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International Journal of Mathematics and Mathematical Sciences
Volume 2018, Article ID 7019868, 18 pages
https://doi.org/10.1155/2018/7019868
Research Article

Mathematical Model for Hepatocytic-Erythrocytic Dynamics of Malaria

Institute of Mathematical Sciences, Strathmore University, P.O. Box 59857-00200, Nairobi, Kenya

Correspondence should be addressed to Titus Okello Orwa; ude.eromhtarts@awrot

Received 23 October 2017; Accepted 20 May 2018; Published 2 July 2018

Academic Editor: Niansheng Tang

Copyright © 2018 Titus Okello Orwa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Human malaria remains a major killer disease worldwide, with nearly half (3.2 billion) of the world’s population at risk of malaria infection. The infectious protozoan disease is endemic in tropical and subtropical regions, with an estimated 212 million new cases and 429,000 malaria-related deaths in 2015. An in-host mathematical model of Plasmodium falciparum malaria that describes the dynamics and interactions of malaria parasites with the host’s liver cells (hepatocytic stage), the red blood cells (erythrocytic stage), and macrophages is reformulated. By a theoretical analysis, an in-host basic reproduction number is derived. The disease-free equilibrium is shown to be locally and globally asymptotically stable. Sensitivity analysis reveals that the erythrocyte invasion rate , the average number of merozoites released per bursting infected erythrocyte , and the proportion of merozoites that cause secondary invasions at the blood phase are the most influential parameters in determining the malaria infection outcomes. Numerical results show that macrophages have a considerable impact in clearing infected red blood cells through phagocytosis. Moreover, the density of infected erythrocytes and hence the severity of malaria are shown to increase with increasing density of merozoites in the blood. Concurrent use of antimalarial drugs and a potential erythrocyte invasion-avoidance vaccine would minimize the density of infected erythrocytes and hence malaria disease severity.

1. Introduction

Human malaria remains a major killer disease worldwide, with nearly half (3.2 billion) of the world’s population at risk of malaria infection [1]. The infectious disease is endemic in tropical and subtropical regions, with an estimated 212 million new cases (uncertainty range: 148–304 million) and 429,000 malaria-related deaths (range: 235,000–639,000) in 2015 [2]. 92% of the deaths and 90% of the cases occurred in sub-Saharan Africa. 70% of the reported deaths occurred among children below the age of five. Despite existing vector control measures and tremendous progress in the development of antimalarial therapy accompanied with worldwide decline in incidence rate (fell by 21% in 2015) and mortality rate (fell by 29% in 2015), malaria remains one of the greatest global health challenges to date [2].

The protozoan disease is caused by parasites of the genus Plasmodium which are transmitted to humans by the bite of female Anopheles mosquito. Plasmodium falciparum, which is predominant in sub-Saharan Africa, New Guinea, and Haiti [3], is the major cause of malaria infections. The other Plasmodium species that cause malaria are P. vivax, P. ovale, P. malariae, and P. knowlesi [4]. P. vivax and P. ovale can hide in the liver for prolonged periods as hypnozoites, causing relapsing malaria months or even years after the initial infection [5]. P. vivax has the greatest geographical range of the disease and hence is the main contributor to worldwide malaria morbidity [3]. Our study focuses on the dynamics of Plasmodium falciparum in the human host.

During their obligatory blood meals, infected female Anopheles mosquitoes inject sporozoites belonging to Plasmodium falciparum species into the human dermis [6]. The motile sporozoites travel through the blood vessels and enter the host’s liver. Hepatocyte invasion is accompanied by the formation of parasitophorous vacuole (PV) around the sporozoite [7]. They form preerythrocytic schizonts and multiply by schizogony, culminating in the production of 8–24 first generation merozoites that are released into the blood when the liver schizonts burst open [8]. The released merozoites invade susceptible erythrocytes and undergo another phase of schizogony, which is relatively faster compared to that at the exoerythrocytic stage [9].

Within a period of two days, the infected red blood cells rupture to release about 16 daughter merozoites [10]. Most of the released merozoites quickly invade susceptible erythrocytes, leading to another cycle of infections. The waves of bursting erythrocytes and the invasion of fresh erythrocytes by the newly released merozoites increase parasitemia and produce malaria’s characteristic symptoms [11]. In the absence of adequate protective immune response or antimalarial therapy, the host is likely to suffer severe anaemia or even die [12]. The rest of the daughter merozoites develop into sexual forms called gametocytes [10]. These gametocytes are later taken up by other female Anopheles mosquitoes during feeding [13]. This marks the beginning of the sporogenic cycle that occurs within the mosquito vector.

