The emergence of parasite resistance to antimalarial drugs has contributed significantly to global human mortality and morbidity due to malaria infection. The impacts of multiple-strain malarial parasite infection have further generated a lot of scientific interest. In this paper, we demonstrate, using the epidemiological model, the effects of parasite resistance and competition between the strains on the dynamics and control of Plasmodium falciparum malaria. The analysed model has a trivial equilibrium point which is locally asymptotically stable when the parasite’s effective reproduction number is less than unity. Using contour plots, we observed that the efficacy of antimalarial drugs used, the rate of development of resistance, and the rate of infection by merozoites are the most important parameters in the multiple-strain P. falciparum infection and control model. Although the drug-resistant strain is shown to be less fit, the presence of both strains in the human host has a huge impact on the cost and success of antimalarial treatment. To reduce the emergence of resistant strains, it is vital that only effective antimalarial drugs are administered to patients in hospitals, especially in malaria-endemic regions. Our results emphasize the call for regular and strict surveillance on the use and distribution of antimalarial drugs in health facilities in malaria-endemic countries.
Deutscher Akademischer AustauschdienstST32-PKZ:91711149Kenya National Research Fund1. Introduction
The emergence of parasite resistance [1–4] to antimalarial drugs has contributed significantly to human mortality and morbidity due to malaria infection, worldwide [5–7]. A global malaria control strategy of 1992 [8] that advocated for early diagnosis and prompt treatment has been heavily compromised by the emergence of parasite resistance to antimalarial drugs. The evolution of parasite resistance has been described in [9] as an example of a Darwinian evolution. Parasites undergo mutations in their genome in response to the drug-treated human host. These mutations reduce the rate of parasite elimination from the host and increase their survival chances [9]. The most extensively used antimalarial drugs against the deadly Plasmodium falciparum malaria are chloroquine (CQ) and sulfadoxine-pyrimethamine (SP) [10, 11]. These drugs are cheap, easily available, and slowly eliminated from the human body [11]. However, the extensive use of CQ and SP has resulted in P. falciparum resistance. This has led to global increase in malaria cases and mortality [12]. In response, the World Health Organization (WHO) in 2006 recommended the use of artemisinin-based combination therapies (ACTs) as a first-line treatment for uncomplicated P. falciparum malaria [13]. Resistance to ACTs which are currently the standard treatment for P. falciparum is likely to cause global health crisis especially in African regions where P. falciparum malaria is endemic [11].
The emergence of parasite resistance to malaria therapy dates back to the 19th century. Quinine (1963) was the first-line antimalarial drug against P. falciparum [14]. High mortality cases coupled with high parasite resistance led to the introduction of a second drug, chloroquine (CQ), in 1934 [15]. A decade later, CQ was considered the first-line antimalarial drug by several countries until 1957, when the first focus of P. falciparum resistance was detected along the Thai-Cambodia border [16]. In Africa, P. falciparum resistance to CQ was first discovered among travelers from Kenya to Tanzania [17]. By 1983, CQ resistance had spread to Sudan, Uganda [18], Zambia [19], and Malawi [20]. Unlike Africa, CQ was replaced for the first time with sulfadoxine-pyrimethamine (SP) as a first-line antimalarial drug in Thailand in 1967. Several other countries in Asia and South America followed thereafter [10]. Resistance to SP was, however, reported the same year [21] in the region. In 1988, CQ was replaced for the first time in Africa. KwaZulu-Natal Province of South Africa replaced CQ with SP [22]. In 1993, the Malawian government changed the treatment policy from CQ to SP. Other African countries followed thereafter: Kenya, South Africa, and Botswana (in 1998); Cameroon and Tanzania (in 2001); and Zimbabwe (in 2000) [23]. The effectiveness of SP was equally undermined by resistance. Unlike CQ, P. falciparum resistance to SP was mainly attributed to the long half-life of the drug [24]. Confirmed resistance to the artemisinin derivatives was first reported in Cambodia and Mekong regions in 2008 [25].
To leverage on parasite resistance, cost of treatment, and burden of malaria infection to communities and governments, the WHO recommends the use of artemisinin-based combination therapies (ACTs) as the first- and second-line treatment drugs for uncomplicated P. falciparum malaria [25]. ACT is a combination of artemisinin derivatives and a partner monotherapy drug. Artemisinin derivatives include artemether, artesunate, and dihydroartemisinin. These derivatives reduce the parasite biomass within the first three days of therapy, while the partner drug, with longer half-life, eliminates the remaining parasites [26]. The WHO currently recommends five different ACTs: (1) artesunate-amodiaquine (AS + AQ), (2) artesunate-mefloquine (AS + MQ), (3) artesunate + sulfadoxine-pyrimethamine (AS + SP), (4) artemether-lumefantrine (AM-LM), and (5) dihydroartemisinin-piperaquine (DHA + PPQ). Additionally, artesunate-pyronaridine may be used in regions where ACT treatment response is low [26]. Access to ACT has been tremendous in the last 8 years, with a recorded increase of 122 million procured treatment courses for the period 2010–2016. However, resistance to currently used ACTs has important public health consequences, especially in the African region, where resistant P. falciparum is predominant.
Numerous cross-sectional studies [27, 28] have revealed the possible impacts of multiple strains of P. falciparum on the development of resistance to ACTs. In [29] and citations therein, drug-sensitive parasites are shown to strongly suppress the growth and transmission of drug-resistant P. falciparum parasites. Although high-transmission settings such as sub-Saharan Africa account for about 90% of all global malaria deaths, resistance to antimalarial drugs has been shown to emerge from low-transmission settings, such as Southeast Asia and South America [29]. Causes of parasite resistance to ACTs are diverse. Historical studies [30, 31] indicate that antimalarial-resistant parasites could emerge from a handful of lineages. It is argued elsewhere [32, 33] that recombination during sexual reproduction in the mosquito vector could be responsible for the delayed appearance of multilocus resistance in high-transmission regions. Moreover, owing to repeated exposure for many years, individuals in high-transmission settings are likely to develop clinical immunity to malaria, leading to stronger selection for resistance [34]. Studies in [29] also support the hypothesis that in-host competition between drug-sensitive and drug-resistant parasites could inhibit the spread of resistance in high-transmission settings. Owing to their integral role in the recent success of global malaria control, the protection of efficacy of ACTs should be a global health priority [35].
Mathematical models of in-host malaria epidemiology and control constitute important tools in guiding strategies for malaria control [36, 37] and the associated financial planning [38]. While some researchers have focussed on probabilistic models [39, 40], others have investigated the effects of drug treatment and resistance development using dynamic models [41, 42]. A deterministic model by Esteva et al. [43] monitored the impact of drug resistance on the transmission dynamics of malaria in a human population. In [29], the impacts of within-host parasite competition are shown to inhibit the spread of resistance [44, 45]. On the contrary, some models [39, 46] have suggested that within-host competition is likely to speed up the spread of resistance in high-transmission settings due to a phenomenon called “competitive release.” In this paper, we provide theoretical insights using mathematical modelling of the impacts of multiple-strain infections on resistance, dynamics, and antimalarial control of P. falciparum malaria.
The rest of the paper is organized as follows: In Section 2, we formulate the within-human malaria model that has both the drug-sensitive and drug-resistant P. falciparum parasite strains subject to antimalarial therapy. In Section 3, we analyze the model based on epidemiological theorems. Within-host competition between parasite strains and the effects of antimalarial drug efficacy on parasite clearance are discussed in Section 4. Sensitivity analysis and multiple-strain infection and its effects on resistance and malaria dynamics are demonstrated in Section 5. We conclude the paper in Section 6 by emphasizing the need for antimalarial therapy with the potential to eradicate multiple-strain infection due to P. falciparum.
2. Model Formulation
We present in this paper a deterministic model that describes the within-human-host competition and transmission dynamics of two strains of P. falciparum parasites during malaria infection. The compartmental model considers the coinfection and competition between the drug-sensitive (dss) and the drug-resistant (drs) P. falciparum strains in the presence of antimalarial therapy. The drs arise presumably from the dss. The rare mechanism here could possibly be due to single point mutation [47]. Both drs and dss initiate immune responses that follow density-dependent kinetics.
