The Dynamics of Naturally Acquired Immunity to Plasmodium falciparum Infection

Severe malaria occurs predominantly in young children and immunity to clinical disease is associated with cumulative exposure in holoendemic settings. The relative contribution of immunity against various stages of the parasite life cycle that results in controlling infection and limiting disease is not well understood. Here we analyse the dynamics of Plasmodium falciparum malaria infection after treatment in a cohort of 197 healthy study participants of different ages in order to model naturally acquired immunity. We find that both delayed time-to-infection and reductions in asymptomatic parasitaemias in older age groups can be explained by immunity that reduces the growth of blood stage as opposed to liver stage parasites. We found that this mechanism would require at least two components – a rapidly acting strain-specific component, as well as a slowly acquired cross-reactive or general immunity to all strains. Analysis and modelling of malaria infection dynamics and naturally acquired immunity with age provides important insights into what mechanisms of immune control may be harnessed by malaria vaccine strategists.

these parameters.
We assumed that the parasite multiplication rate (PMR) has a normal distribution within each age group i with the mean i and the standard deviation i = p i (i = 1,…,4) proportional to the mean, where p is a positive number and is the same in all groups. A PMR <1 implies that for each currently infected RBC, less than one newly infected RBC will be produced in the next round of infection -so the parasite will not grow. The following mathematical model describes the exponential growth of parasites during the early blood stage.

Let
denote the concentration of parasites at a time t after emerging from the liver. It is easy to see that where r is the average PMR, and constant A is the concentration of parasites in blood at the beginning of the blood stage .
Let us denote the probability density function (PDF) and cumulative density function (CDF) of the normal distribution by f N (. ) and F N ( .) respectively. Knowing the PDF of the PMR and the relation between PMR and the delay to detection, we can find g( t )-the PDF of the delay, for the fraction of population of 1 -F N ( x ) which has the PMR >1. To do this we used the formula for the distribution of function of random variable. It requires the inverse function to the delay function of r. The inverse function

() Ct
Let us introduce a time constant as the earliest possible moment of blood stage infection in our model. It means that all blood stage infections that had been present in the blood before the -th day were killed by the anti-malaria drug.
Because infective mosquito bites occur continuously and independently of each other, we are to expect that the waiting time until an infective bite and therefore the time until successful initiation of blood stage will have an exponential distribution, and we can assume that the average number of successful initiations of blood stage infections per day is equal to k. Let us denote the CDF of an exponential distribution with parameter k by F E ( . ). Now we can find the infection function S(t) , which includes the convolution of the CDF of the exponential distribution with variable initial plateau (delay to detection) and the distribution of the delays.
Parameters of the model are in the square brackets.  The values of all constants in the model were taken from previously published papers.
To estimate the initial concentration of parasites in i-th age group A i ,we need to know the number of initially infected RBC and the average blood volume in the age group. The average blood volumes in age groups V i , i=1,...,4, were found from the Chart 1 in reference [1] that gave us the values V 1 =1.1x10 6 l, V 2 =2 x10 6 l, V 3 =3.3 x10 6 l, V 4 =5 x10 6 l. The initial number of parasitized RBC for a single bite was estimated as 5.6x10 4 knowing that after 5 simultaneous bites the initial number of infected RBC was 28*10 4 [2]. The first possible moment of blood stage infection was taken as equal to 7 days, since the blood concentration of lumefantrin even by day 7 post treatment is high enough (more than 280 ng/ l) to kill a relatively small number of parasites released from liver (reported in [3,4]). The detection threshold T was taken as equal to 40 parasites per microlitre, since this was the minimal concentration in the analysed data table which completely conforms to the range of 20-50 parasites/microlitre reported in [2,5]. However, the variation of the above-mentioned constants does not significantly affect the optimal values of PMR. This is easy to see from the relation 2/ / t r TA where t is the time between the beginning of the blood stage and the detection of parasitaemia. We observe that t is more than 6 days. Consequently, the exponent 2/t<1/3 and the variation of the constant T/A will not significantly affect the PMR.
The best fit parameters are in the [Text S.4].