# Mathematical Model for Hepatocytic-Erythrocytic Dynamics of Malaria

Authors: Titus Okello Orwa; Rachel Waema Mbogo; Livingstone Serwadda Luboobi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## 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 ofPlasmodium 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 R0 is derived. The disease-free equilibrium is shown to be locally and globally asymptotically stable. Sensitivity analysis reveals that the erythrocyte invasion rate βr, the average number of merozoites released per bursting infected erythrocyte K, and the proportion of merozoites that cause secondary invasions at the blood phase ζ 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.


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## Body

## 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 genusPlasmodium which are transmitted to humans by the bite of femaleAnopheles mosquito.Plasmodium falciparum, which is predominant in sub-Saharan Africa, New Guinea, and Haiti [3], is the major cause of malaria infections. The otherPlasmodium species that cause malaria areP. vivax,P. ovale,P. malariae, andP. knowlesi [4].P. vivax andP. 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 ofPlasmodium falciparum in the human host.During their obligatory blood meals, infected femaleAnopheles mosquitoes inject sporozoites belonging toPlasmodium 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 femaleAnopheles 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 thePlasmodium 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 toPlasmodium [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 Section2, 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., [25–27]) focused on improvingPlasmodium 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 ofPlasmodium 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 [32–36].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 ofPlasmodium 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 S(t), susceptible hepatocytes H(t), infected hepatocytes HX(t), susceptible red blood cells (RBCs) R(t), infected red blood cells (IRBCs) RX(t), merozoites M(t), and macrophages Z(t) at any time t. The dynamics of malaria parasites and host-cell populations in each compartment are described as follows.Sporozoites ( S ). The femaleAnopheles mosquito is assumed to inject sporozoites into the human system during blood meal at a constant rate Λ. The sporozoites molt through the blood stream and reach the liver in about 2 hours, where they invade the hepatocytes at the rate βs. We assume that the sporozoites can die naturally at a rate δs.Susceptible Hepatocytes ( H ). 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 λh. 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 ρ1HX/κ1+HX=ψ1(HX), where HX and ρ1, respectively, represent the concentration of infected hepatocytes and their rates of generation. The parameter κ1 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 βs, susceptible hepatocytes get infected and progress to subpopulation HX. In addition, hepatocytes in compartment H are assumed to have a natural life expectancy and may hence die naturally at the rate μ1.Infected Hepatocytes ( H X ). Infected hepatocytes mature into liver-stage schizonts. These schizonts burst open releasing 2000–40000 uninucleate merozoites into the blood stream [39]. The term Nμ2HX represents the total population of merozoites released upon bursting of infected hepatocytes. The parameter μ2 represents the death rate of the infected hepatocytes.Susceptible Red Blood Cells ( R ). Similar to malaria models in [21, 23, 30], we have assumed that the susceptible RBCs get recruited at a constant rate λr from the bone marrow. We further assume that, during infection, the erythrocyte production is accelerated owing to the presence of IRBCs at the rate ρ2. This increase is denoted by the term ρ2RX/κ2+RX=ψ2(RX), where κ2 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, βrMR. The positive constant βr describes the rate of successful invasion by a malaria merozoite. The susceptible RBCs die naturally at a rate μ3.Infected Red Blood Cells ( R X ). Upon invasion by merozoites, the healthy RBCs get infected, leading to the formation of infected red blood cells RX. Although the RBCs die at a constant rate μ4, they can similarly be killed through phagocytosis by the macrophages at the rate η. At maturity, the IRBCs burst open, releasing free merozoites into the blood system, causing secondary invasion and disease progression.Merozoites ( M ). After 2–15 days, the infected hepatocytes burst open and release merozoites into the blood system. This is represented by the term Nμ2HX, where N is the average number of merozoites released per bursting infected hepatocytes. An average of K merozoites is released per each bursting IRBC. These free parasites suffer a natural death at a rate δm and invade susceptible RBCs at a rate βr. 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 ζ of the merozoites contribute to secondary invasion of the susceptible RBCs. The rest of the merozoites (1-ζ) transform into gametocytes that are later picked up by femaleAnopheles mosquitoes during feeding.Macrophages ( Z ). 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 λz from the bone marrow. Moreover, they proliferate at a rate ρ3 in the sites of infection in proportion to the density of IRBCs. This is represented by the term ρ3RX/κ3+RX=ψ3(RX), where κ3 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 δz.The variables and parameters that describe in-host malaria dynamics are as in Tables1 and 2, respectively.Table 1
Symbols and definition of state variables considered in the model.
Variable Description H(t) The population of susceptible hepatocytes at timet HX(t) The population of infected hepatocytes at timet R(t) The population of susceptible red blood cells (erythrocytes) at timet RX(t) The population of infected red blood cells at timet Z(t) The density of macrophages in the human body at timet S(t) The population of sporozoites at timet M(t) The population of merozoites at timetTable 2
Symbols and description of parameters used in the model.
Parameter Description Λ The total rate of injection of sporozoites into liver due to mosquito bites δs The death rate of sporozoites λh Recruitment rate of susceptible hepatocytes from the bone marrow μ1 Natural death rate of susceptible hepatocytes βs The invasion rate of hepatocytes by sporozoites μ2 Death rate of infected hepatocytes λr Recruitment rate of susceptible RBCs by the bone marrow μ3 The natural death rate of RBCs βr The invasion rate of RBCs by merozoites μ4 Death rates of IRBCs δm The death rate of merozoites λz Recruitment rate of macrophages from the bone marrow δz The death rate of a macrophage η Elimination rate of IRBCs by macrophages ρ1 Production rate of hepatocytes due to presence of infected hepatocytes ρ2 Production rate of RBCs due to presence of IRBCs ρ3 Immunogenicity of IRBCs κ1 Number ofHX at which the recruitment of H is a half of the maximum rate κ2 Number ofRX at which the recruitment of R is a half of the maximum rate κ3 Number ofRX at which the recruitment of Z is a half of the maximum rate ζ The proportion of the merozoites that cause secondary infections K The average number of merozoites released per bursting IRBCs N The average number of merozoites released per bursting infected hepatocytesThe above transmission dynamics of malaria are summarised in the compartmental diagram in Figure1.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 Figure1, we derive the following system of ordinary differential equations:(1)dHdt=λh+ρ1HXκ1+HX-μ1H-βsSH,dHXdt=βsSH-μ2HX,dRdt=λr+ρ2RXκ2+RX-μ3R-βrRM,dRXdt=βrRM-μ4RX-ηRXZ,dZdt=λz+ρ3RXκ3+RX-δzZ,dSdt=Λ-δsS-βsSH,dMdt=Nμ2HX+Kζμ4RX-δmM-βrRM,where H(0)≥0, HX(0)≥0, R(0)≥0, RX(0)≥0, Z(0)≥0, S(0)≥0, and M(0)≥0.


