Research article

Polymer physics-based mathematical models for the correlation of DNA and mRNA in a eukaryotic cell

  • Received: 08 February 2022 Revised: 01 May 2022 Accepted: 19 July 2022 Published: 27 September 2022
  • DNA and mRNA are essential aspects of cells. They are responsible for much of the genomic activity that takes place in a cell, and are significant macromolecules for research in cell and molecular biology. DNA and mRNA are polymers, molecules that are composed of repeating subunits known as monomers. In the past, a number of theoretical models that elucidate the physical properties of polymers have been proposed to the scientific community. These models include the Freely-Jointed Chain, Freely-Rotating Chain, Worm-Like Chain, and Gaussian Chain Models. In this paper, I make use of such theoretical models in polymer physics, and derive a number of theoretical models that correlate DNA, its respective pre-mRNA strand, and the corresponding post-mRNA strand in a eukaryotic cell. Furthermore, graphical representations of some of the mathematical models derived in the paper are also rendered. Based on this, the theoretical models formulated in this paper can be applied to research in the fields of mathematical biology, biophysics, biochemistry, and cell/molecular biology.

    Citation: Archit Chaturvedi. Polymer physics-based mathematical models for the correlation of DNA and mRNA in a eukaryotic cell[J]. Mathematical Modelling and Control, 2022, 2(3): 131-138. doi: 10.3934/mmc.2022014

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  • DNA and mRNA are essential aspects of cells. They are responsible for much of the genomic activity that takes place in a cell, and are significant macromolecules for research in cell and molecular biology. DNA and mRNA are polymers, molecules that are composed of repeating subunits known as monomers. In the past, a number of theoretical models that elucidate the physical properties of polymers have been proposed to the scientific community. These models include the Freely-Jointed Chain, Freely-Rotating Chain, Worm-Like Chain, and Gaussian Chain Models. In this paper, I make use of such theoretical models in polymer physics, and derive a number of theoretical models that correlate DNA, its respective pre-mRNA strand, and the corresponding post-mRNA strand in a eukaryotic cell. Furthermore, graphical representations of some of the mathematical models derived in the paper are also rendered. Based on this, the theoretical models formulated in this paper can be applied to research in the fields of mathematical biology, biophysics, biochemistry, and cell/molecular biology.



    Malaria is a vector-borne disease of global public health concern with high level of morbidity and mortality in the tropical regions of the world. The disease is caused by several species of parasites of the Plasmodium genus type and transmitted to humans by the bite of a female anopheles mosquito when taking the blood meal necessary for egg production. In 2018, there was an estimated 228 million cases of malaria worldwide with about 405,000 deaths attributable to malaria[39]. Various mathematical models of the transmission dynamics of malaria and its control have been proposed in the literature [2,3,5,8,10,15,16,20,34,38]. The very first model is that of Ross-MacDonald who laid the foundations for modeling malaria [26,35]. Models that include therapeutic (treatment and vaccination) and non-therapeutic (insecticide-treated bed net) measures have also flourished in the literature [12,17,19,33]. Because insecticide-treated nets (ITNs) reduce human/mosquito contacts, distribution campaigns have been organized in affected countries, including Cameroon. However, the use of these mosquitoes treated bednets have not always been satisfactory as several people let holes in the bednets, do not use them every night (poor adherence), or use these bednets for other activities such as fishing [36]. Mosquitoes insecticide-treated bednets could influence the force of infection, the rate of recruitment of new females mosquito, or the death rate of mosquitoes [6,10,11,17,19]. Moreover, the use of these treated bednets could influence the rate of loss of immunity.

    We formulate a mathematical model for the transmission dynamics of malaria in human populations, which takes the (good or poor) use of bednets as a control measure. First, we formulate the autonomous model with a constant proportion of bednets usage as control strategy. Next, we compute the basic reproduction number T0, and investigate the existence and stability of the equilibria. Analytical results show that both model equilibria; the disease-free and the endemic states are locally asymptotically stable when T0< and when T0>1, respectively. However, the model could exhibit the phenomenon of backward bifurcation when T0<1, an epidemiological situation where, although necessary, having the basic reproduction number less than unity is not sufficient for malaria elimination [38].