The presence of the malaria parasites in the human body elicits response from numerous immune cells. The innate immune system and the adaptive immune system form the first and the second lines of defence, respectively [14]. Adaptive immune system further provides protection against future exposures to malaria pathogens. Innate immune cells such as the Plasmodium falciparum DNA, natural killer cells (NK cells), dendritic cells (DCs), macrophages, natural killer T (NKT) cells, and T cells are involved in the clearance of circulating parasites, infected erythrocytes, and infected hepatocytes [14]. Subject to parasite strain, the DCs and NK cells may prompt or restrain inflammatory responses [15]. The NKT cells also help regulate DCs and T cell responses to Plasmodium [14]. Moreover, studies in [16] have demonstrated that malaria infection induces activation of Toll-like receptors (TLRs): TLR1, TLR2, TLR4 (which are located on the cell surface), and TLR9 which is not expressed on the cell surface. TLR2 and TLR9 are also activated by malarial glycosylphosphatidylinositol (GPI) anchors and parasite-derived DNA bound to hemozoin [16].

Unlike the NK cells, the macrophages have been shown to effectively phagocytose malaria-infected red blood cells during the erythrocytic phase [17]. A part from its ability to wholly ingest infected red blood cells, the macrophages can also selectively extract malaria parasites from recently infected erythrocytes [18]. The parasite-extraction capability of macrophage therefore leaves the surviving erythrocytes to continue circulating like the other healthy red blood cells.

The rest of the paper is organized as follows: in Section 2, we formulate the in-host malaria model and state the invariant region in which the model is defined. In Section 3, we compute and describe the model in-host reproduction number. The results on model equilibrium points (disease-free equilibrium and endemic equilibrium points) and the stability of the disease-free equilibrium point are also considered in Section 3. Section 4 is devoted to numerical solution of the in-host model under different conditions of the threshold parameter (in-host reproduction number). Parameter sensitivity analysis and the effects of parameter variation on different populations are investigated in Section 4. A conclusion and discussion complete the paper in Section 5.

2. In-Host Malaria Model

Several studies on mathematical modelling of in-host malaria and its dynamics within the human host have been done. Nearly all the earlier mathematical models (see, e.g., [2527]) focused on improving Plasmodium falciparum control while focusing on the blood stage of parasite development. These models have been found to be useful in explaining in-host observations by means of biologically plausible assumptions such as parasite diversity, predicting the impact of interventions or the use of antimalarials [28], and estimating hidden parameter values [29]. Although the models in [19, 21, 23, 30] have considered the impact of immune response and treatment, the modelling is only limited to the blood stage of Plasmodium falciparum development. In [20, 22, 31], the liver stage is incorporated in the malaria model. However, the contribution of immune system is ignored in [20, 31]. Moreover, all the immune cells are assumed to play an active role during malaria infection in [22]. This may not be entirely true. The specific impacts of immune responses to malaria infection are well discussed in [3236].

In the following sections, we extended the model in [21] by incorporating the liver stage of parasite development. The reformulated in-host malaria model focuses on the erythrocytic and hepatocytic stages and describes the dynamics of interactions between the malaria parasites, the liver hepatocytes, the red blood cells, and the macrophages (immune system cells). Unlike the work in [20, 22], we ignored the vector stage of parasite development and assumed a twofold process in the generation of hepatocytes: from the bone marrow and from self-replication of the existing hepatocytes. Again, we have assumed that the generation of macrophages and the susceptible red blood cells from the bone marrow increase with increasing density of the infected erythrocytes. However, whatever density of the infected erythrocytes, there is a limit on the rate at which cells can be released from the bone marrow.

2.1. Model Formulation

The hepatocytic-erythrocytic malaria model describes the dynamics of Plasmodium falciparum parasite during the hepatocytic and erythrocytic stages and their interactions with the host’s red blood cells, liver hepatocytes, and the macrophages. The compartmental model assumes seven interacting populations of sporozoites , susceptible hepatocytes , infected hepatocytes , susceptible red blood cells (RBCs) , infected red blood cells (IRBCs) , merozoites , and macrophages at any time . The dynamics of malaria parasites and host-cell populations in each compartment are described as follows.

Sporozoites . The female Anopheles mosquito is assumed to inject sporozoites into the human system during blood meal at a constant rate . The sporozoites molt through the blood stream and reach the liver in about 2 hours, where they invade the hepatocytes at the rate . We assume that the sporozoites can die naturally at a rate .