## 2.1. Model Formulation

The hepatocytic-erythrocytic malaria model describes the dynamics ofPlasmodium 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 S(t), susceptible hepatocytes H(t), infected hepatocytes HX(t), susceptible red blood cells (RBCs) R(t), infected red blood cells (IRBCs) RX(t), merozoites M(t), and macrophages Z(t) at any time t. The dynamics of malaria parasites and host-cell populations in each compartment are described as follows.Sporozoites ( S ). The femaleAnopheles mosquito is assumed to inject sporozoites into the human system during blood meal at a constant rate Λ. The sporozoites molt through the blood stream and reach the liver in about 2 hours, where they invade the hepatocytes at the rate βs. We assume that the sporozoites can die naturally at a rate δs.Susceptible Hepatocytes ( H ). 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 λh. 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 ρ1HX/κ1+HX=ψ1(HX), where HX and ρ1, respectively, represent the concentration of infected hepatocytes and their rates of generation. The parameter κ1 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 βs, susceptible hepatocytes get infected and progress to subpopulation HX. In addition, hepatocytes in compartment H are assumed to have a natural life expectancy and may hence die naturally at the rate μ1.Infected Hepatocytes ( H X ). Infected hepatocytes mature into liver-stage schizonts. These schizonts burst open releasing 2000–40000 uninucleate merozoites into the blood stream [39]. The term Nμ2HX represents the total population of merozoites released upon bursting of infected hepatocytes. The parameter μ2 represents the death rate of the infected hepatocytes.Susceptible Red Blood Cells ( R ). Similar to malaria models in [21, 23, 30], we have assumed that the susceptible RBCs get recruited at a constant rate λr from the bone marrow. We further assume that, during infection, the erythrocyte production is accelerated owing to the presence of IRBCs at the rate ρ2. This increase is denoted by the term ρ2RX/κ2+RX=ψ2(RX), where κ2 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, βrMR. The positive constant βr describes the rate of successful invasion by a malaria merozoite. The susceptible RBCs die naturally at a rate μ3.Infected Red Blood Cells ( R X ). Upon invasion by merozoites, the healthy RBCs get infected, leading to the formation of infected red blood cells RX. Although the RBCs die at a constant rate μ4, they can similarly be killed through phagocytosis by the macrophages at the rate η. At maturity, the IRBCs burst open, releasing free merozoites into the blood system, causing secondary invasion and disease progression.Merozoites ( M ). After 2–15 days, the infected hepatocytes burst open and release merozoites into the blood system. This is represented by the term Nμ2HX, where N is the average number of merozoites released per bursting infected hepatocytes. An average of K merozoites is released per each bursting IRBC. These free parasites suffer a natural death at a rate δm and invade susceptible RBCs at a rate βr. 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 ζ of the merozoites contribute to secondary invasion of the susceptible RBCs. The rest of the merozoites (1-ζ) transform into gametocytes that are later picked up by femaleAnopheles mosquitoes during feeding.Macrophages ( Z ). 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 λz from the bone marrow. Moreover, they proliferate at a rate ρ3 in the sites of infection in proportion to the density of IRBCs. This is represented by the term ρ3RX/κ3+RX=ψ3(RX), where κ3 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 δz.The variables and parameters that describe in-host malaria dynamics are as in Tables1 and 2, respectively.Table 1
Symbols and definition of state variables considered in the model.
Variable Description H(t) The population of susceptible hepatocytes at timet HX(t) The population of infected hepatocytes at timet R(t) The population of susceptible red blood cells (erythrocytes) at timet RX(t) The population of infected red blood cells at timet Z(t) The density of macrophages in the human body at timet S(t) The population of sporozoites at timet M(t) The population of merozoites at timetTable 2
Symbols and description of parameters used in the model.
Parameter Description Λ The total rate of injection of sporozoites into liver due to mosquito bites δs The death rate of sporozoites λh Recruitment rate of susceptible hepatocytes from the bone marrow μ1 Natural death rate of susceptible hepatocytes βs The invasion rate of hepatocytes by sporozoites μ2 Death rate of infected hepatocytes λr Recruitment rate of susceptible RBCs by the bone marrow μ3 The natural death rate of RBCs βr The invasion rate of RBCs by merozoites μ4 Death rates of IRBCs δm The death rate of merozoites λz Recruitment rate of macrophages from the bone marrow δz The death rate of a macrophage η Elimination rate of IRBCs by macrophages ρ1 Production rate of hepatocytes due to presence of infected hepatocytes ρ2 Production rate of RBCs due to presence of IRBCs ρ3 Immunogenicity of IRBCs κ1 Number ofHX at which the recruitment of H is a half of the maximum rate κ2 Number ofRX at which the recruitment of R is a half of the maximum rate κ3 Number ofRX at which the recruitment of Z is a half of the maximum rate ζ The proportion of the merozoites that cause secondary infections K The average number of merozoites released per bursting IRBCs N The average number of merozoites released per bursting infected hepatocytesThe above transmission dynamics of malaria are summarised in the compartmental diagram in Figure1.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 Figure1, we derive the following system of ordinary differential equations:(1)dHdt=λh+ρ1HXκ1+HX-μ1H-βsSH,dHXdt=βsSH-μ2HX,dRdt=λr+ρ2RXκ2+RX-μ3R-βrRM,dRXdt=βrRM-μ4RX-ηRXZ,dZdt=λz+ρ3RXκ3+RX-δzZ,dSdt=Λ-δsS-βsSH,dMdt=Nμ2HX+Kζμ4RX-δmM-βrRM,where H(0)≥0, HX(0)≥0, R(0)≥0, RX(0)≥0, Z(0)≥0, S(0)≥0, and M(0)≥0.


## 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 t≥0. 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 (H(t),HX(t),R(t),RX(t),Z(t),S(t),M(t)) exists for all the state variables with nonnegative initial conditions {H(0)=H0≥0, HX(0)=HX0≥0, R(0)=R0≥0, RX(0)=RX0≥0, Z(0)=Z0≥0, S(0)=S0≥0, M(0)=M0≥0}∀t≥0.Proof.
Considering the first equation in system (1), let ψ1(t)=ρ1HX/κ1+HX, so that(2)dHdt=λh+φt-μ1H-βsSH,dHdt≥-μ1+βsSH,which yields(3)Ht≥H0exp⁡-∫0tβsSsds+μ1t>0.In 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:(4)HXt≥HX0exp⁡-μ2t>0,Rt≥R0exp⁡-∫0tβrMsds+μ3t>0,RXt≥RX0exp⁡-∫0tηZsds+μ4t>0,Zt≥Z0exp⁡-δzt>0,St≥S0exp⁡-∫0tβsHsds+δst>0,Mt≥M0exp⁡-∫0tβrRsds+δmt>0.Therefore, state variables (H,HX,R,RX,Z,S,M) of model system (1) are nonnegative for all time t>0.


#### 3.1.2. Invariant Region

LetNH(t) represent the total hepatocyte population, so that NH(t)=H(t)+HX(t).On substituting the derivatives in system (1) and simplifying, we have(5)dNHdt≤λh+ψ1t-μhNH,where ψ1(t)=ρ1HX/κ1+HX and μh=min⁡{μ1,μ2}.Using integrating factoreμht,(6)NHt≤λhμh+e-μht∫0tψ1τeμhτdτ+c1e-μht,where c1 is a constant of integration. By applying the initial condition NH(0)=NH0>0 in (6), we obtain(7)c1=NH0-λhμh-∫0tψ1τe-μhτdτ.Substituting the value of c1 into NH(t) in (6) and simplifying, we get(8)NHt≤λhμh+e-μhtNH0-λhμh.There are two possible cases in analyzing the behaviour of NH(t) in (8). In the first case, we consider NH(0)>λh/μh so that, at time t=0, the right-hand side (RHS) of (8) experiences the largest possible value of NH(0). That is, NH(t)≤NH(0) for all time t>0.In the second case, we considerNH(0)<λh/μh, so that the largest possible value of the RHS of (8) approaches λh/μh as time t goes to infinity. Thus, NH(t)≤λh/μh, ∀t>0. From these two cases, we conclude that NH(t)≤max⁡NH(0),λh/μh for all time t>0.Using the above approach, let the total red blood cells population beNR(t), so that NR(t)=R(t)+RX(t). From the model equations in system (1), we have(9)dNRdt≤λr+ψ2t-μrNRt,where ψ2(t)=ρ2RX/κ2+RX and μr=min⁡{μ3,μ4}. Upon solving for NR in (9), we have NR(t)≤max⁡NR(0),λr/μr, ∀t>0.For the macrophage compartmentZ(t), we have(10)dZdt=λz+ψ3t-δzZ,forψ3t=ρ3RXκ3+RX.By integration, the solution of (10) is presented as(11)Zt≤λzδz+e-δztZ0-λzδz.By inspection, Z(t)≤max⁡Z(0),λz/δz for all time t>0.Finally, letNP(t) represent the total population of malaria parasites at any time t. That is, NP(t)=S(t)+M(t) and, from system (1),(12)dNPdt=Λ-δsS-βsSH+Nμ2HX+Kζμ4RX-δmM,≤Λ+Kζμ4RX+Nμ2HX-δpNp,whereδp=min⁡δs,δm.Let (Kζμ4RX+Nμ2HX)=ψ4(t), so that on solving for NP(t) we get(13)Npt≤Λδp+e-δptNp0-ΛδpClearly, the malaria parasite populations S(t) and M(t) are bounded above. That is,N P ( t ) ≤ max ⁡ N P ( 0 ) , Λ / δ p for all time t>0.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 Ω∈R+7, where(14)Ω=H,HX,R,RX,Z,S,M∈R+7:NPt≤max⁡NP0,Λδp,NHt≤max⁡NH0,λhμh,NRt≤max⁡NR0,λrμr,Zt≤max⁡Z0,λzδz.Moreover, any solution of our system (1) which commences in Ω at any time t≥0 will always remain confined in that region. We therefore deduce that the region Ω 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,E0, is the state in which the human host is free of malaria infection. At E0, the sporozoite recruitment rate, Λ=0, and parasite and host-infected compartments have zero values; that is, S∗=M∗=RX∗=HX∗=0. Therefore,(15)E0=H∗,HX∗,R∗,RX∗,Z∗,S∗,M∗=λhμ1,0,λrμ3,0,λzδz,0,0.