Our model is composed of eight compartments: susceptible/healthy/unparasitized erythrocytes (red blood cells) Xt, parasitized/infected erythrocytes (Yrt and Yst), merozoites (Mst and Mrt), gametocytes (Gst and Grt), and immune cells Wt. The healthy erythrocytes (RBCs) make up the resource for competition between the drug-resistant and drug-sensitive parasite strains. The infected red blood cells (IRBCs) and different erythrocytic parasite life cycles are categorized based on the strain of the infecting parasite. The merozoites are therefore categorized into drug-sensitive and drug-resistant strains, denoted by Mst and Mrt, respectively. The merozoites invade the healthy erythrocytes during the erythrocytic stage, leading to formation of infected erythrocytes. The variable Yst denotes the red blood cells (RBCs) infected with drug-sensitive merozoites, whereas Yrt refers to the RBCs infected with drug-resistant merozoites. Similarly, the variables Gst and Grt represent drug-sensitive and drug-resistant gametocytes, respectively. Owing to saturation in cell and parasite growth, we consider the nonlinear Michaelis–Mented–Monod function described in [48, 49] and used in [50–53] to model the reductive effects of the immune cells on the parasite and infected-cell populations.
The density of the healthy RBCs is increased at the rate λx per healthy RBC per unit time from the host’s bone marrow, and healthy RBCs die naturally at a rate μx. Following parasite invasion by free floating merozoites, the healthy erythrocytes get infected by both drug-sensitive and drug-resistant merozoite strains at the rates β and δrβ, respectively. The parameter δr (with 0<δr<1) accounts for the reduced fitness (infectiousness) of the resistant parasite strains in relation to the drug-sensitive strains. The destruction of the healthy red blood cells is however limited by the adaptive immune cells W. This is represented by the term 1/1+γW, where γ is a measure of the efficacy of the immune cells. The equation that governs the evolution of the healthy RBCs is hence given by(1)dXdt=λx−μxX−βX1+γWMs+δrMr.
The parasitized erythrocytes are generated through mass action contact (invasion) between the susceptible healthy erythrocytes X and the blood floating merozoites (Mr and Ms). The merozoites subdivide mitotically, within the infected erythrocytes, into thousands of other merozoites, leading to cell burst and emergence of characteristic symptoms of malaria. Additionally, a single infected erythrocyte undergoes hemolysis at the rate μys to produce P secondary merozoites, sustaining the erythrocytic cycle. The drug-sensitive IRBCs Ys burst open to generate more drug-sensitive merozoites or drug-sensitive gametocytes at the rate σs. Similar dynamics are observed with the drug-resistant IRBCs, where the drug-resistant gametocytes are generated at the rate σr from IRBCs. Treatment with ACT is assumed to disfranchise the development of the merozoite within the infected erythrocyte. The drug-infested erythrocytes are hence likely to die faster. This is represented by the term 1−ωs−1, where 0<ωs<1 represents the antimalarial-specific treatment efficacy. In this paper and for purposes of illustration and simulations, ωs corresponds to the efficacy of artemether-lumefantrine (AL), which is the recommended first-line antimalarial ACT drug for P. falciparum infection in Kenya. We assume that no treatment is available for erythrocytes infected with the resistant parasite strains. The time rate of change for Ys and Yr takes the following form:(2)dYsdt=βXMs1+γW−kyYsW1+aYs−11−ωsμysYs−σsYs,dYrdt=δrβXMr1+γW−kyYrW1+aYr−μyrYr−σrYr.
The drug-resistant merozoites Mr and the drug-resistant gametocytes Gr die naturally at the rates μmr and μgr, respectively. It is further assumed that drug-sensitive merozoites Ms and gametocytes Gs may develop into drug-resistant merozoites Mr and gametocytes Gr at the rates Ψ1 and Ψ2, respectively. The cost of resistance associated with AL is represented by the parameter αs. Parasite resistance to antimalarial drugs exacerbates the erythrocytic cycle and increases the cost of treatment [54, 55]. The higher the resistance to antimalarial therapy, the higher the density of malarial parasites in blood. We therefore model this decline in drug effectiveness by rescaling the density of merozoites produced per bursting parasitized erythrocyte P by the factor 1−αs, where αs=1 implies no resistance; that is, the ACT is highly effective in eradicating the parasites. If αs=0 corresponds to maximum resistance, the used ACT drug is least effective in treating the infection. The converse of these descriptions applies to the drug-resistant P. falciparum parasite strains. The equations that govern the rate of change of the infected red blood cells and the merozoites take the following form:(3)dMsdt=1−αsPμysYs−βMsX1+γW−kmMsW1+aMs−Ψ1+μms+ζMs,dMrdt=1−αrPμyrYr+Ψ1Ms−δrβMrX1+γW−kmMrW1+aMr−μmrMr,dGsdt=σsYs−kgWGs1+aGs−Ψ2+μgs+ηGs,dGrdt=σrYr+Ψ2Gs−kgWGr1+aGr−μgrGr.
Antimalarial therapy increases the rate of elimination of drug-sensitive merozoites and gametocytes. This is represented by the nonnegative enhancement parameters ζ and η, respectively.
Although the innate immunity is faster, it is often limited by the on and off rates in its response to invading pathogens [56, 57]. The adaptive immunity, on the contrary, is very slower at the beginning but lasts long enough to ensure no parasite growth in subsequent infections [27]. We assume an immune system that is independent of the invading parasite strain. For purposes of simplicity, we only consider the adaptive immune system, which is mainly composed of the CD8+T cells [58]. We adopt the assumption that the background recruitment of immune cells is constant (at the rate λw). Additionally, the production of the immune cells is assumed to be boosted by the infective and infected cells Gr,Gs, Mr,Ms, and Yr,Ys at constant rates hg, hm, and hy, respectively. Circulating gametocytes, infective merozoites, and infected erythrocytes are removed phagocytotically by the immune cells at the rates kgW, kmW, and kyW, respectively. The immune cells also get depleted through natural death at the rate μw. The equation for the immune cells takes the following form:(4)dWdt=λw+hgGs+GrGs+Gr+eg+hyYs+YrYs+Yr+ey+hmMs+MrMs+Mr+emW−μwW.
Following invasion by the merozoites, the IRBCs either produce merozoites or differentiate into gametocytes upon bursting. The total erythrocyte population at any time t, denoted by Ct, is therefore given by(5)Ct=Xt+Yst+Yrt.
Similarly, the sum total of P. falciparum parasites, denoted by Pt, within the host at any time t is described by the following equation:(6)Pt=Mst+Mrt+Gst+Grt.
The above dynamics can be represented by the schematic diagram in Figure 1. The list of model variables and model parameters is provided in Tables 1 and 2, respectively.
A model flow diagram. Drug-sensitive variables are shown in green colours while the drug-resistant variables are indicated in orange colours. Non-strain-specific variables like susceptible RBCs and immune cells are shown in blue colour. Solid lines indicate the movement of populations from one compartment to another. Dotted lines show possible interactions between the different populations.
Description of the state variables of model system (11)–(18).
Variable
Description
X
Population of uninfected/unparasitized red blood cells (erythrocytes)
Ys
Population of red blood cells infected by drug-sensitive merozoites
Yr
Population of red blood cells infected by drug-resistant merozoites
Ms
Population of drug-sensitive merozoites
Mr
Population of drug-resistant merozoites
Gs
Population of drug-sensitive gametocytes
Gr
Population of drug-resistant gametocytes
W
Population of strain-independent immune cells
Description of model parameters.