Susceptible Hepatocytes . We consider the bone marrow and self-replication as the main sources of the liver hepatocytes. The recruitment of hepatocytes from the bone marrow is assumed to occur at a constant rate . Just like during liver transplant [37], we argue that, during severe malaria infections, the rate of generation of healthy hepatocytes is likely to increase tremendously and in proportion to the concentrations of the infected liver cells [38]. This additional increase is represented by the term , where and , respectively, represent the concentration of infected hepatocytes and their rates of generation. The parameter represents the number/concentration of the infected hepatocytes at which the recruitment of the healthy hepatocytes is a half of the maximum rate. Owing to invasion by sporozoites at the rate , susceptible hepatocytes get infected and progress to subpopulation . In addition, hepatocytes in compartment are assumed to have a natural life expectancy and may hence die naturally at the rate .

Infected Hepatocytes . Infected hepatocytes mature into liver-stage schizonts. These schizonts burst open releasing 2000–40000 uninucleate merozoites into the blood stream [39]. The term represents the total population of merozoites released upon bursting of infected hepatocytes. The parameter represents the death rate of the infected hepatocytes.

Susceptible Red Blood Cells . Similar to malaria models in [21, 23, 30], we have assumed that the susceptible RBCs get recruited at a constant rate from the bone marrow. We further assume that, during infection, the erythrocyte production is accelerated owing to the presence of IRBCs at the rate . This increase is denoted by the term , where represents number/concentration of the infected red blood cells at which the recruitment of susceptible red blood cells is a half of the maximum rate. The particular mechanisms involved in this accelerated process are, however, still poorly understood [40]. The susceptible RBCs get infected by merozoites at a rate proportional to the contact rate of their density, . The positive constant describes the rate of successful invasion by a malaria merozoite. The susceptible RBCs die naturally at a rate .

Infected Red Blood Cells . Upon invasion by merozoites, the healthy RBCs get infected, leading to the formation of infected red blood cells . Although the RBCs die at a constant rate , they can similarly be killed through phagocytosis by the macrophages at the rate . At maturity, the IRBCs burst open, releasing free merozoites into the blood system, causing secondary invasion and disease progression.

Merozoites . After 2–15 days, the infected hepatocytes burst open and release merozoites into the blood system. This is represented by the term , where is the average number of merozoites released per bursting infected hepatocytes. An average of merozoites is released per each bursting IRBC. These free parasites suffer a natural death at a rate and invade susceptible RBCs at a rate . Within the red blood cells, the merozoites mature either into uninucleate gametocyte or into erythrocytic stage schizont containing 10–36 merozoites [39]. After about 48–72 hours, the erythrocytic stage schizont ruptures, releasing more merozoites into blood stream to cause further invasion of healthy RBCs. We assume that a proportion of the merozoites contribute to secondary invasion of the susceptible RBCs. The rest of the merozoites transform into gametocytes that are later picked up by female Anopheles mosquitoes during feeding.

Macrophages . Owing to their effectiveness in elimination of infected erythrocytes and infective malaria parasites, we have considered the innate macrophage cells as the main part of the immune response in malaria infection. Consequently, we have assumed that the macrophage cells are recruited at a constant rate from the bone marrow. Moreover, they proliferate at a rate in the sites of infection in proportion to the density of IRBCs. This is represented by the term , where denotes the number/concentration of the infected red blood cells at which the recruitment of the macrophages is a half of the maximum rate. We further assume that they can die naturally at a constant rate .

The variables and parameters that describe in-host malaria dynamics are as in Tables 1 and 2, respectively.

Table 1: Symbols and definition of state variables considered in the model.
Table 2: Symbols and description of parameters used in the model.

The above transmission dynamics of malaria are summarised in the compartmental diagram in Figure 1.

Figure 1: Schematic diagram for hepatocytic-erythrocytic and malaria parasite dynamics. The dotted lines without arrows indicate cell-parasite interaction and the solid lines show progression from one compartment to another.

From the above description of the in-host dynamics of malaria and the representation in Figure 1, we derive the following system of ordinary differential equations:where , , , , , , and .

3. Model Analysis

3.1. Basic Properties

In this section, we study whether the formulated model (1) is biologically and mathematically meaningful. We establish model equilibrium points and investigate their stability properties.

3.1.1. Well-Posedness of the Model

For the in-host malaria model (1) to be mathematically and biologically meaningful, we need to prove that all the solutions of model system (1) with nonnegative initial conditions would remain nonnegative for all time . Positivity in the model is shown by proving the following theorem.