    We then extend our autonomous model by considering a time-dependent control of the proportion of bednets usage. Optimal control theory is used to establish conditions under which the spread of malaria can be mitigated. The characterization of the optimal control is obtained by the application of Pontryagin's maximum principle. We use the Forward-backward fourth-order Runge-Kutta method for numerical simulations to determine an optimal control strategy. In addition, we focus on a bednet control strategy since the other controls measures are expensive. By other vector controls, we mean outdoor application of larvicides (chemical or biological), breeding habitat reduction (e.g., draining standing water), outdoor vector control (mosquito fogging, attractive toxic sugar bait (ATSB)), indoor residual spraying (IRS), repellents, including topical repellents, mosquito coils, etc, rapid diagnosis and treatment (RDT), preventative drugs like seasonal malaria chemo-prevention (SMC), intermittent preventative treatment (IPT) [19]. Generally, the bednet control in the literature concerns the bednets usage, including insecticide-treated bed nets (ITNs), long-lasting insecticide-treated nets (LLINs), and untreated bednets (UBNs) [19]. Some of these other vector controls do not respectful the ecological population environment. But, the use of these other vector control require periodic actions on a short time (1 day, 1 week, 1 month, 1 year) that is less effective and practicable than the three years' use of the LLNs. We observe that the disadvantage (loss of immunity, even at its maximum) in the use of bednets is compensated by the benefit of the number of susceptible/infected individuals excluded from the malaria disease dynamics, the only danger being the poor use of the long-lasting bednets. In fact, the model suggests that if one family fails to use accurately the LLNs, then the number of infectious will increase (sometimes exponentially) since the protection of the bednets would drop down. However, it is possible to come close to the optimal results with a realistic strategy.

    The rest of the paper is organized as follows. In Section 2, we present the mathematical model for malaria transmission dynamics with a parameter w that represents the proportion of persons having and using the treated mosquito bednets correctly. In Section 4, we propose an optimal control problem for the minimization of the number of infected humans while controlling the cost of control interventions with bednets. Finally, in Section 4.4, some numerical simulations provided to support the analytical results are interpreted from the epidemiological point of view. Section 5 is the conclusion.

    In this section, we consider two populations, namely human hosts and female mosquitoes that are assumed homogeneously distributed. We suppose that female mosquitoes only feed with human blood. The human population is subdivided into three classes, namely the susceptible Sh, the infectious Ih and the immune Rh as shown in Figure 1. According to [20], we do not consider an exposed compartment because it does not significantly influence qualitatively the evolution of infections in the human population. The recruitment is done only in the susceptible class at a rate Λh. In all compartments, there is an output of μh due to natural death, with an additional death rate μd in the infectious compartment. When in contact with an infectious mosquito, a susceptible humans become infected at a rate αh representing the force of infection. Infectious humans recover and gain immunity at a rate δh, while the rate of loss of immunity is γh.

    Figure 1.  Compartment diagram of the human component of the model.

    Following the work in [17,19,37], we distinguish the questing anopheles (looking for the blood meal) from the resting one. As depicted in Figure 2, the new mosquitoes arrive through the compartment of questing susceptible at rate Λv. The natural death rate is μv. When in contact with an infectious human, a susceptible mosquito can become infected at a rate of αv corresponding to the strength of infection of the mosquitoes. Once infected, a mosquito will alternate (at most 12 times) during its period of latency between the resting status and the questing status. Hence, we consider 6 resting infected compartments and 6 questing infected compartments. The transition rate from resting status to questing status is χ, and the transition rate from questing to resting is β. Assuming that the number of bites on a human by female mosquitoes per day is a, the fraction of persons having a long-lasting insecticide-treated net is b[0,1], and the sub-fraction of persons using it effectively is u[0,1]. Hence, the proportion of people who own a mosquito net and use it adequately is w:=b×u[0,1]. We suppose that the dependence of some parameters with respect to w can be express as

    a(w)=amax(1w)+wamin, (1)
    Λv(w)=Λmaxv(1w)+wΛminv, (2)
    γh(w)=γminh(1w)+wγmaxh, (3)
    ~μv(w)=μv+wΔμv. (4)
    Figure 2.  Compartment diagram of the mosquito component of the model.

    Indeed, the adequate use of mosquito bednet reduces transmission forces αh and αv through the factor a. It also limits indirectly the horizontal immigration of mosquitoes in the human environment via the rate Λv. Finally, w indirectly increases the rate of death in the population of mosquitoes since it reduces mosquito-human contacts, and then there is a reduction of mosquito's blood meal leading to additional death rate.

    The variables and parameters of the model are presented in detail in Table 1, Table 2 and Table 3.