Parameter
Description
λx
The rate of recruitment of red blood cells
ωs
Antimalarial treatment efficacy
αs,αr
Parasite strain-specific fitness cost
λw
Background recruitment rate of immune cells
eg,em,ey
Hill parameters in Gi, Mi, and Yi dynamics i=s,r
μx
Per capita natural mortality rate of unparasitized erythrocytes
μys
Natural mortality rate of drug-sensitive IRBCs
μyr
Natural death rate of drug-resistant IRBCs
ζ,η
Rate of antimalarial eradication of merozoites and gametocytes, respectively
μms
Death rate of drug-sensitive merozoites
μmr
Mortality rate of drug-resistant merozoites
μgs
Per capita mortality rate of drug-sensitive gametocytes
μgr
Mortality rate of drug-resistant gametocytes
μw
Natural mortality rate of immune cells (CD8 + T cells)
β
The rate of infection of susceptible RBCs by blood floating merozoites
σr, σs
Rate of formation of gametocytes from the infected RBCs
P
Number of merozoites produced per dying infected RBC
hy
Immune cell proliferation rate due to IRBCs
hm
Immune cell proliferation rate due to asexual merozoites
hg
Immune cell proliferation rate due to gametocytes
ky
Phagocytosis rate of IRBCs by immune cell
km
Phagocytosis rate of merozoites by immune cell
kg
Phagocytosis rate of gametocytes by immune cell
Ψ1
Rate of development of resistance by drug-sensitive merozoites
Ψ2
Rate of development of resistance by drug-sensitive gametocytes
δr
Accounts for the reduced fitness of the resistant parasite strains
γ
Efficiency of immune effector to inhibit merozoite infection
1/a
Half-saturation constant for Yt, Mt, and Gt
2.1. Model Equations
Based on the above model descriptions and schematic diagram shown in Figure 1, the model in this paper consists of the following nonlinear system of ordinary differential equations:(7)dXdt=λx−μxX−βX1+γWMs+δrMr,(8)dYsdt=βXMs1+γW−kyYsW1+aYs−11−ωsμysYs−σsYs,(9)dYrdt=δrβXMr1+γW−kyYrW1+aYr−μyrYr−σrYr,(10)dMsdt=1−αsPμysYs−βMsX1+γW−kmMsW1+aMs−Ψ1+μms+ζMs,(11)dMrdt=1−αrPμyrYr+Ψ1Ms−δrβMrX1+γW−kmMrW1+aMr−μmrMr,(12)dGsdt=σsYs−kgWGs1+aGs−Ψ2+μgs+ηGs,(13)dGrdt=σrYr+Ψ2Gs−kgWGr1+aGr−μgrGr,(14)dWdt=λw+hgGs+GrGs+Gr+eg+hyYs+YrYs+Yr+ey+hmMs+MrMs+Mr+emW−μwW,subject to the following initial conditions:(15)X0>0,Yi0≥0,Mi0≥0,Gi0≥0,W0>0,fori=s,r.
3. Model Analysis3.1. Positivity and Uniqueness of Solutions
The consonance between a formulated epidemiological model and its biological reality is key to its usefulness. Given that all the model parameters and variables are nonnegative, it is only sound that the model solutions be nonnegative at any future time t≥0 within a given biological space.
Theorem 1.
The region ℝ+8 with solutions of system (7)–(14) is positively invariant under the flow induced by system (7)–(14).
Proof.
We need to show that every trajectory from the region ℝ+8 will always remain within it. By contradiction, assume ∃t∗ (where t∗ refers to time) in the interval 0,∞, such that Xt∗=0, X′t∗<0 but for 0<t<t∗, Xt>0, and Yit>0, Mit>0, Git>0, and Wit>0 for i=r,s. Notice that, at t=t∗, Xt is declining from the original zero value. If such an X exists, then it should satisfy the differential equation (7). That is,(16)dXdt=λx−μxXt∗−βXt∗1+γWt∗Mst∗+δrMrt∗=λx>0.
We arrive at a contradiction, i.e., X′t∗>0. This shows the nonexistence of such t∗. This argument can be extended to all the remaining seven variables Ys,Yr,Ms,Mr,Gs,Gr,W. The process of verification is however simpler. We can follow the steps as presented in [59, 60]. Let the total erythrocyte population Ct evolve according to the following formulation:(17)dCdt≤λx−μcC,where μc=minμx,μys,μyr. Similarly, the total density of malarial parasites Pt is described by(18)dPdt≤P1−αsμysYs+1−αrμyrYr+σsYs+σrYr−μpP,where μp=minμms,μmr,μgs,μgr.
The solutions of equations (14), (17), and (18) are, respectively, given as(19)Wt≤λwμw+W0−λwμwe−μwt,Ct≤λxμc+C0−λxμce−μct,Pt≤σs∫0tYstΔIFdt+σr∫0tYrtΔIFdtΔIF+P0−σs+σrμp1−αsμys+1−αrμyr1ΔIF,where(20)ΔIF=exp−1−αsμys∫0tYstdt+1−αrμyr∫0tYrtdt−∫0tμpdt.
Here, C0=X0+Ys0+Yr0 and P0=Ms0+Mr0+Gs0+Gr0 represent the initial total populations of erythrocytes and malarial parasites, respectively. We observe that all the solutions of equations (14), (17), and (18) remain nonnegative for all future time, t≥0. Moreover, the total populations are bounded: 0≤Ct≤maxC0,λx/μc, 0≤Wt≤maxW0,λw/μw and Pt≤maxP0,σs+σrμp/1−αsμys+1−αrμyr. Thus, all the state variables of model system (7)–(14) and all their corresponding solutions are nonnegative and bounded in the phase space φ, where(21)φ=X,Ys,Yr,Ms,Mr,Gs,Gr,W∈ℝ+8:Ct≤maxC0,λxμc,Wt≤maxW0,λwμw,Pt≤maxP0,σs+σrμp1−αsμys+1−αrμyr.
It is obvious that φ is twice continuously differentiable function. That is, φi∈ℂ2. This is because its components φi,i=1,2,…,8, are rational functions of state variables that are also continuously differentiable functions. We conclude that the domain φ is positively invariant. It is therefore feasible and biological meaningful to study model system (7)–(14).
Theorem 2.
The model system (7)–(14) has a unique solution.
Proof.
Let x=X,Ys,Yr,Ms,Mr,Gs,Gr,WT∈ℝ+8 so that x1=X and x2=Ys as presented in system (7)–(14). Similarly, let gx=gix,i=1,…,8T be a vector defined in ℝ+8. The model system (7)–(14) can hence be written as(22)dxdt=gx,x0=x0,where x:0,∞⟶ℝ+8 denotes a column vector of state variables and g:ℝ+8⟶ℝ+8 represents the right-hand side (RHS) of system (7)–(14). The result is as follows.
Lemma 1.
The function g is continuously differentiable in x.
Proof.
All the terms in g are either linear polynomials or rational functions of nonvanishing polynomials. Since the state variables X,Ys,Yr,Ms,Mr,Gs,Gr,W are all continuously differentiable functions of t, all the elements of vector g are continuously differentiable. Moreover, let Lx,n,θ=x+θn−x:0≤θ≤1. By the mean value theorem,(23)gn−gx∞=g′m;n−x∞,where m∈Lx,n,θ denotes the mean value point and g′ the directional derivative of the function g at m. However,(24)g′m,n−x∞=∑i=18▽gim⋅n−xei∞≤∑i=18▽gim∞n−x∞,where ei is the ith coordinate unit in ℝ+8. We can clearly see that all the partial derivatives of g are bounded and that there exists a nonnegative U such that(25)∑i=18▽gim∞≤U,for allm∈L.
Therefore, there exists U>0 such that(26)gn−gx∞≤Un−x∞.
This shows that the function g is Lipschitz continuous. Since g is Lipschitz continuous, model system (7)–(14) has a unique solution by the uniqueness theorem of Picard [61].
3.2. Stability Analysis of the Parasite-Free Equilibrium Point (PFE)
The in-host malaria dynamics are investigated by studying the behaviour of the model at different model equilibrium points. Knowledge on model equilibrium points is useful in deriving parameters that drive the infection to different stability points. The model system (7)–(14) has a parasite-free equilibrium point E0 given by(27)E0=X∗,Ys∗,Yr∗,Ms∗,Mr∗,Gs∗,Gr∗,W∗=λxμx,0,0,0,0,0,0,λwμw.
Using the next-generation operator method by van den Driessche and Watmough [62] and matrix notations therein, we obtain a nonsingular matrix Q showing the terms of transitions from one compartment to the other and a nonnegative matrix F of new infection terms as follows:(28)F=00βλxμwγλw+μwμx000000δrβλxμwγλw+μwμx00000000000000000000000000,(29)Q=v1000000v20000−P1−αsμys0v30000−P1−αrμyr−Ψ1v400−σs000v500−σr00−Ψ2v6,where v1=kyλw/μw+σs+μys/1−ωs, v2=kyλw/μw+σr+μyr, v4=μmr+kmλw/μw+δrβλxμw/γλw+μwμx, v3=ζ+μms+Ψ1+kmλw/μw+βλxμw/γλw+μwμx, and v5=η+μgs+kgλw/μw+Ψ2, v6=μgr+kgλw/μw.