#### 3.1.4. In-Host Basic Reproduction Number

The in-host reproduction number of model (1) denoted by R0 is computed using the technique of the next-generation matrix approach described in [41]. We consider HX, RX, S, and M as the parasite infested compartments. Adopting the notations in [41], we generate a nonnegative matrix F of new infections and a nonsingular matrix V, showing the transfer of infections from one compartment to the other as follows:(16)F=00βsλhμ10000βrλrμ300000000and(17)V=μ20000μ4+ηλzδz0000δs+βsλhμ10-Nμ2-Kζμ40δm+βrλrμ3.The inverse of matrix V is hence given by (18)V-1=1μ20000δzηλz+δzμ400001δs+βsλh/μ10Nμ3βrλr+δmμ3Kζδzμ3μ4βrλr+δmμ3ηλz+δzμ401δm+βrλr/μ3.The next-generation matrix G, which is the product of matrices F and V-1, works out to be (19)G=00βsλhβsλh+δsμ10Nβrλrβrλr+δmμ3Kζβrδzλrμ4βrλr+δmμ3ηλz+δzμ40βrλrβrλr+δmμ300000000.The in-host basic reproduction number R0 is the spectral radius of the next-generation matrix G. It can clearly be seen that three of the four eigenvalues of matrix G in (19) have zero values; that is, λ1=λ2=λ3=0. The fourth and largest nonnegative eigenvalue λ4 becomes the in-host model reproduction number. We therefore have(20)R0=Kβrλrβrλr+δmμ3·ζδzμ4ηλz+δzμ4.The terms in model R0 can be interpreted as follows:(1)
The termKβrλr/(βrλr+δmμ3) represents the expected number of infectious merozoite parasites resulting from bursting blood schizonts at the blood stage of malaria infection.(2)
The second termζδzμ4/(ηλz+δzμ4) represents the expected proportion of merozoites that participate in the cycle of erythrocytic schizogony.(3)
Observe that the terms(βrλr)/(βrλr+δmμ3)<1 and (δzμ4)/(ηλz+δzμ4)<1. So our R0≤Kζ. This implies that the number of secondary infections during malaria infections is largely influenced by the average number of merozoites released K, 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,E0

The Jacobian matrix of model system (1) evaluated at the disease-free equilibrium E0 is given by(21)J1E0=-μ1ρ1κ1000-βsλhμ100-μ2000βsλhμ1000-μ3ρ2κ200-βrλrμ3000-ηλzδz-μ400βrλrμ3000ρ3κ3-δz0000000-βsλhμ1-δs00Nμ20Kζμ400-βrλrμ3-δm.It is clear from the first, third, and fifth columns of matrix (21) that the Jacobian matrix has negative eigenvalues λ1=-μ1,λ2=-μ3, and λ3=-δz. Upon deleting the first, third, and fifth rows and columns, matrix (21) is reduced to the following 4×4 matrix:(22)J2E0=-μ20βsλhμ100-ηλzδz-μ40βrλrμ300-βsλhμ1-δs0Nμ2Kζμ40-βrλrμ3-δm.From row three in (22), λ4=-βsλh/μ1-δs. We further reduce matrix (22) by deleting row three and column three. So,(23)J3E0=-μ2000-ηλzδz-μ4βrλrμ3Nμ2Kζμ4-βrλrμ3-δm.Note from row one of (23) that the fifth eigenvalue λ5=-μ2<0.The remaining two eigenvalues can be obtained by reducing matrix (23) into the following 2×2 matrix:(24)J6E0=-ηλzδz-μ4βrλrμ3Kζμ4-βrλrμ3-δm.Using the variable λ, the characteristic polynomial associated with matrix (24) is(25)pλ=λ2+Aλ+B,where(26)A=δm+ηλzδz+βrλrμ3+μ4and(27)B=ηδmλzδz+ηβrλrλzδzμ3+δmμ4+βrλrμ4μ3-Kζβrλrμ4μ3.The characteristic polynomial (25) has negative roots (eigenvalues) if A>0 and B>0. The coefficient A in (26) is clearly positive. We now need to show that B in (27) is strictly positive if R0<1. This is done by expressing the coefficient term B in terms of model R0 as follows:(28)B=1δzμ3μ4δz+ηλzβrλr+δmμ3-Kζβrδzλrμ4,=1δzμ3μ4δz+ηλzβrλr+δmμ31-Kζβrδzλrμ4μ4δz+ηλzβrλr+δmμ3,=μ4δz+ηλzβrλr+δmμ3δzμ31-R0.It can clearly be seen from (28) that the coefficient B is positive if and only if R0<1. We have thus established the following result.Theorem 2.
The disease-free equilibriumE0 is locally asymptotically stable in Ω if R0<1. If R0>1, then E0 is unstable.Biologically, Theorem2 implies that malaria infection can be eliminated from the human host when R0<1. To ensure that elimination of malaria is independent of the initial sizes of the subpopulations, it is necessary to show that E0 is globally asymptotically stable in Ω, 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 E0 is globally asymptotically stable when R0<1. We begin by rewriting the model system (1) in pseudotriangular form as follows:(29)X˙1=A1XX1-X1∗+A2XX2,X˙2=A3XX2,where X1 is the vector representing the state of different compartment of liver and blood cells that are not infected and do not transmit malaria infections. X2 represents the states of malaria parasites and host’s cells that are responsible for disease transmission. Hence, (30)X=X1,X2,X1=H,R,Z,X2=HX,RX,S,MandX1∗=λhμ1,λrμ3,λzδz.From the subsystem X1, we have(31)A1X=-μ1000-μ3000-δzandA2X=ρ1κ10-λhμ1βs00ρ2κ20-λrμ3βr0ρ3κ300.A direct computation indicates that the eigenvalue of matrix A1(X) is real and negative. This shows that the system X˙1=A1(X)(X1-X1∗)+A2(X)X2 is globally asymptotically stable at the disease-free equilibrium, E0. Similarly, the subsystem X2 gives rise to the following matrix A3(X):(32)A3X=-μ20βsλhμ100-ηλzδz+μ40βrλrμ300-βsλhμ1+δs0Nμ2Kζμ40-βrλrμ3+δm.It can clearly be seen that A3(X) is a Metzler matrix: all the off-diagonal elements of A3(X) are nonnegative. In order to establish the global stability of the disease-free equilibrium, we need to show that the matrix A3(X) is Metzler stable by providing a proof of the following lemma.Lemma 3.
LetM be a square Metzler matrix that is block decomposed:(33)M=ABCD,where A and D are square matrices. The matrix M is Metzler stable if and only if A and D-CA-1B are Metzler stable.In our case, matrixM is represented by matrix A3 in (32), so that(34)A=-μ200-ηλzδz+μ4,B=βsλhμ100βrλrμ3,C=00Nμ2Kζμ4andD=-βsλhμ1+δs00-βrλrμ3+δm.Upon computation in Mathematica software, we obtain(35)D-CA-1B=-ω10ω2-ω3,where ω1=δs+βsλh/μ1, ω2=Nβsλh/μ1, and ω3=δm+βrλr(ηλh+(1-Kζ)δzμ4)/μ3(ηλz+δzμ4).For the matrixD-CA-1B to be Metzler stable, ω3 should be strictly nonnegative. Therefore, the expression in the numerator(36)βrλrηλh+1-Kζδzμ4≥0.Upon simplification of (36),(37)Kζβrλrδzμ4≤βrλrηλh+δzμ4,(38)βrλr+δmμ3βrλrKζβrλrδzμ4βrλr+δmμ3ηλz+δzμ4≤1,(39)βrλr+δmμ3βrλrR0≤1(40)R0≤βrλrβrλr+δmμ3<1.Clearly, matrix A in (34) is Metzler stable. However, the matrix D-CA-1B is Metzler stable if and only if R0<1. From Lemma 3, we deduce the following theorem.Theorem 4.
The malaria-free equilibriumE0 of model system (1) is globally asymptotically stable if the threshold quantity R0<1.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 numberR0 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

WhenR0>1, 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 R, H, Z, S, and M in terms of the infected states HX and RX, we obtain the endemic state E1=(H∗,HX∗,R∗,RX∗,Z∗,S∗,M∗), where(41)H∗=1μ1λh+ρ1HX∗κ1+HX∗-μ2HX∗,S∗=μ1μ2HX∗βsλh+ρ1HX∗/κ1+HX∗-μ2HX∗,(42)R∗=1μ3λr+ρ2RX∗κ2+RX∗-μ4RX∗,M∗=μ3RX∗μ4+ηRX∗ρ3RX∗/κ3+RX∗+λzμ1(43)Z∗=1δzλz+ρ3RX∗κ3+RX∗Substituting (41) into the 2nd equation in (1) and simplifying, we obtain the following cubic equation:(44)α3HX∗3+α2HX∗2+α1HX∗+α0=0,where(45)α3=μ22βs>0,α2=μ2μ1-δs-βsλh-κ1μ2+Λ+ρ1,α1=βsλhΛ-κ1μ2+Λρ1-κ1μ2-κ1μ1μ2δs,andα0=κ1Λλhβs>0.The number and nature of the roots of (44) are determined by the following discriminant:(46)Δ=18α3α2α1α0-4α23α0+α22α12-4α3α13-27α32α02.So(i)
ifΔ=0, then (44) has multiple real roots and only one endemic equilibrium would exist,(ii)
ifΔ<0, then (44) has 1 real root and a complex conjugate root and hence only one endemic equilibrium,(iii)
ifΔ>0, then (44) has 3 distinct real roots and so there is more than one endemic equilibrium when R0>1 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 RX at equilibrium by substituting expressions in (42) and (43) into the 4th equation in (1). After simplification in Mathematica software, we obtain the following cubic equation:(47)RX∗θ3RX∗3+θ2RX∗2+θ1RX∗+θ0=0,where (48)θ3=-μ4βrμ1μ4+ηρ3+λz<0,θ2=ηρ3βr-κ2μ4+ρ2+λr-1+μ1μ4+ηλzβr-κ2+κ3μ4+ρ2+λr-1,θ1=ηκ2ρ3βrλr-1-μ1μ4+ηλzκ3+κ2βrκ3μ4-λr+1+κ3-βrρ2+λr,θ0=κ2κ3βrλr-1μ1μ4+ηλz.Clearly, RX∗=0 or(49)θ3RX∗3+θ2RX∗2+θ1RX∗+θ0=0.The state RX∗=0 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 (H∗∗,HX∗∗,R∗∗,0,0,S∗∗,0) exists. Alternatively, RX∗=0 could correspond to the disease-free equilibrium point (15) for system (1).The roots of the cubic equation (49) are given as(50)RX1∗=-κ3μ1μ4+ηλzηρ3+μ1μ4+ηλz<0,RX2,3∗=βrλr-κ2μ4βr+ρ2βr-1±Θ2μ4βr,where(51)Θ=4μ4βrκ2βrλr-κ2+βrλr-κ2μ4βr+ρ2βr-12.The root RX1∗<0 should be ignored, since all the model state variables are nonnegative for all time t≥0. This leaves RX2,3∗ 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 RX2,3∗ 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.


