying biology and simple analytical calculations.

During the initial growth phase, the number of infected cells *Y*(*t*) increases exponentially as . This is simply understood as follows: a single infected cell produces *m* merozoites after a time *t*_{m} and, at the onset of infection, target cells are in excess and essentially all merozoites produced go on to productively infect RBCs. This explains why the initial growth rate depends only on these two parameters, which are similar for AS and AJ, and not on the infectivity profile *β*(*τ*), which differs for AS and AJ.

The model not only accurately describes the initial growth rate of the parasite but also the peak parasite density and the minimum RBC density, suggesting that RBC limitation is sufficient to explain the key features of the initial dynamics, and that differences in the age range of RBCs that are susceptible to infection can account for the differences in the dynamics of the two clones. In the appendix we show that the peak parasite density and the maximum anaemia are linearly dependent on the width, *σ*, of the age range of RBCs that are susceptible to infection.

We observe that the model begins to fail subsequent to day 12 after infection. Following this time, the parasite density drops to a lower level than predicted by the simulations. This suggests that immunity plays an important role in controlling the parasite after this time. This is consistent with the experimental observations of the presence of parasite-specific antibodies beginning at approximately two weeks after infection (Mota *et al*. 1998).

The next step is to test or validate our model. We do so by asking whether without any modification it can reproduce the dynamics of competing strains in mixed infections. In other words, we used the parameters we have obtained for AS and AJ and change only the initial conditions to simulate the introduction of the two strains at the appropriate times. Figure 3 shows that the model agrees closely with all the mixed infection data. In particular, in simultaneous infections of AS and AJ parasites, the density of both initial peaks is somewhat lower than that in single-strain infections and AJ subsequently outcompetes AS. When AS is introduced 3 days before AJ, the initial peak of AS is higher than that of AJ, but AJ invades and takes over and AS is not seen subsequently. The results also show that the duration of anaemia is substantially extended. Finally, following the introduction of AJ 3 days prior to AS, the initial peak of AJ is not substantially altered by AS, and AS has only a small primary peak several logs lower than that during single-strain infection and it is subsequently eliminated by AJ. The anaemia is similar to that of the infection with AJ alone. As in the case for infections with a single parasite strain, the onset of humoral immunity at day 12 results in the simulations departing from the data after this time.

## 3. Discussion

Our results suggest that the dynamics of parasites and the loss of RBCs during the initial acute stages of *P. chabaudi* infections can be understood in terms of a simple resource limitation model with the RBCs as the resource for the parasite. The differences in the dynamics of the parasite and the loss of RBCs following infection with different (AS and AJ) strains of *P. chabaudi* can be explained by the differences in the age preferences of the parasite strains.

We developed this model based on the data for the densities of parasites and RBCs following infection of mice with a single parasite strain. We tested this model by showing that it could describe the dynamics of competition following co-infections with both strains under a number of different conditions. For simultaneous co-infection, we found that since the two strains' age preferences overlap, in the first peak of infected RBCs the density of each parasite is lower than that in a single-strain infection. As new RBCs are produced, AS is outcompeted as its potential target RBCs are infected by AJ before AS parasites can infect them. In mice infected with AS first, anaemia is extended as there are effectively two waves of depletion (AS induces a 50% anaemia, which is prolonged by the depletion induced by AJ peaking 3 days later and infecting a wider range of RBCs). In mice infected with AJ first, AS target cells are substantially depleted by the time AS is introduced and infection dynamics are similar to those of AJ alone. The model thus provides an explanation for the observed correlation between virulence and competitive ability in mixed infections with the AS and AJ strains (de Roode *et al*. 2005*a*; Bell *et al*. 2006). In ecological terms, the extent of competition depends on the degree to which the different populations share resources, and strains with a larger resource (RBC age range) are less affected by a resource loss due to competing strains.

Thus far, we have considered the dynamics of the AS and AJ strains, which have the same growth rate. In the appendix, we use our model to explore how the initial growth rate of strains affects their dynamics in single and mixed infections. For the case of single-strain infections, we find that our principal result, namely that the age preference determines the maximum parasite density and maximum loss of RBCs, holds independent of the growth rate of the strain. For the case of multiple-strain infections, we find that the extent of competition is dependent on the overlap in the age preference of the two strains for RBCs and that strains with a higher growth rate and a wider age preference for RBCs have a competitive advantage.

We now consider the previous hypotheses for the dynamics of the parasite and the loss of RBCs during malaria infections. We have shown that the maximum parasite density and anaemia are insensitive to the initial growth rate (which affects the time of the peak but not its magnitude), rejecting the hypothesis that virulence is proportional to the initial multiplication rate (Antia *et al*. 1994; Mason *et al*. 1999; Chotivanich *et al*. 2000; Gandon *et al*. 2001). We also describe the dynamics of RBCs entirely as a consequence of loss following infection by merozoites. This suggests that the role of bystander destruction of uninfected RBCs (Jakeman *et al*. 1999; Haydon *et al*. 2003) may need to be reconsidered, at least for the initial dynamics of *P. chabaudi* infections of mice.