The effective reproduction number RE of model system (7)–(14) associated with the parasite-free equilibrium is the spectral radius of the next-generation matrix FQ−1, where(30)Q−1=1v10000001v20000P1−αsμysv1v301v3000P1−αsμysΨ1v1v3v4P1−αrμyrv2v401v400σs/v1v50001v50σsΨ2/v1v5v6σrv2v600Ψ2/v5v61v6.
It follows that(31)RE=ρFQ−1=maxRs,Rr,where(32)Rs=P1−αsμysβλxμwkyλw/μw+σs+μys/1−ωsζ+μms+kmλw/μw+Ψ1+βλxμx/γλw+μwμxγλw+μwμx,Rr=P1−αrμyrδrβλxμwkyλw/μw+σr+μyrμmr+kmλw/μw+δrβλxμw/γλw+μwμxγλw+μwμx.
From equation (31), it is evident that, in a multiple-strain P. falciparum malaria infection, the progression of the disease depends on the reproduction number of different parasite strains. If the threshold quantity Rs>Rr, the drug-sensitive parasite strains will dominate the drug-resistant strains and hence the driver of the infection. To manage the infection in this case, the patient should be given antimalarials that can eradicate the drug-sensitive parasites. Conversely, if Rr>Rs, the infection is mainly driven by the drug-resistant parasite strains. In this scenario, the used antimalarial drugs should be highly efficacious and effective enough to kill both the drug-resistant and drug-sensitive parasite strains in the blood of the human host. This result is quite instrumental in improving antimalarial therapy for P. falciparum infections. The best antimalarials should be sufficient enough to eradicate both parasite strains within the human host.
Based on Theorem 2 in [63], we have the following lemma.
Lemma 2.
The parasite-free equilibrium point E0 is locally asymptotically stable if RE<1Rs<1 and Rr<1 and unstable otherwise.
The Jacobian matrix associated with the in-host model system (7)–(14) at E0 is given by(33)JE0=−μx00−βλxμwγλw+μwμx−δrβλxμwγλw+μwμx0000−v10βλxμwγλw+μwμx000000−v20βλxμwγλw+μwμx0000P1−αsμys0−v3000000P1−αrμyrΨ1−v40000σs000−v50000σr00Ψ2−v600hyλweyμwhyλweyμwhmλwemμwhmλwemμwhgλwegμwhgλwegμw−μw,where the terms v1,…,v6 are as defined in (30). It is clear from matrix (33) that the first four eigenvalues are −μx (from column 1), −μw (from column 8), −μgr+kgλw/μw=−v6 (from column 7), and −η+μgs+kgλw/μw=−v5 (from column 6). They are all negative. The remaining four eigenvalues are obtained from the roots of the following quartic equation:(34)Pλ=λ4+p1λ3+p2λ2+p3λ+p4,where(35)p1=v1+v2+v3+v4>0,(36)p2=v3v4+v2v3+v4+v1v2+v3+v4−Pβλxμwγλw+μwμx1−αsμys−1−αrμyrδr,(37)p3=1Kv3v2v4K−P1−αrμyrδrβλxμw−1KP1−αsμysβλxμwv2+v4+v1Kv3v4+v2v3+v4K−P1−αrμyrδrβλxμw,(38)p4=v2v4K−P1−αrμyrδrβλxμwv1v3K−P1−αsμysβλxμwK.
Due to complexity in the coefficients of the polynomial (34), we shall rely on the Routh–Hurwitz stability criterion [64], which provides sufficient condition for the existence of the roots of the given polynomial on the left half of the plane.
Definition 1.
The solutions of the quartic equation (34) are negative or have negative real parts provided that the determinants of all Hurwitz matrices are positive [64].
Based on the Routh–Hurwitz criterion, the system of inequalities that describe the stability region E0 is presented as follows:
p1>0
p3>0
p4>0
p1p2p3>p32+p12p4
From (35), it is clear that p1>0. Upon simplifying p2 in (36), we obtain(39)p2=v3v4+v2v3+v1v2+v1v4+v1v3+λxμwβB1K+v2v4+λxμwδrβB2K,where B1=−P1−αsμys and B2=−P1−αrμyr.
Thus,(40)p2=v3v4+v2v3+v1v2+v1v4+v1v31−B1βλxμwv1v3K+v2v41−B2δrβλxμwv2v4K=v1+v3v2+v4+v1v31−Rs+v2v41−Rr>0,if and only ifRs,Rr<1.
Similarly, the expression for p4 can be rewritten as follows:(41)p4=v1v3+B1βλxμxKv2v4+B2δrβλxμwK=v1v31+B1βλxμwv1v3Kv2v41+B2δrβλxμwv2v4K=v1v31−Rsv2v41−Rr>0,if and only ifRs,Rr<1.
Lastly, upon simplifying equation (37), we obtain(42)p3=v2v3v4+v1v3v4+v1v2v3+v4+βB1λxμwv2+v4K+δrβB2λxμwv1+v3K=v1v2v3v41v41+βB1λxμwv1v3K+1v21+βB1λxμwv1v3K+1v11+δrβB2λxμwv2v4K+1v31+δrβB2λxμwv2v4K=v1v2v3v4v2+v4v2v41−Rs+v1+v3v1v31−Rr=v1v3v2+v41−Rs+v2v4v1+v31−Rr>0,if and only ifRs,Rr<1.
Since all the coefficients of the quartic equation (34) are nonnegative, all its roots are therefore negative or have negative real parts. Hence, the Jacobian matrix (33) has negative eigenvalues or eigenvalues with negative real parts if and only if the effective reproduction number RE is less than unity. Equilibrium point E0 is therefore locally asymptotically stable when RE<1 (when both Rs<1 and Rr<1). This implies that an effective antimalarial drug would cure the costrain infected human host, provided that the drug reduces the effective reproduction number to less than 1.
Lemma 2 shows that P. falciparum malaria can be eradicated/controlled within the human host if the initial parasite and cell populations are within the basin of attraction of the trivial equilibrium point E0. To be certain to eradicate/control the infection irrespective of the initial parasite and cell populations, we need to prove the global stability of the parasite-free equilibrium point. This is presented in the following section.
3.3. Global Asymptotic Stability Analysis of the Parasite-Free Equilibrium Point
Following the work by Kamgong and Sallet [65], we begin by rewriting system (7)–(14) in a pseudotriangular form:(43)X˙1=D1XX−X1∗+D2XX2,X˙2=D3XX2,,where X1 is a vector representing the densities of noninfective population groups (unparasitized erythrocytes and immune cells) and X2 represents the densities of infected/infective groups (infective P. falciparum parasites and/or infected host cells) that are responsible for disease transmissions. For purposes of clarity and simplicity to the reader, we shall represent X1,0 with X1 and 0,X2 with X2 in ℝ+8×ℝ+8. We assume the existence of a parasite-free equilibrium in φ: X∗=X1∗,0. Thus,(44)X=X1,X2,X1=X,W,X2=Ys,Yr,Ms,Mr,Gs,Gr,X1∗=λxμx,λwμw.
We analyze system (43) based on the assumption that it is positively invariant and dissipative in φ. Moreover, the subsystem X¯1 is globally asymptotically stable at X1∗ on the projection of φ on ℝ+8. This implies that whenever there are no infective malarial parasites, all cell populations will settle at the parasite-free equilibrium point E0. Finally, D2 in (43) is a Metzler matrix that is irreducible for any X∈φ. We assume adequate interactions between and among different parasites and cell compartments in the model.
The matrices D1X and D2X are easily computed from subsystem X˙1 in (43) so that we have(45)D1X=−μx00−μw,D2X=00−βλxμwγλw+μwμx−δrβλxμwγλw+μwμx00hyλweyμwhyλweyμwhmλwemμwhmλwemμwhgλwegμwhgλwegμw.