## 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 t≥0. 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 (H(t),HX(t),R(t),RX(t),Z(t),S(t),M(t)) exists for all the state variables with nonnegative initial conditions {H(0)=H0≥0, HX(0)=HX0≥0, R(0)=R0≥0, RX(0)=RX0≥0, Z(0)=Z0≥0, S(0)=S0≥0, M(0)=M0≥0}∀t≥0.Proof.
Considering the first equation in system (1), let ψ1(t)=ρ1HX/κ1+HX, so that(2)dHdt=λh+φt-μ1H-βsSH,dHdt≥-μ1+βsSH,which yields(3)Ht≥H0exp⁡-∫0tβsSsds+μ1t>0.In 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:(4)HXt≥HX0exp⁡-μ2t>0,Rt≥R0exp⁡-∫0tβrMsds+μ3t>0,RXt≥RX0exp⁡-∫0tηZsds+μ4t>0,Zt≥Z0exp⁡-δzt>0,St≥S0exp⁡-∫0tβsHsds+δst>0,Mt≥M0exp⁡-∫0tβrRsds+δmt>0.Therefore, state variables (H,HX,R,RX,Z,S,M) of model system (1) are nonnegative for all time t>0.


### 3.1.2. Invariant Region

LetNH(t) represent the total hepatocyte population, so that NH(t)=H(t)+HX(t).On substituting the derivatives in system (1) and simplifying, we have(5)dNHdt≤λh+ψ1t-μhNH,where ψ1(t)=ρ1HX/κ1+HX and μh=min⁡{μ1,μ2}.Using integrating factoreμht,(6)NHt≤λhμh+e-μht∫0tψ1τeμhτdτ+c1e-μht,where c1 is a constant of integration. By applying the initial condition NH(0)=NH0>0 in (6), we obtain(7)c1=NH0-λhμh-∫0tψ1τe-μhτdτ.Substituting the value of c1 into NH(t) in (6) and simplifying, we get(8)NHt≤λhμh+e-μhtNH0-λhμh.There are two possible cases in analyzing the behaviour of NH(t) in (8). In the first case, we consider NH(0)>λh/μh so that, at time t=0, the right-hand side (RHS) of (8) experiences the largest possible value of NH(0). That is, NH(t)≤NH(0) for all time t>0.In the second case, we considerNH(0)<λh/μh, so that the largest possible value of the RHS of (8) approaches λh/μh as time t goes to infinity. Thus, NH(t)≤λh/μh, ∀t>0. From these two cases, we conclude that NH(t)≤max⁡NH(0),λh/μh for all time t>0.Using the above approach, let the total red blood cells population beNR(t), so that NR(t)=R(t)+RX(t). From the model equations in system (1), we have(9)dNRdt≤λr+ψ2t-μrNRt,where ψ2(t)=ρ2RX/κ2+RX and μr=min⁡{μ3,μ4}. Upon solving for NR in (9), we have NR(t)≤max⁡NR(0),λr/μr, ∀t>0.For the macrophage compartmentZ(t), we have(10)dZdt=λz+ψ3t-δzZ,forψ3t=ρ3RXκ3+RX.By integration, the solution of (10) is presented as(11)Zt≤λzδz+e-δztZ0-λzδz.By inspection, Z(t)≤max⁡Z(0),λz/δz for all time t>0.Finally, letNP(t) represent the total population of malaria parasites at any time t. That is, NP(t)=S(t)+M(t) and, from system (1),(12)dNPdt=Λ-δsS-βsSH+Nμ2HX+Kζμ4RX-δmM,≤Λ+Kζμ4RX+Nμ2HX-δpNp,whereδp=min⁡δs,δm.Let (Kζμ4RX+Nμ2HX)=ψ4(t), so that on solving for NP(t) we get(13)Npt≤Λδp+e-δptNp0-ΛδpClearly, the malaria parasite populations S(t) and M(t) are bounded above. That is,N P ( t ) ≤ max ⁡ N P ( 0 ) , Λ / δ p for all time t>0.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 Ω∈R+7, where(14)Ω=H,HX,R,RX,Z,S,M∈R+7:NPt≤max⁡NP0,Λδp,NHt≤max⁡NH0,λhμh,NRt≤max⁡NR0,λrμr,Zt≤max⁡Z0,λzδz.Moreover, any solution of our system (1) which commences in Ω at any time t≥0 will always remain confined in that region. We therefore deduce that the region Ω 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,E0, is the state in which the human host is free of malaria infection. At E0, the sporozoite recruitment rate, Λ=0, and parasite and host-infected compartments have zero values; that is, S∗=M∗=RX∗=HX∗=0. Therefore,(15)E0=H∗,HX∗,R∗,RX∗,Z∗,S∗,M∗=λhμ1,0,λrμ3,0,λzδz,0,0.


### 3.1.4. In-Host Basic Reproduction Number

The in-host reproduction number of model (1) denoted by R0 is computed using the technique of the next-generation matrix approach described in [41]. We consider HX, RX, S, and M as the parasite infested compartments. Adopting the notations in [41], we generate a nonnegative matrix F of new infections and a nonsingular matrix V, showing the transfer of infections from one compartment to the other as follows:(16)F=00βsλhμ10000βrλrμ300000000and(17)V=μ20000μ4+ηλzδz0000δs+βsλhμ10-Nμ2-Kζμ40δm+βrλrμ3.The inverse of matrix V is hence given by (18)V-1=1μ20000δzηλz+δzμ400001δs+βsλh/μ10Nμ3βrλr+δmμ3Kζδzμ3μ4βrλr+δmμ3ηλz+δzμ401δm+βrλr/μ3.The next-generation matrix G, which is the product of matrices F and V-1, works out to be (19)G=00βsλhβsλh+δsμ10Nβrλrβrλr+δmμ3Kζβrδzλrμ4βrλr+δmμ3ηλz+δzμ40βrλrβrλr+δmμ300000000.The in-host basic reproduction number R0 is the spectral radius of the next-generation matrix G. It can clearly be seen that three of the four eigenvalues of matrix G in (19) have zero values; that is, λ1=λ2=λ3=0. The fourth and largest nonnegative eigenvalue λ4 becomes the in-host model reproduction number. We therefore have(20)R0=Kβrλrβrλr+δmμ3·ζδzμ4ηλz+δzμ4.The terms in model R0 can be interpreted as follows:(1)
The termKβrλr/(βrλr+δmμ3) represents the expected number of infectious merozoite parasites resulting from bursting blood schizonts at the blood stage of malaria infection.(2)
The second termζδzμ4/(ηλz+δzμ4) represents the expected proportion of merozoites that participate in the cycle of erythrocytic schizogony.(3)
Observe that the terms(βrλr)/(βrλr+δmμ3)<1 and (δzμ4)/(ηλz+δzμ4)<1. So our R0≤Kζ. This implies that the number of secondary infections during malaria infections is largely influenced by the average number of merozoites released K, 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,E0