    Table 1.  Variables of the model.
    Variable Description
    Humans
    Sh Number of susceptible humans within the population
    Ih Number of infectious humans within the population
    Rh Number of immune humans within the population
    Nh=Sh+Ih+Rh Total number of humans in the population
    Mosquitoes
    Sq Number of questing susceptible mosquitoes
    Eiq Number of questing infected mosquitoes in step i
    Eir Number of resting infected mosquitoes in step i
    Iq Number of questing infectious mosquitoes
    Ir Number of resting infectious mosquitoes
    Nqv=Sq+6i=1Eiq+Iq Total number of questing mosquitoes
    Nrv=7i=1Eir+Ir Total number of resting mosquitoes
    Nv=Nqv+Nrv Total number of mosquitoes

     | Show Table
    DownLoad: CSV
    Table 2.  Composite parameters of the model.
    Parameter Formula Description
    αh a(w)mIqNh Incidence rate of susceptible human
    αv a(w)(cIhNh+˜cRhNh) Incidence rate of susceptible mosquitoes
    fr χχ+μv Resting frequency of mosquitoes
    fq ββ+~μv Questing frequency of mosquitoes

     | Show Table
    DownLoad: CSV
    Table 3.  Atomic parameters of the model.
    Parameter Description Value Reference
    Human
    Λh Immigration in the host population 1000059365 [6,11]
    γmaxh Maximal transmission rate of loss of immunity within the host population 0.0146 [8]
    γminh Minimal transmission rate of loss of immunity within the host population 0.00055 [8]
    δh Rate of recovery in the host population 0.0035 [8]
    μh Death rate in the host population 159×365 [6,11]
    μd Disease-induced death rate within the host population [105,103] [6]
    amax Maximal number of bites on humans by one female mosquito per day 190.5 [8]
    amin Minimal number of bites on humans by one female mosquito per day 4.30.33 [8]
    m Infectivity coefficient of hosts due to a bite of infectious vector 0.022 [13]
    Mosquitoes
    Λmaxv Maximun immigration rate of vectors 10421+1 Assumed
    Λminv Minimun immigration rate of vectors 10421 [11]
    χ Rate at which resting vectors move to the questing state 15 [37]
    β Rate at which questing vectors move to the resting state 23 [37]
    μv=μminv Natural death rate of vectors 121 [6,11]
    Δμv Death rate of vectors due to bednet 121 [6,11]
    c Infectivity coefficient of vector due to a bite of infectious host 0.48 [13]
    ˜c Infectivity coefficient of vector due to a bite of removed host group 0.048 [13]

     | Show Table
    DownLoad: CSV

    According to the above description and assumptions, we proposed the following system of non-linear ordinary differential equations (5).

    {Sh=Λh+γh(w)Rh(αh(w)+μh)Sh,Sq=Λv(w)(αv(w)+~μv(w))Sq,E1r=αv(w)Sq(χ+μv)E1r,Eir=βEi1q(χ+μv)Eir,2i6,Eiq=χEir(β+~μv(w))Eiq,1i6,E7r=βE6q(χ+μv)E7r,,Ir=βIq(χ+μv)Ir,Iq=χ(E7r+Ir)(β+~μv(w))IqIh=αh(w)Sh(δh+μh+μd)Ih,Rh=δhIh(γh(w)+μh)Rh. (5)

    By setting xS=(Sh,Sq)R2 and xI((Eir,Eiq)1i6,E7r,Ir,Iq,Ih,Rh)R17, the system (5) can be rewritten as

    x=A(x)x+b. (6)

    More precisely, we have

    {xS=AS(x)xS+AS,I(x)xI+bS,xI=AI(x)xI, where (7)
    bS=(ΛhΛv), AS=(μh00~μv) and

    ASI is the 2×17 matrix with all its coefficients null, except ASI(1,15)=amShNh, ASI(1,17)=γh, ASI(2,16)=acSqNh and ASI(2,17)=~acSqNh.

    AI=(A11A12A21A22).

    A11 is a 13×13 matrix satisfying

    {A11(i,i)=χμv,i=1,3,,13A11(i,i)=β~μv,i=2,4,,12A11(i+1,i)=χ,i=1,3,,11A11(i+1,i)=β,i=2,4,,12A11(i+1,i)=0, otherwise.

    A12 is the 13×4 matrix defined by A12(1,3)=acSqNh, A12(1,4)=a˜cSvNh and A12(i,j)=0 for other cases. The matrix A21 is the 4×13 matrix defined by A21(2,13)=χ and A21(i,j)=0 for the other cases. The matrix A22 is the 4×4 square matrix defined by

    A22=((χ+μv)β00χ(β+~μv)000amShNh(δh+μh+μd)000δh(γh+μh)).

    Proposition 2.1. The non-negative cone (R+)19 is positively invariant for system (7).