We can easily see that the eigenvalues of matrix D1 are both real and negative (−μx<0, −μw<0). This shows that the subsystem X˙1=D1XX−X1∗+D2XX2 is globally asymptotically stable at the trivial equilibrium X1∗. Additionally, from subsystem X˙2=D3XX2, we obtain the following matrix:(46)D3X=−v10βλxμwγλw+μwμx0000−v20βλxμwγλw+μwμx00P1−αsμys0−v30000P1−αrμyrΨ1−v400σs000−v500σr00Ψ2−v6.
Notice that all the off-diagonal entries of D3X are nonnegative (equal to or greater than zero), showing that D3X is a Metzler matrix. To show the global stability of the parasite-free equilibrium E0, we need to show that the square matrix D3X in (46) is Metzler stable. We therefore need to prove the following lemma.
Lemma 3.
Let K be a square Metzler matrix that is block decomposed:(47)K=K11K12K21K22,where K11 and K22 are square matrices. The matrix K is Metzler stable if and only if K11 and K22−K21K11−1K12 are Metzler stable.
Proof.
The matrix K in Lemma 3 refers to D3X in our case. We therefore let(48)K11=−v10βλxμwγλw+μwμx0−v20P1−αsμys0−v3,K12=000βλxμwγλw+μwμx00000,K21=0P1−αrμyrΨ1σs000σr0,K22=−v4000−v500Ψ2−v6.
Results from analytical computations based on Maple software give(49)K11−1=−v3v1v3+Pβαs−1λxμwμys/γλw+μwμx0−βλxμwv1v3γλw+μwμx+Pβαs−1λxμwμys0−1v20Pαs−1γλw+μwμxμysv1v3γλw+μwμx+Pβαs−1λxμwμys0−v1v1v3+Pβαs−1λxμwμys/γλw+μwμx,(50)K22−K21K11−1K12=−v4000−v500Ψ2−v6,where v4=μmr+kmλw/μw+δrβλxμw/γλw+μwμx, v5=η+μgs+kgλw/μw+Ψ2, and v6=μgr+kgλw/μw.
From equation (50), it is evident that all the diagonal elements of matrix K22−K21K11−1K12 are negative and the rest of the elements in the matrix are nonnegative. This shows that matrix K22−K21K11−1K12 is Metzler stable, and the parasite-free equilibrium point E0 is globally asymptotically stable in the biologically feasible region φ of model system (7)–(14). Epidemiologically, the above result implies that when there is no malaria infection, different cell populations under consideration will stabilize at the parasite-free equilibrium. However, if there exists a P. falciparum infection, then an appropriate control in forms of effective antimalarial drugs would be necessary to clear the parasites from the human blood and restore the system to the stable parasite-free equilibrium state.
3.4. Coexistence of Parasite-Persistent Equilibrium Point
The existence of a nontrivial equilibrium point represents a scenario in which the P. falciparum parasites are present within the host and the following conditions hold: X∗>0,Ys∗≥0,Yr∗≥0,Ms∗≥0,Mr∗≥0,Gs∗≥0,Gr∗≥0, and W∗>0. Upon equating the right-hand side of system (7)–(14) to zero and solving for the state variables, we obtain the parasite-persistent equilibrium point E1=X∗,Ys∗,Yr∗,Ms∗,Mr∗,Gs∗,Gr∗,W∗, where(51)X∗=1+γW∗λxβMs∗+δrMr∗+1+γW∗μx,Ys∗=b¯+b¯2−4a¯c¯−2a¯,Yr∗=b¯+b¯2−4a¯c¯−2a¯,(52)a¯=−a1−ωsσs+μysβMs∗+βMr∗δr+γW∗+1μx<0,(53)b¯=−βMs∗−a1−ωsλx−ωsσs+σs+μys−W∗1−ωskyβMs∗+βMr∗δr+γW∗μx+μx,(54)c¯=βMs∗1−ωsλx>0,(55)a¯=−aσ2+μyrβMs∗+βMr∗δr+γW∗+1μx<0,(56)b¯=βMr∗δraλx−σ2−μyr−W∗kyβMs∗+βMr∗δr+γW∗μx+μx−σ2+μyrβMs∗+γW∗μx+μx,(57)c¯=βMr∗δrλx>0,(58)Gs∗=b1+b12−4a1c1−2a1,Gr∗=b2+b22−4a2c2−2a2,(59)a1=−aη+μg1+Ψ2<0,b1=aσ1Ys∗−W∗kg−η−μg1−Ψ2,c1=σ1Ys∗>0,(60)a2=−aμg2<0,b2=aG1Ψ2+aσ2Yr∗−W∗kg−μg2,c2=G1Ψ2+σ2Yr∗>0,(61)Ms∗=b3+b32−4a3c3−2a3,Mr∗=b4+b42−4a4c4−2a4,(62)a3=−aβMr∗δrζ+μms+Ψ1+aγW∗μmsμx+aμmsμx+aβP1−αsμysYs∗+Ψ1aγW∗+1μx+β+aγζW∗μx+aβλx+aζμx+βζ+βW∗km+βμms,(63)b3=−βMr∗δraαs−1PYs∗μys+ζ+W∗km+μms+Ψ1−αs−1βPYs∗μys−βλx−γW∗+1μxaαs−1PYs∗μys+ζ+W∗km+μms+Ψ1,(64)c3=P1−αsYs∗μysβMr∗δr+γW∗+1μx>0,(65)a4=−aβMs∗Ψ1δr+μmr+μmraγW∗+1μx+βδr+a1−αrβPY2δrμy2+βW∗kmδr,(66)b4=aβMs∗2Ψ1+Ms∗−βaδrλx+μmr+a1−αrβPY2μy2+Ψ1aγW∗+1μx+βδr+1−αrPY2μy2aγW∗+1μx+βδr−W∗kmβMs∗+γW∗+1μx−μmrγW∗+1μx,(67)c4=βMs∗2Ψ1+Ms∗1−αrβPY2μy2+βδrλx+Ψ1γW∗+1μx+1−αrPY2γW∗+1μxμy2>0,(68)W∗=ΔμwΔ−hgGs∗+Gr∗+hmMs∗+Mr∗+hyYs∗+Yr∗,where Δ=eg+Gs∗+Gr∗em+Ms∗+Mr∗ey+Ys∗+Yr∗.
Using Descartes’ “Rule of Signs” [66], it is evident that irrespective of the sign of b¯ in (53), b¯ in (56), b1 in (59), b2 in (60), b3 in (63), and b4 in (66), the state variables Ys∗,Yr∗,Ms∗,Mr∗,Gs∗, and Gr∗ can only have one real positive solution. This shows that the model system (7)–(14) has a unique parasite-persistent equilibrium point E1.
3.5. Stability of the Coexistence of Parasite-Persistent Equilibrium Point
Here, we shall prove that the coexistence of parasite-persistent equilibrium E1 is locally asymptotically stable when RE>1or whenRs>1 and Rr>1. We shall follow the methodology by Esteva and Vargus presented in [67], which is based on the Krasnoselskii technique [68]. This methodology requires that we prove that the linearization of system (7)–(14) about the coexistence of parasite-persistent equilibrium does not have a solution of the form(69)S¯t=S¯0eξt,where S¯0=S1,S2,…,S7, Si,ξ∈ℂ, and the real part of ξ is nonnegative (Reξ≥0). Note that ℂ is a set of complex numbers.
Next, we substitute a solution of the form (69) into the linearized system (7)–(14) about the coexistence of parasite-persistent equilibrium. We obtain(70)ξS1=−βMs1+γW+kyW1+γW+μys1−ωs+σsS1−βMs1+γWS2+βC∗−Ys−Yr1+γWS3,ξS2=−δrβMr1+γWS1−δrβMr1+γW+kyW1+aYr+μyr+σrS2+δrβC∗−Ys−Yr1+γWS4,ξS3=−βMs1+γW+P1−αsμysS1+βMs1+γWS2−kmW1+aMs+βC∗−Ys−Yr1+γW+k1S3,ξS4=Ψ1S3+δrβMr1+γW+P1−αrμyrS2−δrβC∗−Ys−Yr1+γW+kmW1+aMr+μmrS4+δrβMr1+γWS1,ξS5=σsS1−kgW1+aGs+k2S5,ξS6=σrS2+Ψ2S4−kgW1+aGr+μgrS5,ξS7=λ+hgGs+GrGs+Gr+eg+hyYs+YrYs+Yr+ey+hmMs+MrMs+Mr+emS7−μwS7,where C∗−Ys−Yr=X, k1=Ψ1+μms+ζ, and k2=Ψ2+μgs+η.