Theorem 1. Let the parameters in model (1) be positive constants. A nonnegative solution exists for all the state variables with nonnegative initial conditions , , , , , , .

Proof. Considering the first equation in system (1), let , so thatwhich yieldsIn a similar fashion, this procedure can be applied to all the remaining six equations in model system (1), so that we have the following solutions:Therefore, state variables of model system (1) are nonnegative for all time .

3.1.2. Invariant Region

Let represent the total hepatocyte population, so that .

On substituting the derivatives in system (1) and simplifying, we havewhere and

Using integrating factor ,where is a constant of integration. By applying the initial condition in (6), we obtainSubstituting the value of into in (6) and simplifying, we getThere are two possible cases in analyzing the behaviour of in (8). In the first case, we consider so that, at time , the right-hand side (RHS) of (8) experiences the largest possible value of . That is, for all time .

In the second case, we consider , so that the largest possible value of the RHS of (8) approaches as time goes to infinity. Thus, , . From these two cases, we conclude that for all time .

Using the above approach, let the total red blood cells population be , so that . From the model equations in system (1), we havewhere and . Upon solving for in (9), we have , .

For the macrophage compartment , we haveBy integration, the solution of (10) is presented asBy inspection, for all time .

Finally, let represent the total population of malaria parasites at any time . That is, and, from system (1),Let , so that on solving for we getClearly, the malaria parasite populations and are bounded above. That is,

for all time .

Based on this discussion, we have shown the existence of a bounded positive invariant region for our model system (1). Let us denote this region as , whereMoreover, any solution of our system (1) which commences in at any time will always remain confined in that region. We therefore deduce that the region is positively invariant and attracting with respect to malaria model (1). Our in-host malaria model (1) is hence well posed mathematically and biologically.

3.1.3. Disease-Free Equilibrium Point

The disease-free equilibrium point, , is the state in which the human host is free of malaria infection. At , the sporozoite recruitment rate, , and parasite and host-infected compartments have zero values; that is, . Therefore,

3.1.4. In-Host Basic Reproduction Number

The in-host reproduction number of model (1) denoted by is computed using the technique of the next-generation matrix approach described in [41]. We consider , , , and as the parasite infested compartments. Adopting the notations in [41], we generate a nonnegative matrix of new infections and a nonsingular matrix , showing the transfer of infections from one compartment to the other as follows:andThe inverse of matrix is hence given by The next-generation matrix , which is the product of matrices and , works out to be The in-host basic reproduction number is the spectral radius of the next-generation matrix . It can clearly be seen that three of the four eigenvalues of matrix in (19) have zero values; that is, . The fourth and largest nonnegative eigenvalue becomes the in-host model reproduction number. We therefore haveThe terms in model can be interpreted as follows:(1)The term represents the expected number of infectious merozoite parasites resulting from bursting blood schizonts at the blood stage of malaria infection.(2)The second term represents the expected proportion of merozoites that participate in the cycle of erythrocytic schizogony.(3)Observe that the terms and . So our . This implies that the number of secondary infections during malaria infections is largely influenced by the average number of merozoites released , from a bursting blood schizont, most of which are responsible for secondary infections at the blood stage.

Despite the inclusion of the liver stage dynamics, it is interesting to observe that the above in-host reproduction number and hence the disease progression are heavily driven by the dynamics at the erythrocytic stage.

In the sections that follow, we shall establish both the local stability and global stability of disease-free equilibrium point (15) of model system (1).

3.1.5. Local Stability of the Disease-Free Equilibrium Point,

The Jacobian matrix of model system (1) evaluated at the disease-free equilibrium is given byIt is clear from the first, third, and fifth columns of matrix (21) that the Jacobian matrix has negative eigenvalues , and . Upon deleting the first, third, and fifth rows and columns, matrix (21) is reduced to the following matrix:From row three in (22), . We further reduce matrix (22) by deleting row three and column three. So,Note from row one of (23) that the fifth eigenvalue .

The remaining two eigenvalues can be obtained by reducing matrix (23) into the following matrix:Using the variable , the characteristic polynomial associated with matrix (24) iswhereThe characteristic polynomial (25) has negative roots (eigenvalues) if and . The coefficient in (26) is clearly positive. We now need to show that in (27) is strictly positive if . This is done by expressing the coefficient term in terms of model as follows:It can clearly be seen from (28) that the coefficient is positive if and only if . We have thus established the following result.

Theorem 2. The disease-free equilibrium is locally asymptotically stable in if . If , then is unstable.