Many models have indicated the importance of RBC limitation (Anderson *et al*. 1989; Hellriegel 1992; Gravenor *et al*. 1995; Hetzel & Anderson 1996; Mason *et al*. 1999; Haydon *et al*. 2003) but are not able to explain the observation that strains with identical growth rates generate different parasite densities and hence different degrees of anaemia. The need to consider the RBC age preference to explain infection dynamics has been emphasized in the modelling studies of McQueen & McKenzie (2004, 2006). However, they suggested that following infection with a single parasite species, the extent of anaemia depends on the youngest age of RBCs that the strain infects. This is because in the absence of immunity, the killing of young cells induces greater anaemia, owing to the downstream impact on the RBC age profile. However, our study suggests that this does not happen in the *P. chabaudi* infections modelled here because immunity must be considered shortly after the peak in parasite density and this limits the subsequent aggravation of anaemia predicted by these models.

Our model is more parsimonious than others that invoke innate or specific immunity (Dietz *et al*. 2006) as it does not involve an additional term for innate immunity. More work will need to be done in order to determine the relative contributions of RBC limitation and innate immunity. At present, there are a number of difficulties with modelling the innate immunity, including the lack of quantitative measurements of the magnitude of innate immunity and uncertainty in the term for the killing of merozoites and infected cells by innate immunity.

Finally, we note that the observation that the model begins to fail after 12 days post-infection suggests a role for specific immune responses arising at this time. This is the time at which antibody responses develop in mice (Mota *et al*. 1998). Models considering the dynamics following this time will need to explicitly consider the effects of the adaptive immune responses of the host and antigenic variation of the parasite (Frank 1999; Recker *et al*. 2004). This is, however, beyond the scope of the current paper, which focuses on the initial or acute stage following infection.

Because our model was based on data from rodent malaria, its key predictions can be subject to further experimental tests. The most important prediction concerns the age ranges of RBCs that different malaria strains can infect. These could be determined by transferring labelled RBCs of different ages into singly infected mice and determining their loss following infection. An indirect approach would be to compare competition between malaria strains in untreated mice, and mice treated with erythropoietin (the hormone that stimulates erythropoiesis, the production of RBCs) just prior to infection (Suzuki *et al*. 2006). Specifically, we predict that such a treatment would result in an increase in the relative density of strains that can infect younger RBCs.

Several experimental and theoretical studies have suggested a role for RBC age preference in the dynamics of human malaria infections (Kitchen 1949*b*; Hutagalung *et al*. 1999; Simpson *et al*. 1999; Chotivanich *et al*. 2000; Miller *et al*. 2002; Duraisingh *et al*. 2003; McQueen & McKenzie 2004, 2006) as well as in the experimental *Plasmodium berghei* infections of mice (Cromer *et al*. 2006). For example, *P. falciparum* isolates able to invade a larger proportion of RBCs maintained higher parasitaemias and caused more disease (Simpson *et al*. 1999; Chotivanich *et al*. 2000); similarly, an aberrant red cell phenotype in haemoglobin E carriers resulted in a reduced number of suitable cells for parasite invasion (Chotivanich *et al*. 2002), and was associated with a lower incidence of severe malaria (Hutagalung *et al*. 1999). It has also been suggested that the difference in virulence between *P. vivax* (mild) and *P. falciparum* (virulent) in humans could be due to the latter's ability to invade a wider range of erythrocytes than the former (Kitchen 1949*b*; Simpson *et al*. 1999; Miller *et al*. 2002). There are a number of difficulties associated with linking our model with these observations regarding the dynamics of anaemia in humans. Our results describe the initial stages of untreated acute infections, and many, if not most, of the experimental studies mentioned above consider the dynamics of the parasite and the virulence it causes during the persistent phase of infection or possibly following reinfection of semi-immune hosts. In these circumstances, immunity and antigenic variation play a major role in determining the dynamics of the parasite (Phillips *et al*. 1997; Bruce *et al*. 2000; Bruce & Day 2002; Recker *et al*. 2004) and anaemia is likely to be multifactorial (Menendez *et al*. 2000).

In conclusion, we note that much remains to be done to extend the model of parasite and RBCs to incorporate factors such as immunity and antigenic variation. A key problem will be bringing the resulting models into close contact with experimental data in a way which allows the models to be tested. While this may be essential for understanding the complex dynamics of chronic infections and the pathology they cause, our results suggest that many features of acute malaria infections (at least in the *P. chabaudi* model system) can be explained by RBC age preference.

## Acknowledgments

We thank S. Altizer, M. Choisy, J. Davies, A. Graham, G. Long, A. Read, A. Pedersen, P. Rohani and H. Wearing for their helpful comments and discussion. R.A. and A.Y. acknowledge support from the NIH, and J.d.R. acknowledges support from a Netherlands organization for scientific research (NWO) TALENT fellowship and a European Commission Marie Curie Outgoing International fellowship.

## Footnotes

- Received February 11, 2008.
- Accepted March 3, 2008.

- © 2008 The Royal Society