Upon simplifying the equations in (70), we obtain(71)1+1+γW1+aYs1−ωsΔ1ξS1=1+γW1+aYs1−ωsΔ1−βMs1+γWS2+βC∗−Ys−Yr1+γWS3,1+ξ1+γW1+aYrΔ2S2=1+γW1+aYrΔ2−δrβMr1+γWS1+δrβC∗−Ys−Yr1+γWS4,1+1+γW1+aMsΔ3ξS3=1+γW1+aMsΔ3βMs1+γW+P1−αsμysS1+βMs1+γWS2,1+ξ1+γW1+aMrΔ4S4=1+γW1+aMrΔ4Ψ1S3+δrβMr1+γW+P1−αrμyrS2+δrβMr1+γWS4,1+1+aGskgW+k2ξS5=σs1+aGskgW+k2S1,1+1+aGrkgW+μgrξS6=1+aGrkgW+μgrσrS2+Ψ2S4,1+1μwξS7=λwμw+WμwhgS5+S6Gs+Gr+eg+hyS1+S2Ys+Yr+ey+hmS3+S4Ms+Mr+em,where(72)Δ1=βMs1+aYs1−ωs+kyW1−ωs1+γW+μys1+aYs1+γW+σs1+aYs1−ωs1+γW,Δ2=δrβMr1+aYr+kyW1+γW+μyr+σr1+aYr1+γW,Δ3=1+aMsβC∗−Ys−Yr+kmW1+γW+k11+aMs1+γW,Δ4=1+aMrδrβC∗−Ys−Yr+kmW1+γW+μmr1+aMr1+γW.
Separating the negative terms, we obtain the following system:(73)1+FjξSj=HS¯j,forj=1,2,…,7,where(74)F1ξ=1+γW1+aYs1−ωsΔ1ξ,F2ξ=ξ1+γW1+aYrΔ2,F3ξ=1+γW1+aMsΔ3ξ,F4ξ=ξ1+γW1+aMrΔ4,F5ξ=1+aGskgW+k2ξ,F6ξ=1+aGrkgW+μgrξ,F7ξ=1μwξ,with(75)H=00βC∗1+γw0000000δrβC∗1+γW000P1−αsμys0βC∗1+γW+k100000P1−αrμyrΨ10000σs0000000σr0000000λw0000.
Note that X∗=C∗−Ys∗−Yr∗ and all the elements in the square matrix H are nonnegative. The coordinates of E1 are all positive, and the jth coordinate of the vector HS¯ is described by the notation HS¯j for j=1,…,7. Additionally, the equilibrium E1=Ys∗,Yr∗,Ms∗,Mr∗,Gs∗,Gr∗,W∗ satisfies E1=HE1. If we assume, for example, that system (73) has a solution of the form S¯, then there exists a small positive real number ϵ, such that S¯≤ϵE1, where S¯=S1,S2,…,S7. Note also that . is a norm in the field of complex numbers.
Next, we show that Reξ<0. To do so, we apply proof by contradiction. We let ξ=0 and ξ≠0. For the case when ξ=0, the determinant (∇) of (70) is given by(76)∇=v5v6μwv2v4γλx+μwμx+Pβ1−αrλxμwμyrv1v3γλx+μwμx+Pβ1−αsλxμwμysγλx+μw2μx2,where the positive terms v1,…,v6 are as defined in matrix (29).
It is clear that the above determinant is nonnegative (∇>0). Consequently, the system (70) can only have the trivial solution S¯=0,0,0,0,0,0,λw/μw. On the contrary, for ξ≠0, we assume Reξ≥0 and define Fξ=min1+Fjξ,j=1,2,…,7. This implies that Fξ>1 and ϵ/Fξ<ϵ. The minimality of ϵ means that S¯>ϵ/FξE1. While considering the nonnegativity property of H, if we assume the norms on the two sides of (73), we shall have(77)FξS¯≤HS¯≤εHE1=εE1.
This implies that S¯≤ϵ/FξE1≤ϵE1, which is a contradiction. Therefore, Reξ<0 and E1 is locally asymptotically stable when RE>1.
In this section, we show by means of numerical simulation the existence and stability of a positive parasite-persistent equilibrium point that involves only one of the parasite strains under study.
4.1.1. Drug-Sensitive-Only Persistent Equilibrium Point <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M317"><mml:mrow><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mtext>s</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>
This is an equilibrium point where only the drug-sensitive parasite strains are present in the infected human host. That is, the populations Yr=Mr=Gr=0. This steady state is only feasible if no resistant parasites emerge from infected red blood cells and the use of antimalarial treatment does not lead to resistance development; that is, Ψ1=Ψ2=0. The original model (7)–(14) is thus reduced to(78)dXdt=λx−μxX−βXMs1+γW,dYsdt=βXMs1+γW−kyYsW1+aYs−11−ωsμysYs−σsYs,dMsdt=1−αsPμysYs−βMsX1+γW−kmMsW1+aMs−μms+ζMs,dGsdt=σsYs−kgWGs1+aGs−μgs+ηGs,dWdt=λw+hgGsGs+eg+hyYsYs+ey+hmMsMs+emW−μwW.
Numerically, this equilibrium point is illustrated, as shown in Figure 2.
Simulations of model system (11)–(18) showing the existence of drug-sensitive-only equilibrium point. All parameter values are as presented in Table 3.
4.1.2. Drug-Resistant-Only Persistent Equilibrium Point <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M321"><mml:mrow><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mtext>r</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>
In this case, the population of the drug-sensitive parasite strains declines to zero as the density of the resistant strains grows and stabilizes at an optimal population size. This is also illustrated numerically, as shown in Figure 3.
Simulations of model system (11)–(18) showing the existence of drug-resistant-only equilibrium point. All parameter values are as presented in Table 3.
4.2. Within-Host Competition between Parasite Strains
We investigate the competitive exclusion principle by simulating the model system (7)–(14) under different values of the threshold quantities Rs and Rr in (31). Model (7)–(14) is simulated so that Rs=4.022 and Rr=0.3131, and we achieve a convergence to the drug-sensitive-only endemic equilibrium point Es, as shown in Figure 4(a). Again, using the parameter values in Table 3 with Ψ1=0.9 and (Rs=0.022, Rr=3.0098), the solutions of Ys and Yr converge to the drug-resistant-only endemic equilibrium point Er (Figure 4(b)).
Simulations of model system (11)–(18). The figures show the dynamics of drug-sensitive and drug-resistant infected red blood cells under different conditions of the threshold values Rs and Rr. In Figure 4a, Rs>Rr. In Figure 4b, Rr>Rs, Ψ1=0 and all other parameter values are as presented in Table 3.
Baseline values and range for parameters of model (11)–(18).