The Jacobian matrix of model system (1) evaluated at the disease-free equilibrium E0 is given by(21)J1E0=-μ1ρ1κ1000-βsλhμ100-μ2000βsλhμ1000-μ3ρ2κ200-βrλrμ3000-ηλzδz-μ400βrλrμ3000ρ3κ3-δz0000000-βsλhμ1-δs00Nμ20Kζμ400-βrλrμ3-δm.It is clear from the first, third, and fifth columns of matrix (21) that the Jacobian matrix has negative eigenvalues λ1=-μ1,λ2=-μ3, and λ3=-δz. Upon deleting the first, third, and fifth rows and columns, matrix (21) is reduced to the following 4×4 matrix:(22)J2E0=-μ20βsλhμ100-ηλzδz-μ40βrλrμ300-βsλhμ1-δs0Nμ2Kζμ40-βrλrμ3-δm.From row three in (22), λ4=-βsλh/μ1-δs. We further reduce matrix (22) by deleting row three and column three. So,(23)J3E0=-μ2000-ηλzδz-μ4βrλrμ3Nμ2Kζμ4-βrλrμ3-δm.Note from row one of (23) that the fifth eigenvalue λ5=-μ2<0.The remaining two eigenvalues can be obtained by reducing matrix (23) into the following 2×2 matrix:(24)J6E0=-ηλzδz-μ4βrλrμ3Kζμ4-βrλrμ3-δm.Using the variable λ, the characteristic polynomial associated with matrix (24) is(25)pλ=λ2+Aλ+B,where(26)A=δm+ηλzδz+βrλrμ3+μ4and(27)B=ηδmλzδz+ηβrλrλzδzμ3+δmμ4+βrλrμ4μ3-Kζβrλrμ4μ3.The characteristic polynomial (25) has negative roots (eigenvalues) if A>0 and B>0. The coefficient A in (26) is clearly positive. We now need to show that B in (27) is strictly positive if R0<1. This is done by expressing the coefficient term B in terms of model R0 as follows:(28)B=1δzμ3μ4δz+ηλzβrλr+δmμ3-Kζβrδzλrμ4,=1δzμ3μ4δz+ηλzβrλr+δmμ31-Kζβrδzλrμ4μ4δz+ηλzβrλr+δmμ3,=μ4δz+ηλzβrλr+δmμ3δzμ31-R0.It can clearly be seen from (28) that the coefficient B is positive if and only if R0<1. We have thus established the following result.Theorem 2.
The disease-free equilibriumE0 is locally asymptotically stable in Ω if R0<1. If R0>1, then E0 is unstable.Biologically, Theorem2 implies that malaria infection can be eliminated from the human host when R0<1. To ensure that elimination of malaria is independent of the initial sizes of the subpopulations, it is necessary to show that E0 is globally asymptotically stable in Ω, 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 E0 is globally asymptotically stable when R0<1. We begin by rewriting the model system (1) in pseudotriangular form as follows:(29)X˙1=A1XX1-X1∗+A2XX2,X˙2=A3XX2,where X1 is the vector representing the state of different compartment of liver and blood cells that are not infected and do not transmit malaria infections. X2 represents the states of malaria parasites and host’s cells that are responsible for disease transmission. Hence, (30)X=X1,X2,X1=H,R,Z,X2=HX,RX,S,MandX1∗=λhμ1,λrμ3,λzδz.From the subsystem X1, we have(31)A1X=-μ1000-μ3000-δzandA2X=ρ1κ10-λhμ1βs00ρ2κ20-λrμ3βr0ρ3κ300.A direct computation indicates that the eigenvalue of matrix A1(X) is real and negative. This shows that the system X˙1=A1(X)(X1-X1∗)+A2(X)X2 is globally asymptotically stable at the disease-free equilibrium, E0. Similarly, the subsystem X2 gives rise to the following matrix A3(X):(32)A3X=-μ20βsλhμ100-ηλzδz+μ40βrλrμ300-βsλhμ1+δs0Nμ2Kζμ40-βrλrμ3+δm.It can clearly be seen that A3(X) is a Metzler matrix: all the off-diagonal elements of A3(X) are nonnegative. In order to establish the global stability of the disease-free equilibrium, we need to show that the matrix A3(X) is Metzler stable by providing a proof of the following lemma.Lemma 3.
LetM be a square Metzler matrix that is block decomposed:(33)M=ABCD,where A and D are square matrices. The matrix M is Metzler stable if and only if A and D-CA-1B are Metzler stable.In our case, matrixM is represented by matrix A3 in (32), so that(34)A=-μ200-ηλzδz+μ4,B=βsλhμ100βrλrμ3,C=00Nμ2Kζμ4andD=-βsλhμ1+δs00-βrλrμ3+δm.Upon computation in Mathematica software, we obtain(35)D-CA-1B=-ω10ω2-ω3,where ω1=δs+βsλh/μ1, ω2=Nβsλh/μ1, and ω3=δm+βrλr(ηλh+(1-Kζ)δzμ4)/μ3(ηλz+δzμ4).For the matrixD-CA-1B to be Metzler stable, ω3 should be strictly nonnegative. Therefore, the expression in the numerator(36)βrλrηλh+1-Kζδzμ4≥0.Upon simplification of (36),(37)Kζβrλrδzμ4≤βrλrηλh+δzμ4,(38)βrλr+δmμ3βrλrKζβrλrδzμ4βrλr+δmμ3ηλz+δzμ4≤1,(39)βrλr+δmμ3βrλrR0≤1(40)R0≤βrλrβrλr+δmμ3<1.Clearly, matrix A in (34) is Metzler stable. However, the matrix D-CA-1B is Metzler stable if and only if R0<1. From Lemma 3, we deduce the following theorem.Theorem 4.
The malaria-free equilibriumE0 of model system (1) is globally asymptotically stable if the threshold quantity R0<1.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 numberR0 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.


## 3.1.1. 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 t≥0. 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 (H(t),HX(t),R(t),RX(t),Z(t),S(t),M(t)) exists for all the state variables with nonnegative initial conditions {H(0)=H0≥0, HX(0)=HX0≥0, R(0)=R0≥0, RX(0)=RX0≥0, Z(0)=Z0≥0, S(0)=S0≥0, M(0)=M0≥0}∀t≥0.Proof.
Considering the first equation in system (1), let ψ1(t)=ρ1HX/κ1+HX, so that(2)dHdt=λh+φt-μ1H-βsSH,dHdt≥-μ1+βsSH,which yields(3)Ht≥H0exp⁡-∫0tβsSsds+μ1t>0.In 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:(4)HXt≥HX0exp⁡-μ2t>0,Rt≥R0exp⁡-∫0tβrMsds+μ3t>0,RXt≥RX0exp⁡-∫0tηZsds+μ4t>0,Zt≥Z0exp⁡-δzt>0,St≥S0exp⁡-∫0tβsHsds+δst>0,Mt≥M0exp⁡-∫0tβrRsds+δmt>0.Therefore, state variables (H,HX,R,RX,Z,S,M) of model system (1) are nonnegative for all time t>0.


## 3.1.2. Invariant Region

LetNH(t) represent the total hepatocyte population, so that NH(t)=H(t)+HX(t).On substituting the derivatives in system (1) and simplifying, we have(5)dNHdt≤λh+ψ1t-μhNH,where ψ1(t)=ρ1HX/κ1+HX and μh=min⁡{μ1,μ2}.Using integrating factoreμht,(6)NHt≤λhμh+e-μht∫0tψ1τeμhτdτ+c1e-μht,where c1 is a constant of integration. By applying the initial condition NH(0)=NH0>0 in (6), we obtain(7)c1=NH0-λhμh-∫0tψ1τe-μhτdτ.Substituting the value of c1 into NH(t) in (6) and simplifying, we get(8)NHt≤λhμh+e-μhtNH0-λhμh.There are two possible cases in analyzing the behaviour of NH(t) in (8). In the first case, we consider NH(0)>λh/μh so that, at time t=0, the right-hand side (RHS) of (8) experiences the largest possible value of NH(0). That is, NH(t)≤NH(0) for all time t>0.In the second case, we considerNH(0)<λh/μh, so that the largest possible value of the RHS of (8) approaches λh/μh as time t goes to infinity. Thus, NH(t)≤λh/μh, ∀t>0. From these two cases, we conclude that NH(t)≤max⁡NH(0),λh/μh for all time t>0.Using the above approach, let the total red blood cells population beNR(t), so that NR(t)=R(t)+RX(t). From the model equations in system (1), we have(9)dNRdt≤λr+ψ2t-μrNRt,where ψ2(t)=ρ2RX/κ2+RX and μr=min⁡{μ3,μ4}. Upon solving for NR in (9), we have NR(t)≤max⁡NR(0),λr/μr, ∀t>0.For the macrophage compartmentZ(t), we have(10)dZdt=λz+ψ3t-δzZ,forψ3t=ρ3RXκ3+RX.By integration, the solution of (10) is presented as(11)Zt≤λzδz+e-δztZ0-λzδz.By inspection, Z(t)≤max⁡Z(0),λz/δz for all time t>0.Finally, letNP(t) represent the total population of malaria parasites at any time t. That is, NP(t)=S(t)+M(t) and, from system (1),(12)dNPdt=Λ-δsS-βsSH+Nμ2HX+Kζμ4RX-δmM,≤Λ+Kζμ4RX+Nμ2HX-δpNp,whereδp=min⁡δs,δm.Let (Kζμ4RX+Nμ2HX)=ψ4(t), so that on solving for NP(t) we get(13)Npt≤Λδp+e-δptNp0-ΛδpClearly, the malaria parasite populations S(t) and M(t) are bounded above. That is,N P ( t ) ≤ max ⁡ N P ( 0 ) , Λ / δ p for all time t>0.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 Ω∈R+7, where(14)Ω=H,HX,R,RX,Z,S,M∈R+7:NPt≤max⁡NP0,Λδp,NHt≤max⁡NH0,λhμh,NRt≤max⁡NR0,λrμr,Zt≤max⁡Z0,λzδz.Moreover, any solution of our system (1) which commences in Ω at any time t≥0 will always remain confined in that region. We therefore deduce that the region Ω 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,E0, is the state in which the human host is free of malaria infection. At E0, the sporozoite recruitment rate, Λ=0, and parasite and host-infected compartments have zero values; that is, S∗=M∗=RX∗=HX∗=0. Therefore,(15)E0=H∗,HX∗,R∗,RX∗,Z∗,S∗,M∗=λhμ1,0,λrμ3,0,λzδz,0,0.