Biologically, Theorem 2 implies that malaria infection can be eliminated from the human host when . To ensure that elimination of malaria is independent of the initial sizes of the subpopulations, it is necessary to show that is globally asymptotically stable in , where the model is mathematically and biologically sensible.

3.1.6. Global Asymptotic Stability of the Disease-Free Equilibrium

Using the results obtained in [42], we show that the malaria-free equilibrium state is globally asymptotically stable when . We begin by rewriting the model system (1) in pseudotriangular form as follows:where is the vector representing the state of different compartment of liver and blood cells that are not infected and do not transmit malaria infections. represents the states of malaria parasites and host’s cells that are responsible for disease transmission. Hence, From the subsystem , we haveA direct computation indicates that the eigenvalue of matrix is real and negative. This shows that the system is globally asymptotically stable at the disease-free equilibrium, . Similarly, the subsystem gives rise to the following matrix :It can clearly be seen that is a Metzler matrix: all the off-diagonal elements of are nonnegative. In order to establish the global stability of the disease-free equilibrium, we need to show that the matrix is Metzler stable by providing a proof of the following lemma.

Lemma 3. Let be a square Metzler matrix that is block decomposed:where and are square matrices. The matrix is Metzler stable if and only if and are Metzler stable.

In our case, matrix is represented by matrix in (32), so thatUpon computation in Mathematica software, we obtainwhere , , and .

For the matrix to be Metzler stable, should be strictly nonnegative. Therefore, the expression in the numeratorUpon simplification of (36),Clearly, matrix in (34) is Metzler stable. However, the matrix is Metzler stable if and only if . From Lemma 3, we deduce the following theorem.

Theorem 4. The malaria-free equilibrium of model system (1) is globally asymptotically stable if the threshold quantity .

The above result is quite significant in malaria control. The global stability of the disease-free status would be guaranteed if and only if the in-host basic reproduction number is less than one. Malaria intervention should therefore focus on eliminating infected erythrocytes and/or malaria merozoites that are responsible for erythropoiesis cycle and invasions at the blood stage.

3.2. The Endemic Equilibrium Analysis

When , the stability of the disease-free equilibrium (15) is violated. A different equilibrium state termed the endemic equilibrium is achieved. Equating to zero the RHS of system (1) and solving for the state variables , , , , and in terms of the infected states and , we obtain the endemic state , whereSubstituting (41) into the equation in (1) and simplifying, we obtain the following cubic equation:whereThe number and nature of the roots of (44) are determined by the following discriminant:So(i)if , then (44) has multiple real roots and only one endemic equilibrium would exist,(ii)if , then (44) has 1 real root and a complex conjugate root and hence only one endemic equilibrium,(iii)if , then (44) has 3 distinct real roots and so there is more than one endemic equilibrium when for model system (1).

Analysis under (46) implies that, in the absence of external interventions in the form of antimalarial treatment, there will always be some infected hepatocytes during malaria infection. We then evaluate the possible values of the state variable at equilibrium by substituting expressions in (42) and (43) into the equation in (1). After simplification in Mathematica software, we obtain the following cubic equation:where Clearly, orThe state corresponds to a scenario in which there are no parasite-infected red blood cells. This could signify the liver stage of parasite development so that an endemic state exists. Alternatively, could correspond to the disease-free equilibrium point (15) for system (1).

The roots of the cubic equation (49) are given aswhereThe root should be ignored, since all the model state variables are nonnegative for all time . This leaves as the only two possible roots of (49).

From the above discussion, model (1) could experience a single endemic state or multiple endemic states subject to the roots of (44) and (47). If are real and positive, then one or two endemic equilibrium points are possible for model (1). It is thus evident that the explicit form of the endemic equilibrium state for model (1) is cumbersome. We shall therefore show its existence numerically based on a certain choice of parameter values in Section 4. Note that case (iii) of (46) indicates the possibility of having multiple endemic equilibria and hence the likelihood of experiencing a backward bifurcation phenomenon. This will be investigated in another research paper.

4. Numerical Simulations and Discussions

In this section, we provide some numerical simulations to illustrate the behaviour of model system (1). We carry out model sensitivity analysis and investigate parameter influence on the dynamics of red blood cells, macrophages, and malaria parasites under different conditions on the in-host reproduction number, .

4.1. Sensitivity Analysis

In epidemic modelling, sensitivity analysis is performed to investigate model parameters with significant influence on and hence on the transmission and the spread of the disease under study [43]. Following [44], the normalised forward-sensitivity index of a variable, , which depends differentially on a parameter, , is defined asUsing the formulation in (52) and the parameter values in Table 3, the local sensitivity indices (SI) of (see (20)) relative to the model parameters are calculated in Mathematica software and the results summarised in Table 4. Note that, due to limited data on in-host dynamics, all the parameter values used in evaluating the sensitivity indices are obtained from indicated past literature.