Parameter
Value
Range
Units
Source
λx
3×103
(3×103−3×108)
Cells/μl−1/day
[69]
λw
30
(10–40)
Cells/μl/day
[70]
ωs
0.5
(0-1)
Unitless
Assumed
αs
0.4
(0.1–1)
Unitless
Assumed
αr
0.2
(0.01–1)
Unitless
Assumed
eg,em,ey
104
103−105
Unitless
[71]
μx
1/120
(0.05–0.1)
day−1
[72]
μys
0.5
(0.3–0.8)
day−1
[73]
μyr
0.3
(0.3–0.8)
day−1
Assumed
μms,μmr
48
(46–50)
day−1
[69]
μgs,μgr
0.0625
(0.05–0.1)
day−1
[74]
μw
0.05
(0.02–0.08)
day−1
[74]
δr
0.7
(0.01–0.99)
Unitless
Assumed
ζ,η
0.5
(0-1)
day−1
[73]
P
16
(15–20)
Erythrocytes/day
[34]
β
6.5×10−7
4.8×10−7–6.8×10−7
Merozoites/day
[75]
σr, σs
0.02
(0.01–0.03)
day−1
[75]
hy,hm,hg
0.05
(0.01–0.08)
mm−3/day
[70]
ky,km,kg
0.000001
(0.001–0.9)
day−1
[51]
Ψ1
0.2
(0.01–2.2)
day−1
Assumed
Ψ2
0.01
(0.001–0.1)
day−1
Assumed
δr
0.3
(0-1)
Unitless
Assumed
γ
0.5
(0-1)
Immune cell/μl
Assumed
1/a
0.2
(0-1)
Unitless
[76]
Provided that both Rs and Rr are greater than 1 (as shown in Figure 4(c)), the parasite-infected red blood cells remain persistent in the host. This implies that the merozoites (both drug-sensitive and drug-resistant) continue to multiply in the absence of antimalarial therapy, ωs=0, or in the presence of ineffective antimalarial drugs. Similar results are observed in the dynamics of merozoites (Ms and Mr), as shown in Figure 5. It should be noted that the dominant merozoite strains are likely to drive the infection under these conditions. As the density of one strain increases, the population of the other strain is likely to decrease due to a phenomenon known as competitive exclusion principle. The most fit parasite strain survives as the weaker competitor dies out, as shown in Figure 5(a). Both drug-sensitive and drug-resistant merozoites would remain persistent if poor-quality antimalarial drugs are administered to P. falciparum malaria patients. Thus, in the absence of efficacious antimalarial drugs like ACTs with the potential to eradicate resistant merozoites, we are likely to experience an exponential growth in the density of drug-resistant merozoites, as displayed in Figure 5(b). This may lead to severe malaria and eventual death of the patient.
Simulations of model system (11)–(18). The figures show the dynamics of the merozoites under different conditions of the threshold values Rs and Rr. Competitive exclusion among the parasite strains is shown in (a). In (b), both parasite strains coexists and Rr>Rs, Ψ1=0. Other parameter values are available in Table 3.
The bifurcation analysis of both scenarios is presented in Figure 6 (with and without competition between the parasite strains). When there is competition between the parasite strains, as shown in Figure 6(a), we observe that the strain with a higher threshold quantity R0 would exclude the other strain. A decrease in the population of the drug-sensitive strain would pave way for a surge in the population of the drug-resistant strains, and vice versa. This is despite the fact that some drug-resistant strains emerge from the drug-sensitive strains as a result of mutation [77]. In Figure 6(b), we observe coexistence of the strains that do not compete with each other. Like the resistance strains, the sensitive strains are only present when their threshold quantity, Rs, is greater than unity. Both strains are however present when Rr>1 and Rs>1. Additionally, when Rr<1 and Rs<1, we arrive at the parasite-free equilibrium (PFE) point, as shown in Figures 6(a) and 6(b).
Bifurcation diagrams showing competitive exclusion (a) and coexistence equilibrium (b) for the drug-sensitive and drug-resistant P. falciparum parasite strains under different values of threshold quantities Rs and Rr. Both parasite strains coexists when Rs>1 and Rr>1 (see part (b)).
4.3. Antimalarial Drug Effects and Parasite Clearance
The effects of antimalarial drug treatment are monitored by establishing first and foremost that(79)∂Rs∂ωs=−βμ1μ2P1−αsμwλx1−ωs2μxγλw+μwkyλw/μw+μ2/1−ω1+σs2ζ+kmλw/μw+μms+βλxμx2/γλw+μw+Ψ1<0.
Thus, Rs is a decreasing function of ωs (the efficacy of the antimalarial drug used). Therefore, using a highly efficient antimalarial drug could lead to a scenario where Rs<1 and Rr<1 (disease-free state shown in Figure 7(c)). In Figure 7(a), model system (7)–(14) is simulated by varying the efficacy of the antimalarial drug ωs and other model parameters chosen such that Rr=3.221 and Rs=2.221. The higher the efficacy of the used antimalarial, the lower the density of infected erythrocytes. Thus, governments and ministry of health officers should only roll out or permit the administration of antimalarials or ACTs that can eradicate (totally) both the drug-resistant and the drug-sensitive strains of P. falciparum parasites.
The effect of varying the efficacy of antimalarial drug used ωs and the rate of development of resistance by the drug-sensitive merozoites Ψ1, on the density of infected erythrocytes Ys,Yr. The value of ωs ranges from 0 to 1. The rest of the parameter values are available in Table 3. Figure (c) shows that in the absence of highly effective ACTs, drug-resistant parasite would take a longer time to eradicate.
The rate of development of resistance by the drug-sensitive merozoites, Ψ1, is shown to have very minimal impact on the dynamics of infected red blood cells Yr as long as Rs>1 and Rr>1 (Figure 7(b)). Nevertheless, analytical results indicate that the higher the rate of development of resistance, the lower the severity of future malaria infections. This is presented as(80)∂Rs∂Ψ1=−βμ1P1−αsμwλxμxγλw+μwkyλw/μw+μ2/1−ω1+σsζ+kmλw/μw+μms+βλxμx2/γλw+μw+Ψ12<0.
Other parameters that have direct negative impacts on the progression of malaria infection are the efficacy of the immune effectors, γ, and the rate of therapeutic elimination of drug-sensitive merozoites, ζ:(81)∂Rs∂ζ=−βμ1P1−αsμwλxμxγλw+μwkyλw/μw+μ2/1−ω1+σsζ+kmλw/μw+μms+βλxμx2/γλw+μw+Ψ12<0,(82)∂Rr∂γ=−βμ2P1−αrδrλwμw3λxkmλw+μmrμwμxkyλw+μwμ2+σrγλw+μwkmλw+μmrμw+βδrμwλxμx22<0.
Further simulations based on contour plots (see [78] for theory on contour plots) are used to ascertain the relational effects of selected pairs of model parameters on the disease threshold quantities Rs and Rr. In Figure 8(a), both β and μw increase the reproduction number due to drug-sensitive P. falciparum parasite strains. A direct relationship exists between the two parameters: the higher the decay rate of the immune cells, the higher the rate of infection of healthy erythrocytes.
Contour plot of Rs as a function of (a) β and μw, (b) ωs and μys, (c) P and γ.
In Figure 8(b), we observe the least increase in Rs with respect to an increase in ωs relative to μys. Antimalarial therapy is shown to be very effective in reducing the severity of P. falciparum infection. Conversely, the number of merozoites produced per dying blood schizont, P, is shown in Figure 8(c) to have a very high positive impact on Rs and hence on the severity of malaria infection due to drug-sensitive parasite strains. Clinical control should target and eradicate infected red blood cells to diminish the erythrocytic cycles of infections.
We observe in Figure 9(b) that the rate at which merozoites develop resistance due to treatment failure has no resultant effects on the rate of formation of gametocytes that undergo sexual reproduction within the mosquito vector. The higher the value of Rr, the higher the cost of resistance, as shown in Figure 9(a). The higher the density of drug-resistant parasite strains, the higher the level of resistance and hence the cost of disease control. Unfortunately, highly effective antimalarial drugs (such as ACTs) that can eradicate both parasite strains are slightly expensive in several P. falciparum malaria-endemic regions [79]. Like the parameter P, the parasite infection rate β is shown to have a direct positive effect on the threshold quantity Rr (Figure 9(c)) due to drug-resistant parasite strains. Effective antimalarials should hence target new cell infections and eliminate recrudescence (by killing already infected erythrocytes).
Contour plot of Rr as a function of (a) αr and μyr, (b) Ψ1 and σr, (c) P and β.
5. Effects of Multiple-Strain Infection and Fitness Cost on Parasite Clearance
Numerous studies [27, 80] have suggested the negative impacts of drug resistance on the fitness and ability of the parasite to dominate the P. falciparum infection. Resistance to antimalarial drugs imposes fitness cost on the drug-resistant parasite. The drug-resistant parasite strains are thought to experience impaired growth within the human host [29]. The cost of resistance is further exacerbated due to the competition between parasite strains within an infected human host. In Figure 10(a), the area under the curve for the drug-resistant strain or the number of infected erythrocytes is lower than that of the drug-sensitive strains. However, in a multiple-strain infection (Figure 10(b)), the area difference is much bigger. This implies that competition between the parasite strains within the human host could result in elimination of one of the parasite strains provided that both Rs and Rr are less than unity.