    Proof 2.0.1. The result comes from the fact that A(x) is a Metzler matrix for allx(R+)19, i.e., aij(x)=A(x)(i,j)0 for ij. Indeed, since the solution of the system (7) is continuous, it suffices to check that every trajectory starting at the boundary of (R+)19 remains in (R+)19. That boundary is defined as 19i=1Hi where Hi{x(R+)19,xi=0}. For xHi, xi=19j=1aij(x)xi+bi=19j=1,ijaij(x)xi+bi0. This means that xi moves from 0 to non-negative values.

    Proposition 2.2. The set Ω={(Sh,Sq,(Eir,Eiq)1i6,E7r,Ir,Iq,Ih,Rh)R19+/0Nh Λhμh,0NvΛvμv} is a compact forward-invariant and absorbing set for the model system (5).

    Proof 2.0.2. Adding respectively the human sub-populations equations and then the mosquitoes sub-populations equations, we have

    Nh=ΛhμhNhμdIhΛhμhNh, (8)
    Nv=ΛvμvNv˜μvwNqvΛvμvNv. (9)

    From Proposition 2.1, we have Nh,Nv0. Applying Gronwall's inequality, we have

    Nh(t)Nh(0)eμht+Λhμh(1eμht), (10)
    Nv(t)Nv(0)eμvt+Λvμv(1eμvt). (11)

    Assuming that Nh(0)Λhμh and Nv(0)Λvμv. it follows that NhΛhμh and NvΛvμv.

    Proposition 2.3. At any time t, the model system (5) is well-posed, i.e., it admits a non-negative global unique C1-solution on the set R+.

    Proof 2.0.3. Applying the Cauchy-Lipschitz theorem, Proposition 2.1 and Proposition 2.2, the result is immediate.

    In this section we study the existence of equilibria and their stability using the theory of bifurcation.

    This section is devoted to local and global stability of the disease-free equilibrium which unconditionally exists.

    Proposition 3.1. System (5) admits a disease-free equilibrium (DFE) given by x=(xS,xI) with

    xS=(Sh,Sq)=(Λhμh,Λv~μv) and xI=0R17R17 (12)

    Proof 3.0.1. An equilibrium is obtained by solving x=A(x)x+b=0. A DFE corresponds to any solution satisfying xI=0R17.

    Proposition 3.2. The system x=AS(x).(xxS) is globally asymptotically stable (GAS) at xS on R2+.

    Proof 3.0.2. The proof is immediate since the matrix

    AS(x)=(μh00~μv)

    has all its eigenvalues μh and ~μv negative.

    In the following, we determine a stability threshold condition using a technique well described and used in [17,37]. In our case, this threshold can be biologically interpreted as the basic reproduction number T0 [19].

    Let us define the threshold

    T0=Λvam(frfq)7β~μv(1fqfr)aμh[c(γh+μh)+˜cδh]Λh(δh+μh+μd)(γh+μh). (13)

    Theorem 3.1. T0 is equivalent to the basic reproduction number T0 of the model system (5). Moreover, the DFE is locally asymptotically stable (LAS) if T01.

    Proof 3.1.1. According to Gautier-Sallet's algorithm in [18] and Proposition 3.2, it is sufficient to obtain condition ensuring that AI(x) is stable (all its eigenvalues are negative). Looking carefully at A11 and A22, they are Metzler stable. Thus, letN=A22(x)A21(x)×A111(x)×A12(x). Using formal calculus (under Sagemath software for instance), we have

    N=((χ+μr)β00χ(β+~μv)000amShNh(δh+μh+μd)000δh(γh+μh))(000000χ7β6(χ+μv)7(β+~μv)6acSqNhχ7β6(χ+μv)7(β+~μv)6a˜cSqNh00000000).=((χ+μr)β00χ(β+~μv)acf7rf6qSqNha˜cf7rf6qSqNh0am(δh+μh+μd)000δh(γh+μh))

    Let us consider the 2×2 blocks in N. Since N11 is Metzler stable, let L=N22N21×N111×N12.

    L=((δh+μh+μd)0δh(γh+μh))+(0am00)(β+~μv(χ+μv)(β+~μv)βχβ(χ+μv)(β+~μv)βχχ(χ+μv)(β+~μv)βχχ+μv(χ+μv)(β+~μv)βχ)(00acf7rf6qSqNha˜cf7rf6qSqNh)=((δh+μh+μd)+am(χ+μv)(χ+μv)(β+~μv)βχacf7rf6qSqNham(χ+μv)(χ+μv)(β+~μv)βχa˜cf7rf6qSqNhδh(γh+μh))

    AI(x) is stable if

    amδh(χ+μv)(χ+μv)(β+~μv)βχa˜cf7rf6qSqNh(γh+μh)(δh+μh+μd)+am(χ+μv)(χ+μv)(β+~μv)βχacf7rf6qSqNh<0

    Since Nh=Sh, this is equivalent to

    1>a2mf7rf7qSq(˜cδh+c(γh+μv))βSh(δh+μh+μd)(γh+μv)(1frfq)=T0.