## 3.1.4. In-Host Basic Reproduction Number

The in-host reproduction number of model (1) denoted by R0 is computed using the technique of the next-generation matrix approach described in [41]. We consider HX, RX, S, and M as the parasite infested compartments. Adopting the notations in [41], we generate a nonnegative matrix F of new infections and a nonsingular matrix V, showing the transfer of infections from one compartment to the other as follows:(16)F=00βsλhμ10000βrλrμ300000000and(17)V=μ20000μ4+ηλzδz0000δs+βsλhμ10-Nμ2-Kζμ40δm+βrλrμ3.The inverse of matrix V is hence given by (18)V-1=1μ20000δzηλz+δzμ400001δs+βsλh/μ10Nμ3βrλr+δmμ3Kζδzμ3μ4βrλr+δmμ3ηλz+δzμ401δm+βrλr/μ3.The next-generation matrix G, which is the product of matrices F and V-1, works out to be (19)G=00βsλhβsλh+δsμ10Nβrλrβrλr+δmμ3Kζβrδzλrμ4βrλr+δmμ3ηλz+δzμ40βrλrβrλr+δmμ300000000.The in-host basic reproduction number R0 is the spectral radius of the next-generation matrix G. It can clearly be seen that three of the four eigenvalues of matrix G in (19) have zero values; that is, λ1=λ2=λ3=0. The fourth and largest nonnegative eigenvalue λ4 becomes the in-host model reproduction number. We therefore have(20)R0=Kβrλrβrλr+δmμ3·ζδzμ4ηλz+δzμ4.The terms in model R0 can be interpreted as follows:(1)
The termKβrλr/(βrλr+δmμ3) represents the expected number of infectious merozoite parasites resulting from bursting blood schizonts at the blood stage of malaria infection.(2)
The second termζδzμ4/(ηλz+δzμ4) represents the expected proportion of merozoites that participate in the cycle of erythrocytic schizogony.(3)
Observe that the terms(βrλr)/(βrλr+δmμ3)<1 and (δzμ4)/(ηλz+δzμ4)<1. So our R0≤Kζ. This implies that the number of secondary infections during malaria infections is largely influenced by the average number of merozoites released K, 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,E0

The Jacobian matrix of model system (1) evaluated at the disease-free equilibrium E0 is given by(21)J1E0=-μ1ρ1κ1000-βsλhμ100-μ2000βsλhμ1000-μ3ρ2κ200-βrλrμ3000-ηλzδz-μ400βrλrμ3000ρ3κ3-δz0000000-βsλhμ1-δs00Nμ20Kζμ400-βrλrμ3-δm.It is clear from the first, third, and fifth columns of matrix (21) that the Jacobian matrix has negative eigenvalues λ1=-μ1,λ2=-μ3, and λ3=-δz. Upon deleting the first, third, and fifth rows and columns, matrix (21) is reduced to the following 4×4 matrix:(22)J2E0=-μ20βsλhμ100-ηλzδz-μ40βrλrμ300-βsλhμ1-δs0Nμ2Kζμ40-βrλrμ3-δm.From row three in (22), λ4=-βsλh/μ1-δs. We further reduce matrix (22) by deleting row three and column three. So,(23)J3E0=-μ2000-ηλzδz-μ4βrλrμ3Nμ2Kζμ4-βrλrμ3-δm.Note from row one of (23) that the fifth eigenvalue λ5=-μ2<0.The remaining two eigenvalues can be obtained by reducing matrix (23) into the following 2×2 matrix:(24)J6E0=-ηλzδz-μ4βrλrμ3Kζμ4-βrλrμ3-δm.Using the variable λ, the characteristic polynomial associated with matrix (24) is(25)pλ=λ2+Aλ+B,where(26)A=δm+ηλzδz+βrλrμ3+μ4and(27)B=ηδmλzδz+ηβrλrλzδzμ3+δmμ4+βrλrμ4μ3-Kζβrλrμ4μ3.The characteristic polynomial (25) has negative roots (eigenvalues) if A>0 and B>0. The coefficient A in (26) is clearly positive. We now need to show that B in (27) is strictly positive if R0<1. This is done by expressing the coefficient term B in terms of model R0 as follows:(28)B=1δzμ3μ4δz+ηλzβrλr+δmμ3-Kζβrδzλrμ4,=1δzμ3μ4δz+ηλzβrλr+δmμ31-Kζβrδzλrμ4μ4δz+ηλzβrλr+δmμ3,=μ4δz+ηλzβrλr+δmμ3δzμ31-R0.It can clearly be seen from (28) that the coefficient B is positive if and only if R0<1. We have thus established the following result.Theorem 2.
The disease-free equilibriumE0 is locally asymptotically stable in Ω if R0<1. If R0>1, then E0 is unstable.Biologically, Theorem2 implies that malaria infection can be eliminated from the human host when R0<1. To ensure that elimination of malaria is independent of the initial sizes of the subpopulations, it is necessary to show that E0 is globally asymptotically stable in Ω, 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 E0 is globally asymptotically stable when R0<1. We begin by rewriting the model system (1) in pseudotriangular form as follows:(29)X˙1=A1XX1-X1∗+A2XX2,X˙2=A3XX2,where X1 is the vector representing the state of different compartment of liver and blood cells that are not infected and do not transmit malaria infections. X2 represents the states of malaria parasites and host’s cells that are responsible for disease transmission. Hence, (30)X=X1,X2,X1=H,R,Z,X2=HX,RX,S,MandX1∗=λhμ1,λrμ3,λzδz.From the subsystem X1, we have(31)A1X=-μ1000-μ3000-δzandA2X=ρ1κ10-λhμ1βs00ρ2κ20-λrμ3βr0ρ3κ300.A direct computation indicates that the eigenvalue of matrix A1(X) is real and negative. This shows that the system X˙1=A1(X)(X1-X1∗)+A2(X)X2 is globally asymptotically stable at the disease-free equilibrium, E0. Similarly, the subsystem X2 gives rise to the following matrix A3(X):(32)A3X=-μ20βsλhμ100-ηλzδz+μ40βrλrμ300-βsλhμ1+δs0Nμ2Kζμ40-βrλrμ3+δm.It can clearly be seen that A3(X) is a Metzler matrix: all the off-diagonal elements of A3(X) are nonnegative. In order to establish the global stability of the disease-free equilibrium, we need to show that the matrix A3(X) is Metzler stable by providing a proof of the following lemma.Lemma 3.
LetM be a square Metzler matrix that is block decomposed:(33)M=ABCD,where A and D are square matrices. The matrix M is Metzler stable if and only if A and D-CA-1B are Metzler stable.In our case, matrixM is represented by matrix A3 in (32), so that(34)A=-μ200-ηλzδz+μ4,B=βsλhμ100βrλrμ3,C=00Nμ2Kζμ4andD=-βsλhμ1+δs00-βrλrμ3+δm.Upon computation in Mathematica software, we obtain(35)D-CA-1B=-ω10ω2-ω3,where ω1=δs+βsλh/μ1, ω2=Nβsλh/μ1, and ω3=δm+βrλr(ηλh+(1-Kζ)δzμ4)/μ3(ηλz+δzμ4).For the matrixD-CA-1B to be Metzler stable, ω3 should be strictly nonnegative. Therefore, the expression in the numerator(36)βrλrηλh+1-Kζδzμ4≥0.Upon simplification of (36),(37)Kζβrλrδzμ4≤βrλrηλh+δzμ4,(38)βrλr+δmμ3βrλrKζβrλrδzμ4βrλr+δmμ3ηλz+δzμ4≤1,(39)βrλr+δmμ3βrλrR0≤1(40)R0≤βrλrβrλr+δmμ3<1.Clearly, matrix A in (34) is Metzler stable. However, the matrix D-CA-1B is Metzler stable if and only if R0<1. From Lemma 3, we deduce the following theorem.Theorem 4.
The malaria-free equilibriumE0 of model system (1) is globally asymptotically stable if the threshold quantity R0<1.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 numberR0 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