Table 3: Parameter values used in the numerical simulation and demonstration of the existence of endemic equilibrium point. See Table 2 for detailed parameter descriptions.
Table 4: Sensitivity indices of relative to the model parameters.

A positive sign on the SI indicates that an increase (decrease) in the value of such a parameter increases (decreases) the value of and hence the growth of malaria infection. On the other hand, a negative sign is indicative of a parameter that negatively affects . In order to eliminate in-host malaria infection, the in-host reproduction number should be less than one, that is, .

The average number of merozoites released per bursting infected erythrocyte and the proportion of merozoites that cause secondary invasions at the blood phase are the most sensitive parameters in determining the disease outcomes. They have the highest sensitivity indices of +1.0000. For instance, a 10% increase (decrease) or generates a 10% increase (decrease) on and hence malaria infection severity.

The parameters , , , and occupy the second rank in influencing the model outcomes. An increase in the parameters and is likely to increase the model . On the other hand, an increase in and has a direct negative influence on . Macrophages are highly instrumental in malaria parasite clearance and should be preserved.

The rate of generation of macrophages from the bone marrow, , together with the rate of phagocytosis of infected red blood cells, , is likely to decrease, proportionally, the disease progression when they are increased. With increased , there would be more macrophages to phagocytose and clear the rapidly growing density of blood schizonts. This would negatively affect the erythrocytic schizogony. Decreased clearance rate by macrophages would only guarantee successful multiplication of the merozoites through the erythrocytic schizogonic cycle. The subsequent result is increased concentration of merozoites in the host blood and disease progression to even deadly levels.

The parameters and increase (or decrease) when they are increased (or decreased). Epidemiologically, an improved erythrocyte invasion rate, , is likely to generate even more new blood schizonts. This increases parasitemia in the host. A 10% increase (decrease) in would increase (decrease) the threshold parameter by about +9.2%.

Any therapeutic effort that clears the blood schizonts and the infectious merozoites at the blood stage would definitely guarantee immense reduction in model . Therefore, an increase in the death rate of the infected red blood cells and that of the merozoites is likely to decrease significantly the in-host reproduction number . This can be achieved through the use of effective antimalarials such as the use of artemisinin based combination therapy (ACT) in malaria treatment. Moreover, effective vaccines at the erythrocytic stage could greatly help minimize erythrocyte infection rate .

Since the local sensitivity indices are relatively close, we carry out further investigation on parameter influence on disease progression by generating the partial rank correlation coefficients (PRCCs) for each parameter value in model in the following section.

4.1.1. Global Sensitivity Analysis

A global sensitivity analysis (GSA) is performed to examine the response of an epidemic model to parameter variation within a wider range of parameter space [45]. Applying the approach in [45], the PRCCs between the in-host basic reproduction number and each of the parameters in Table 2 are derived. Using 1000 simulations per run of the Latin Hypercube Sampling (LHS) scheme [46], the established PRCCs are derived and presented in Figure 2.

Figure 2: Tornado plots of PRCCs of parameters that influence model generated using parameter values in Table 3. Parameters with and increase and decrease model , respectively.

Unlike the results in Table 4, the model parameter with the highest influence on according to the PRCCs results shown in Figure 2 is the rate of invasion of red blood cells by merozoites, . This is followed closely by the recruitment rate of susceptible red blood cells from the bone marrow. The second set of parameters that also increase (decrease) model when they are increased (decreased) are , , , and , respectively.

The merozoites’ death rate , the death rate of IRBCs , and the rate of elimination of IRBCs by macrophages are shown to have the highest negative influence on disease progression. Although an increase in was shown to decrease disease progression in Table 4, the results from global sensitivity analysis are contradictory. An increase in the death rate of parasitized erythrocytes decreases parasitemia and hence disease progression.

Based on these results of sensitivity analysis, we make the following remarks: (1) results of global sensitivity analysis are robust and a lot more realistic for implementation, (2) malaria control should target elimination of merozoites and infected red blood cells, (3) an effective and efficient malaria vaccine that deactivates infectious merozoites could be helpful in limiting erythrocyte invasion rate, and (4) a vaccine that is protective of susceptible erythrocytes could further ensure reduced density of second and future generation of merozoites that are responsible for disease progression.