Dynamics of drug-sensitive (blue) and drug-resistant (orange) strains in a single infection (a) and in a multiple infection (b) in a naive human-host with no malaria therapy (ωs=0). The density of the resistant strain is lower than that of drug-sensitive strain for Rs=2.123>1 and Rr=1.912>1 in a multiple-strain P. falciparum infection. The rest of the parameter values are as displayed in Table 3.
The presence of multiple strains of P. falciparum parasites is likely to complicate and worsen the severity of malaria disease infection in humans. Figures 11 and 12 show the simulated model (7)–(14) for single- and multiple-strain infections, in the absence of preexisting immunity and antimalarial drugs. The persistence of gametocytes in Figures 11(b) and 12(b) is consistent with the actual observations of human malaria infection in the absence of antimalarial therapy [81]. Acquired immunity is shown in Figure 11(c) to increase and eventually level-off at higher levels to contain future infections.
Dynamics of infected erythrocytes, gametocytes, and the immune cells with a single-strain P. falciparum infection. Here, we do not have preexisting immunity. The rest of the parameter values are as displayed in Table 3.
Within-human dynamics of single- and multiple-strain dynamics of infected erythrocytes, gametocytes, and the immune cells in the absence preexisting immunity and with no antimalarial treatment (ωs=0). The rest of the model parameter values are in Table 3.
Although the aspect of timing is key in these multiple-strain infections, we assumed here that the two strains are introduced at the same time. In the long run, it is evident in Figures 10 and 12 that the sensitive strain overtakes the resistant strain. We argue that this could be as a result of strain-specific adaptive responses that symmetrically affect the sensitive parasites.
Unlike single-strain P. falciparum parasite infections, data on multiple-strain infections are not readily available. Nevertheless, a multiple-strain infection (drug-sensitive and drug-resistant) as presented in this paper is biologically reasonable and consistent with that of P. Chabaudi described in [82].
5.1. Sensitivity Analysis
In this paper, the primary model output of interest for the sensitivity analysis is the infected erythrocytes (Ys,Yr). However, the effective reproduction number RE is a threshold quantity which represents on overage the number of secondary infected erythrocytes due to merozoite invasions. We can therefore measure the sensitivity indices of the effective reproduction number of model system (7)–(14) relative to model parameters. For example, the sensitivity of RE relative to the parameter Ψ1 is given by the following formulation:(83)ϒΨ1=∂RE∂Ψ1×Ψ1RE.
Using the parameter values in Table 3, the expressions for sensitivity for all the parameters in RE are evaluated and presented in Table 4. The higher the numerical value of the sensitivity index (S.I), the greater the variational impact of the parameter on the disease progression. A parameter with a negative index decreases the model RE when they are increased. On the other hand, a parameter with a positive index would generate a proportional increase in RE when they are magnified. Results shown in Table 4 indicate that the rate of infection of healthy erythrocytes by the merozoites β, the density of merozoites generated from each of the bursting schizonts P, the efficacy of antimalarial drug used ωs, and the rate at which drug-sensitive merozoites develop resistance Ψ1 are the four most influential parameters, in determining the disease dynamics as presented in model system (7)–(14).
Sensitivity indices of RE relative to the model parameters.
Parameter
S.I
β
+0.9988
P
+1.0000
ωs
−0.87513
λx
+0.7199
μx
−0.0016
ky
−0.02701
σs
−0.7619
γ
−0.3333
μmr
−0.433
μms
−0.52123
μyr
−0.232
Ψ1
−0.77534
μys
−0.492
λw
−0.3471
ζ
−0.0041
δr
+0.0023
km
−0.0020
σr
−0.541872
αr
−0.1111
αs
−0.09891
μw
0.3716
Results from sensitivity analysis emphasize the use of highly efficacious antimalarial drugs such as ACTs in malaria-endemic regions. This would mitigate the many cases of malaria in the region and further help to reduce emerging cases of parasite resistance to existing therapies. Drugs with a higher parasite clearance rate would greatly reduce resistance, which is associated with longer parasite exposure to antimalarial drugs. It is imperative, therefore, that governments and ministry of health personnel in malaria-endemic countries enforce the use of efficient antimalarial drugs that not only cure infected malaria patients but also eliminate the chance of P. falciparum parasites to develop resistance to existing therapy.
6. Conclusion
In this paper, a deterministic model of multiple-strain P. falciparum malaria infection has been formulated and analysed. The parasite strains are categorized as either drug-sensitive or drug-resistant. The infected erythrocytes and the malaria gametocytes are similarly grouped according to the strain of the parasite responsible for their existence. The immune cells are incorporated to reduce the invasive characteristic of the malaria merozoites. Antimalarial therapy is applied to the model but only targets red blood cells infected with drug-sensitive merozoites. Based on the next-generation matrix method, we computed the effective reproduction number RE of the formulated model. Based on RE, it is evident that the success of P. falciparum infection in the presence of multiple-parasite strains is directly dependent on the ability of the individual parasite strains to drive the infection. The parasite strain with a higher threshold value, R0, is likely to dominate the infection. Prescribed antimalarial drugs should therefore be effective enough to eradicate both drug-sensitive and drug-resistant parasite strains in vivo. Linearization of the model at the parasite-free equilibrium reveals the local asymptotic stability of the trivial equilibrium point.
By rewriting the model in the pseudotriangular form, the parasite-free equilibrium is also shown to be globally asymptotically stable. Although the parasite-persistent equilibrium exists, its expression based on a single-model variable proved to be mathematically intractable. The use of antimalarial treatment may eradicate one parasite strain so that we arrive at either a drug-sensitive-only persistent equilibrium point or a drug-resistant-only persistent equilibrium point.
To assess the impacts of the different parasite strains to disease dynamics, the model is simulated for different values of the threshold quantities Rs and Rr. We observed that when Rr>1 and Rs>1, then both parasite strains are persistent and the infection becomes severe. If Rr>1 and Rs<1, then the drug-sensitive parasites would decline to zero as the drug-resistant strains continue to multiply and remain persistent, increasing the severity of infections. On the other hand, if Rs>1 and Rr<1, then the drug-resistant parasite strains would be eradicated. Moreover, provided that the threshold quantities Rs and Rr are less than unity, the use of an efficacious antimalarial drug would help eradicate P. falciparum infection.
The efficacy of antimalarial drug is shown to have direct negative impact on the density of infected red blood cells. The higher the efficacy of administered antimalarial drug, the lower the population of infective merozoites and the smaller the density of infected erythrocytes. This ensures prompt recovery from malaria infections. This result is consistent with that in [72, 83]. The efficacy of antimalarial drug is however shown to have least effect on the population of drug-resistant infected erythrocytes. The rate of development of resistance by drug-sensitive parasites is also shown to drive the infection due to resistant parasite strains. Using contour plots and results from sensitivity analysis, we observe that the efficacy of antimalarial drug used ωs, the density of blood floating merozoites produced per infected erythrocyte P, the rate of development of resistance Ψ1, and the rate of infection by merozoites β are the most important parameters in the disease dynamics and control.
Finally, although the drug-resistant strain is shown to be less fit, the presence of both strains in the human host has a huge impact on the cost and success of antimalarial treatment. To reduce the emergence of resistant strains, it is vital that only effective antimalarial drugs are administered to patients in hospitals, especially in malaria-endemic regions. To improve malaria therapy and reduce cases of parasite resistance to existing therapy, our results call for regular and strict surveillance on antimalarial drugs in clinics and hospitals in malaria-endemic countries.
Data Availability
All data used in this study are included in this published article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Authors’ Contributions
All authors contributed to all sections of this manuscript.
Acknowledgments
The authors are thankful to the anonymous referees for their constructive comments. The authors would also like to thank the Strathmore Institute of Mathematical Sciences for its support in the production of this manuscript. The authors acknowledge with gratitude the financial support from the German Academic Exchange Service (DAAD) (ST32-PKZ:91711149) and the Kenya National Research Fund (NRF-Kenya) (NRF-Phd Grant Titus O.O), in the production of this manuscript.
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