WhenR0>1, 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 R, H, Z, S, and M in terms of the infected states HX and RX, we obtain the endemic state E1=(H∗,HX∗,R∗,RX∗,Z∗,S∗,M∗), where(41)H∗=1μ1λh+ρ1HX∗κ1+HX∗-μ2HX∗,S∗=μ1μ2HX∗βsλh+ρ1HX∗/κ1+HX∗-μ2HX∗,(42)R∗=1μ3λr+ρ2RX∗κ2+RX∗-μ4RX∗,M∗=μ3RX∗μ4+ηRX∗ρ3RX∗/κ3+RX∗+λzμ1(43)Z∗=1δzλz+ρ3RX∗κ3+RX∗Substituting (41) into the 2nd equation in (1) and simplifying, we obtain the following cubic equation:(44)α3HX∗3+α2HX∗2+α1HX∗+α0=0,where(45)α3=μ22βs>0,α2=μ2μ1-δs-βsλh-κ1μ2+Λ+ρ1,α1=βsλhΛ-κ1μ2+Λρ1-κ1μ2-κ1μ1μ2δs,andα0=κ1Λλhβs>0.The number and nature of the roots of (44) are determined by the following discriminant:(46)Δ=18α3α2α1α0-4α23α0+α22α12-4α3α13-27α32α02.So(i)
ifΔ=0, then (44) has multiple real roots and only one endemic equilibrium would exist,(ii)
ifΔ<0, then (44) has 1 real root and a complex conjugate root and hence only one endemic equilibrium,(iii)
ifΔ>0, then (44) has 3 distinct real roots and so there is more than one endemic equilibrium when R0>1 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 RX at equilibrium by substituting expressions in (42) and (43) into the 4th equation in (1). After simplification in Mathematica software, we obtain the following cubic equation:(47)RX∗θ3RX∗3+θ2RX∗2+θ1RX∗+θ0=0,where (48)θ3=-μ4βrμ1μ4+ηρ3+λz<0,θ2=ηρ3βr-κ2μ4+ρ2+λr-1+μ1μ4+ηλzβr-κ2+κ3μ4+ρ2+λr-1,θ1=ηκ2ρ3βrλr-1-μ1μ4+ηλzκ3+κ2βrκ3μ4-λr+1+κ3-βrρ2+λr,θ0=κ2κ3βrλr-1μ1μ4+ηλz.Clearly, RX∗=0 or(49)θ3RX∗3+θ2RX∗2+θ1RX∗+θ0=0.The state RX∗=0 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 (H∗∗,HX∗∗,R∗∗,0,0,S∗∗,0) exists. Alternatively, RX∗=0 could correspond to the disease-free equilibrium point (15) for system (1).The roots of the cubic equation (49) are given as(50)RX1∗=-κ3μ1μ4+ηλzηρ3+μ1μ4+ηλz<0,RX2,3∗=βrλr-κ2μ4βr+ρ2βr-1±Θ2μ4βr,where(51)Θ=4μ4βrκ2βrλr-κ2+βrλr-κ2μ4βr+ρ2βr-12.The root RX1∗<0 should be ignored, since all the model state variables are nonnegative for all time t≥0. This leaves RX2,3∗ 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 RX2,3∗ 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, R0.


### 4.1. Sensitivity Analysis

In epidemic modelling, sensitivity analysis is performed to investigate model parameters with significant influence onR0 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, α, is defined as(52)Υα△=∂△∂α×α△.Using the formulation in (52) and the parameter values in Table 3, the local sensitivity indices (SI) of R0 (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 Table2 for detailed parameter descriptions.
Symbol Interpretation Value Source δz Death rate of macrophages 0.05/day [19] δs Death rate of sporozoites 1.2 × 1 0 - 11 /day [20] η Elimination rate of IRBCs by macrophages 10-10cellsμl-1//day [19] λh Recruitment rate ofH 2.5×108cells/μl-1/day [21] ρ1 Production rate ofH due to HX 2.5×10-5/day [19] μ1 Death rate ofH 0.029 /day [20] Λ Rate of injection of sporozoites 20 sporozoites/day [20] ρ2 Production rate of RBCs due to IRBCs 2.5×10-5/day [19] βs Hepatocyte invasion rate 1.0×10-6/sporozoites/day [20] ρ3 Immunogenicity of IRBCs 2.5×10-5/day [19] μ2 Death rate of infected hepatocytes 0.02/day [20] κ1 Inhibition rate 1cells/μl-1 [22] λr Recruitment rate of RBCs 2.5×108cells/μl-1/day [23] ζ Merozoites that cause secondary infections 0.726 (unitless) [24] μ3 Death rate of healthy RBCs 0.0083/day [23] κ2 Inhibition rate 1cells/μl-1 [22] βr Invasion rate of RBCs 2.0×10-9/merozoites/day [23] N Merozoites per liver schizont 10000/day [21] μ4 Death rate of infected RBCs 0.025/day [19] κ3 Inhibition rate 1cells/μl-1 [22] δm Death rate of merozoites 48/day [23] K Merozoites per blood schizont 16 [21] λz Recruitment rate of macrophages 30/μl-1/day [19]Table 4
Sensitivity indices ofR0 relative to the model parameters.
Parameter SI Parameter SI K +1.0000 ζ +1.0000 βr +0.920422 μ 3 -0.920422 λr +0.920422 λ z -0.998585 μ4 +0.998585 η -0.998585 δz +0.998585 δ m -0.920422A positive sign on the SI indicates that an increase (decrease) in the value of such a parameter increases (decreases) the value ofR0 and hence the growth of malaria infection. On the other hand, a negative sign is indicative of a parameter that negatively affects R0. In order to eliminate in-host malaria infection, the in-host reproduction number should be less than one, that is, R0<1.The average number of merozoites released per bursting infected erythrocyteK and the proportion of merozoites that cause secondary invasions at the blood phase ζ 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) ζ or K generates a 10% increase (decrease) on R0 and hence malaria infection severity.The parametersλz, η, μ4, and δz occupy the second rank in influencing the model outcomes. An increase in the parameters μ4 and δz is likely to increase the model R0. On the other hand, an increase in λz and η has a direct negative influence on R0. Macrophages are highly instrumental in malaria parasite clearance and should be preserved.The rate of generation of macrophages from the bone marrow,λz, together with the rate of phagocytosis of infected red blood cells, η, is likely to decrease, proportionally, the disease progression when they are increased. With increased λz, 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 parametersβr and λr increase (or decrease) R0 when they are increased (or decreased). Epidemiologically, an improved erythrocyte invasion rate, βr, is likely to generate even more new blood schizonts. This increases parasitemia in the host. A 10% increase (decrease) in βr would increase (decrease) the threshold parameter R0 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 modelR0. 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 R0. 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 βr.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 modelR0 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 R0 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 modelR0 generated using parameter values in Table 3. Parameters with PRCC>0 and PRCC<0 increase and decrease model R0, respectively.Unlike the results in Table4, the model parameter with the highest influence on R0 according to the PRCCs results shown in Figure 2 is the rate of invasion of red blood cells by merozoites, βr. This is followed closely by the recruitment rate of susceptible red blood cells λr from the bone marrow. The second set of parameters that also increase (decrease) model R0 when they are increased (decreased) are ζ, K, μ3, and δz, respectively.The merozoites’ death rateδm, the death rate of IRBCs μ4, and the rate of elimination of IRBCs by macrophages η are shown to have the highest negative influence on disease progression. Although an increase in μ4 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 μ4 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.ForR0<1 (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 E0 (see Figure 3(a)). As the sporozoites decline to near zero (see Figure 3(b)), infected hepatocytes decline and stabilize at E0 in (15).Figure 3
Graphs showing the simulation of in-host malaria model (1) when the model R0=0.22866<1. Figures (a) and (b) show model dynamics at the liver stage. The chosen initial conditions are H0=300000, HX0=20000, R0=500000, RX0=50, Z0=300000, S0=2000, and M=70. Used parameter values are given in Table 3.
(a)
Hepatocyte dynamics (b)
Sporozoite dynamics (c)
Blood stage dynamicsAt the blood stage, the rising density of infected erythrocytes declines in a similar fashion to that of the infective merozoites whenR0<1 (see Figure 3(c)). The densities of the infected erythrocytes and merozoites approach E0 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).WhenR0>1, 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 whenR0=1.58690>1. Used parameter values can be found in Table 3 with the chosen initial conditions described by H0=300000, HX0=10, R0=500000, RX0=10, Z0=10, S0=2000, and M=20.
(a)
Susceptible hepatocytes (b)
Infected hepatocytes (c)
SporozoitesMalaria infection dynamics are most rapid in the first 2 weeks within the host liver as illustrated in Figures4(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 whenR0=1.58690>1. Simulation parameter values are available in Table 3.
(a)
Susceptible erythrocytes (b)
Infected erythrocytes (c)
Merozoites (d)
MacrophagesAn early sharp rise in the density of merozoites in the first one week of the blood stage is noted in Figure5(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 Figure5(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) ifR0<1, low level malaria infection can easily be contained by the host’s defence mechanism and loss of life is less likely; (2) therapeutically, R0<1 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 δm and βr 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 merozoitesδm from 20 to 80 in steps of 20, while keeping the other parameters (see Table 3) constant.
(a)
Effect ofδm on R (b)
Effect ofδm on RXFigure 7
Graphs showing the behaviour of (a) susceptible RBCs and (b) infected RBCs. They were obtained by varying the merozoite invasion rateβr from 2×10-8 to 2×10-5 in steps of 10-1, while keeping the other parameters in Table 3 constant.
(a)
Effect ofβr on R (b)
Effect ofβr on RXObserve that increased death rate of malaria merozoitesδm 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 Figure7(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 ofPlasmodium 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 RX and the death rate of the macrophages δz. 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,η (in (a)), and the effect of increased decay rate of macrophages, δz (in (b)), on the behaviour of infected erythrocytes RX. All parameter values are in Table 3.
(a)
Effect ofη on RX (b)
Effect ofδz on RXLike 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.