4.2. Numerical Results

Model system (1) is solved numerically using the package scipy.integrate.odeint in Python language. The simulations are performed to illustrate the possible dynamics of the red blood cells, the malaria parasite, and macrophages. For purposes of these simulations, the initial conditions of the variables are hereby assumed. We note that different dynamics could be achieved for a different set of initial conditions.

For (see Figure 3), the density of susceptible hepatocyte initially declines as the density of infected hepatocytes rises due to invasion from sporozoites. The host’s immune system responds to sporozoite invasion by increasing hepatocyte density that levels off at the disease-free equilibrium point (see Figure 3(a)). As the sporozoites decline to near zero (see Figure 3(b)), infected hepatocytes decline and stabilize at in (15).

Figure 3: Graphs showing the simulation of in-host malaria model (1) when the model . Figures (a) and (b) show model dynamics at the liver stage. The chosen initial conditions are , , , , , , and . Used parameter values are given in Table 3.

At the blood stage, the rising density of infected erythrocytes declines in a similar fashion to that of the infective merozoites when (see Figure 3(c)). The densities of the infected erythrocytes and merozoites approach asymptotically. On the other hand, we observe that the density of susceptible red blood cells initially diminishes due to infection by merozoites and later rises before it plateaus as shown in Figure 3(c).

When , a sharp fall in the density of susceptible hepatocytes in the liver is observed (see Figure 4(a)). This is due to rapid invasion of hepatocytes by the sporozoites. An invasion on susceptible hepatocyte generates a corresponding steady rise in the density of infected hepatocytes (see Figure 4(b)). Owing to natural intervention by the immune system cells, the respective decline and rising levels of susceptible and infected hepatocytes level off and remain relatively constant after the third month. More liver cells are generated to replace infected ones. Figure 4(c) indicates a steady decline in sporozoite density at the liver stage during infections. Invaded hepatocytes burst open to produce merozoites instead of sporozoites and hence the steady decline in sporozoite levels.

Figure 4: Graphs showing population dynamics of the liver hepatocytes and malaria sporozoites when . Used parameter values can be found in Table 3 with the chosen initial conditions described by , , , , , , and .

Malaria infection dynamics are most rapid in the first 2 weeks within the host liver as illustrated in Figures 4(a), 4(b), and 4(c). This is similar to results in [22, 31]. In the absence of clinical intervention, some of the sporozoites may remain dormant in the human liver and could cause future malaria infections. As the liver schizonts release merozoites into host’s blood stream, a rapid decline in the density of red blood cells is observed (see Figure 5(a)). However, the density of infected erythrocytes is noted to rise with equal proportion as shown in Figure 5(b).

Figure 5: Graphs showing population dynamics of red blood cells, macrophages, and malaria merozoites when . Simulation parameter values are available in Table 3.

An early sharp rise in the density of merozoites in the first one week of the blood stage is noted in Figure 5(c). The density remains high for several weeks and does not decline for the entire infection period of one month. A second-generation merozoite invades other sets of healthy erythrocytes within minutes, leading to an exponential growth in the density of blood schizonts and hence merozoites in the human blood. Without therapeutic intervention, the density of merozoites stabilizes several weeks after infection at the endemic equilibrium point. This is consistent with the findings in [19, 20, 23].

The invasion of healthy erythrocytes prompts an immune response from host’s macrophages. These macrophages phagocyte on the generated blood schizonts. At the onset of erythrocytic infection, several macrophages are generated. The rise in the density of macrophages is proportional to that of infected erythrocytes as shown in Figure 5(d). This rising density is shown to level off after about 16 days at the endemic equilibrium point. It remains high throughout the infection period.

From these discussions, we make the following observations: (1) if , low level malaria infection can easily be contained by the host’s defence mechanism and loss of life is less likely; (2) therapeutically, may be achieved through quick interventions targeting the blood schizonts and the merozoites responsible for secondary infections during the erythrocytic cycle; (3) Figures 4 and 5 prove the existence of malaria endemic equilibrium point.

Hematological parameters such as the density of healthy and infected erythrocytes in malaria hosts have considerable influence on malaria infection and possible impacts [47]. According to WHO [48], hyperparasitemia causes drastic reduction in concentrations of erythrocytes, leading to anaemia among malaria patients. The impacts of increasing the model parameters and on healthy and infected red blood cells are as shown in Figures 6 and 7, respectively.

Figure 6: Graphs showing the behaviour of (a) susceptible RBCs and (b) infected RBCs. They were obtained by varying the death rate of merozoites from 20 to 80 in steps of 20, while keeping the other parameters (see Table 3) constant.
Figure 7: Graphs showing the behaviour of (a) susceptible RBCs and (b) infected RBCs. They were obtained by varying the merozoite invasion rate from to in steps of , while keeping the other parameters in Table 3 constant.

Observe that increased death rate of malaria merozoites decreases and increases the concentration of parasitized red blood cells and healthy red blood cells, respectively (see Figures 6(a) and 6(b)). Malaria control should thus target the infectious merozoites at the blood stage.

Results in Figure 7(a) indicate that an improved invasion rate by merozoites on susceptible red blood cells causes more loss in healthy erythrocytes. The reverse effect is observed in Figure 7(b), where an increase in the rate of infection of healthy erythrocytes produces a corresponding increase in the density of IRBCs. A keen look at Figure 7(b) reveals that the infected red blood cells begin to appear after about 10–15 days of initial infection. This is consistent with the incubation period of Plasmodium falciparum malaria [49].

The severity of malaria infection can easily increase if the density or production of macrophages is compromised [19]. Figure 8(b) shows a near direct relationship on the density of infected red blood cells and the death rate of the macrophages . An increase in the death of macrophages would propel erythrocytic schizogony and hence increased merozoite numbers in the human blood. A high merozoite density increases the severity of malaria infection. This result is quite vital in malaria intervention, especially with respect to malaria patients who may be suffering from other infections that are deleterious to immune cells. Diseases such as HIV/AIDS greatly weaken the immune system of the patient as crucial immune cells such as macrophages are destroyed. Macrophages are important target cells for HIV-1 virus [50]. During malaria infections, such patients often suffer from severe malaria and should seek immediate medical attention.

Figure 8: Graphs showing the effect of varying the rate of phagocytosis of IRBCs by macrophages, (in (a)), and the effect of increased decay rate of macrophages, (in (b)), on the behaviour of infected erythrocytes . All parameter values are in Table 3.

Like the senescent red blood cells, aberrant infected erythrocytes formed during malaria infection are eliminated phagocytically by the host’s macrophage cells in the red pulp of the spleen [51]. The phagocytic potential of the spleen is vital at the erythrocytic cycle. The higher the phagocytic behaviour of the macrophage, the lower the density of parasitized erythrocytes (see Figure 8(a)). The severity of malaria infection increases with decreasing ability of the host’s phagocytic merozoites to clear infected red blood cells from circulation during the erythrocytic cycle.

5. Conclusion and Discussion

In this paper, a mathematical model of in-host malaria infection in [21] is extended to include the liver stage of parasite development. Unlike the models in [19, 23, 30], we considered the macrophages as the most effective innate immune cells in eliminating malaria parasites from the human blood circulation. In addition, the liver hepatocytes are assumed to be generated from the bone marrow and through a process of self-regeneration from existing hepatocytes.

We proved that the formulated model is biologically and mathematically well posed in an invariant region . The malaria-free equilibrium is shown to be locally asymptotically stable when the in-host reproduction number is less than unity. The global stability of the malaria-free state is only guaranteed if the threshold quantity is less than unity.

Our numerical results show that intervention during malaria infection should focus on minimizing merozoite invasion rate on healthy erythrocytes and the density of merozoites in circulation, which are responsible for secondary invasion at the blood stage. In the absence of malaria treatment, the immune cells (macrophages) are shown to be vital in eliminating infected red blood cells at the blood stage. The higher the rate of phagocytosis of infected erythrocytes by macrophages, the lower the density of infected red blood cells and hence malaria parasitemia. Patients suffering from such infections as HIV/AIDS and TB that have deleterious effect on the protective immune cells should seek immediate medical treatment when infected with malaria. Their compromised immune system exposes them to severe malaria attacks and possible untimely death.

For quick and timely reduction of parasitemia, an increased merozoite death rate using antimalarial drugs such as ACT would be necessary. This would further ensure reduced density of infected red blood cells and hence future generation merozoites. By killing a single blood schizont, we are likely to avoid the production of sixteen merozoites at maturity. Moreover, an appropriate vaccine that targets erythrocyte invasion process may equally guarantee minimal erythropoiesis. The erythrocyte invasion-avoidance vaccine would minimize the density of infected erythrocytes and hence malaria disease severity. This intervention could help terminate the erythrocytic schizont, leading to minimal parasite transmission to mosquito vector for further development and sexual reproduction.

In this study, drug resistance was not analyzed; this can be considered as a potential area for future investigation.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The authors acknowledge with gratitude the support from the Institute of Mathematical Sciences, Strathmore University, and the National Research Fund (NRF), Kenya, for the production of this manuscript.

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