## 4.1. Sensitivity Analysis

In epidemic modelling, sensitivity analysis is performed to investigate model parameters with significant influence onR0 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, α, is defined as(52)Υα△=∂△∂α×α△.Using the formulation in (52) and the parameter values in Table 3, the local sensitivity indices (SI) of R0 (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 Table2 for detailed parameter descriptions.
Symbol Interpretation Value Source δz Death rate of macrophages 0.05/day [19] δs Death rate of sporozoites 1.2 × 1 0 - 11 /day [20] η Elimination rate of IRBCs by macrophages 10-10cellsμl-1//day [19] λh Recruitment rate ofH 2.5×108cells/μl-1/day [21] ρ1 Production rate ofH due to HX 2.5×10-5/day [19] μ1 Death rate ofH 0.029 /day [20] Λ Rate of injection of sporozoites 20 sporozoites/day [20] ρ2 Production rate of RBCs due to IRBCs 2.5×10-5/day [19] βs Hepatocyte invasion rate 1.0×10-6/sporozoites/day [20] ρ3 Immunogenicity of IRBCs 2.5×10-5/day [19] μ2 Death rate of infected hepatocytes 0.02/day [20] κ1 Inhibition rate 1cells/μl-1 [22] λr Recruitment rate of RBCs 2.5×108cells/μl-1/day [23] ζ Merozoites that cause secondary infections 0.726 (unitless) [24] μ3 Death rate of healthy RBCs 0.0083/day [23] κ2 Inhibition rate 1cells/μl-1 [22] βr Invasion rate of RBCs 2.0×10-9/merozoites/day [23] N Merozoites per liver schizont 10000/day [21] μ4 Death rate of infected RBCs 0.025/day [19] κ3 Inhibition rate 1cells/μl-1 [22] δm Death rate of merozoites 48/day [23] K Merozoites per blood schizont 16 [21] λz Recruitment rate of macrophages 30/μl-1/day [19]Table 4
Sensitivity indices ofR0 relative to the model parameters.
Parameter SI Parameter SI K +1.0000 ζ +1.0000 βr +0.920422 μ 3 -0.920422 λr +0.920422 λ z -0.998585 μ4 +0.998585 η -0.998585 δz +0.998585 δ m -0.920422A positive sign on the SI indicates that an increase (decrease) in the value of such a parameter increases (decreases) the value ofR0 and hence the growth of malaria infection. On the other hand, a negative sign is indicative of a parameter that negatively affects R0. In order to eliminate in-host malaria infection, the in-host reproduction number should be less than one, that is, R0<1.The average number of merozoites released per bursting infected erythrocyteK and the proportion of merozoites that cause secondary invasions at the blood phase ζ 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) ζ or K generates a 10% increase (decrease) on R0 and hence malaria infection severity.The parametersλz, η, μ4, and δz occupy the second rank in influencing the model outcomes. An increase in the parameters μ4 and δz is likely to increase the model R0. On the other hand, an increase in λz and η has a direct negative influence on R0. Macrophages are highly instrumental in malaria parasite clearance and should be preserved.The rate of generation of macrophages from the bone marrow,λz, together with the rate of phagocytosis of infected red blood cells, η, is likely to decrease, proportionally, the disease progression when they are increased. With increased λz, 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 parametersβr and λr increase (or decrease) R0 when they are increased (or decreased). Epidemiologically, an improved erythrocyte invasion rate, βr, is likely to generate even more new blood schizonts. This increases parasitemia in the host. A 10% increase (decrease) in βr would increase (decrease) the threshold parameter R0 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 modelR0. 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 R0. 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 βr.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 modelR0 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 R0 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 modelR0 generated using parameter values in Table 3. Parameters with PRCC>0 and PRCC<0 increase and decrease model R0, respectively.Unlike the results in Table4, the model parameter with the highest influence on R0 according to the PRCCs results shown in Figure 2 is the rate of invasion of red blood cells by merozoites, βr. This is followed closely by the recruitment rate of susceptible red blood cells λr from the bone marrow. The second set of parameters that also increase (decrease) model R0 when they are increased (decreased) are ζ, K, μ3, and δz, respectively.The merozoites’ death rateδm, the death rate of IRBCs μ4, and the rate of elimination of IRBCs by macrophages η are shown to have the highest negative influence on disease progression. Although an increase in μ4 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 μ4 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.


## 4.1.1. 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 R0 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 modelR0 generated using parameter values in Table 3. Parameters with PRCC>0 and PRCC<0 increase and decrease model R0, respectively.Unlike the results in Table4, the model parameter with the highest influence on R0 according to the PRCCs results shown in Figure 2 is the rate of invasion of red blood cells by merozoites, βr. This is followed closely by the recruitment rate of susceptible red blood cells λr from the bone marrow. The second set of parameters that also increase (decrease) model R0 when they are increased (decreased) are ζ, K, μ3, and δz, respectively.The merozoites’ death rateδm, the death rate of IRBCs μ4, and the rate of elimination of IRBCs by macrophages η are shown to have the highest negative influence on disease progression. Although an increase in μ4 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 μ4 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.ForR0<1 (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 E0 (see Figure 3(a)). As the sporozoites decline to near zero (see Figure 3(b)), infected hepatocytes decline and stabilize at E0 in (15).Figure 3
Graphs showing the simulation of in-host malaria model (1) when the model R0=0.22866<1. Figures (a) and (b) show model dynamics at the liver stage. The chosen initial conditions are H0=300000, HX0=20000, R0=500000, RX0=50, Z0=300000, S0=2000, and M=70. Used parameter values are given in Table 3.
(a)
Hepatocyte dynamics (b)
Sporozoite dynamics (c)
Blood stage dynamicsAt the blood stage, the rising density of infected erythrocytes declines in a similar fashion to that of the infective merozoites whenR0<1 (see Figure 3(c)). The densities of the infected erythrocytes and merozoites approach E0 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).WhenR0>1, 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 whenR0=1.58690>1. Used parameter values can be found in Table 3 with the chosen initial conditions described by H0=300000, HX0=10, R0=500000, RX0=10, Z0=10, S0=2000, and M=20.
(a)
Susceptible hepatocytes (b)
Infected hepatocytes (c)
SporozoitesMalaria infection dynamics are most rapid in the first 2 weeks within the host liver as illustrated in Figures4(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 whenR0=1.58690>1. Simulation parameter values are available in Table 3.
(a)
Susceptible erythrocytes (b)
Infected erythrocytes (c)
Merozoites (d)
MacrophagesAn early sharp rise in the density of merozoites in the first one week of the blood stage is noted in Figure5(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 Figure5(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) ifR0<1, low level malaria infection can easily be contained by the host’s defence mechanism and loss of life is less likely; (2) therapeutically, R0<1 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 δm and βr 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 merozoitesδm from 20 to 80 in steps of 20, while keeping the other parameters (see Table 3) constant.
(a)
Effect ofδm on R (b)
Effect ofδm on RXFigure 7
Graphs showing the behaviour of (a) susceptible RBCs and (b) infected RBCs. They were obtained by varying the merozoite invasion rateβr from 2×10-8 to 2×10-5 in steps of 10-1, while keeping the other parameters in Table 3 constant.
(a)
Effect ofβr on R (b)
Effect ofβr on RXObserve that increased death rate of malaria merozoitesδm 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 Figure7(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 ofPlasmodium 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 RX and the death rate of the macrophages δz. 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,η (in (a)), and the effect of increased decay rate of macrophages, δz (in (b)), on the behaviour of infected erythrocytes RX. All parameter values are in Table 3.
(a)
Effect ofη on RX (b)
Effect ofδz on RXLike 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 regionΩ. 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 R0 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.



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*Source: 7019868-2018-07-02.xml*