    Theorem 3.2. The DFE is GAS in Ω when T0<μhμh+μdζ.

    Proof 3.2.1. Our proof relies on Theorem 4.3 in [18], which establishes global asymptotic stability (GAS) for epidemiological systems that can be expressed in matrix form (7). The demonstration is completely similar to that made in [37].

    Theorem 3.3. There exists R,Rc,R+R such that the model system (5) has

    (a) a unique endemic equilibrium if R0>1,

    (b) a unique endemic equilibrium if R0=1 and Rc<1,

    (c) two endemic equilibria if Rc<R0<min(1,R) or max(Rc,R+)<R0<1,

    (d) No endemic equilibrium elsewhere.

    Proof 3.3.1. An endemic equilibrium is any non-zero and positive solution of the following system

    {Λh+γhRh(αh+μh)Sh=0,(14)αhSh(δh+μh+μd)Ih=0,(15)δhIh(γh+μh)Rh=0,(16)Λv(w)(αv+(μv+wΔμv))Sq=0,(17)αvSq(χ+μv)E1r=0,(18)χEir(β+(μv+wΔμv))Eiq=0,1i6,(19)βEi1q(χ+μv)Eir=0,2i6,(20)β(E6q+Iq)(χ+μv)Ir=0,(21)χIr(β+(μv+wΔμv))Iq=0.(22)

    The equations (15) and (16) allow us to write Sh and Rh in function of Ih as follows: Sh=δh+μh+μdαhIh and Rh=δhγh+μhIh.

    To simplify the expressions, let D=δh+μh+μd, C=δhγh+μh and F=γhC.

    By subsequently replacing Sh and Rh by their values in (14), we then obtain the expression of Ih with respect to αh.

    Also, αv=a(cIh+˜cRh)Nh=a(C˜c+c)αh(C+1)αh+D. Using equations (17), (18), (19), (20), (21) and (22), we have

    Sq=Λv(w)αv+(μv+wΔμv), E1r=αvSqχ+μv, Eiq=χEirβ+(μv+wΔμv)for1i6, Ejr=βE(j1)qχ+μvfor2j7; Iq=χfqE7rβ(1fqfr) et Ir=βIqχ+μv.

    So all our unknowns are expressed in terms of αh, and it only remains to determine the value of αh.

    By definition, we have αh=amIqNhand by replacing Iq and Nh by their values, and after simplification and re-arrangement, we obtain

    αh[P2(αh)2+P1αh+P0]=0, (23)

    where

    P2=β6χ7μh(C+1)(1fqfr)[a(˜cC+c)+(μv+wΔμv)(C+1)]<0,
    P1=Dβ6χ7(1fqfr)[(μv+wΔμv)T0(DF)μh(a(˜cC+c)+(μv+wΔμv)(C+1))]=Dβ6χ7(1fqfr)(μv+wΔμv)(DF)[T0Tc]withTc=μh[a(˜cC+c)+(μv+wΔμv)(C+1)](μv+wΔμv)(DF) (24)
    P0=D2μh(μv+wΔμv)β6χ7(T01).

    Equation (23) has solution αh=0 and solutions of the equation (E):P2(αh)2+P1αh+P0=0.

    The case αh=0 leads us to the equilibrium without disease. We are interested in the equation (E), of which we are going to analyze the number of positive solutions as a function of the value of T0.

    1. If T0>1 then, P0>0 and since P2<0, the discriminant Δ=P214P2P0 of the equation (E) is positive, hence the equation (E) has two different real solutions. In addition, the product of the solutions is p=P0P2<0. Hence, equation (E) has a unique positive solution.

    2. if T0=1, then, equation (E) has two different real solutions, which are zero and P1P2. But P2<0, so this solution is positive if P1>0, that is to say if T0>Tc.

    3. if T0<1 and Δ=P214P2P0>0 and T0>Tc, then, equation (E) admits two different positive solutions.

    Let P2=b2 P1=b1(T0Tc) and P0=b0(T01); b2,b1andb0 are all positive coefficients. We have, Δ=P214P2P0=b21T20(2b21Tc4b0b2)T04b0b2+b1T2c.

    The last condition can be re-written as follows {Tc<T0<1,Δ=b21T20(2b21Tc4b0b2)T04b0b2+b1T2c>0.

    Let us study the sign of Δ in relation to the values of T0. Consider the equation

    (ET0):b21T20(2b21Tc4b0b2)T04b0b2+b21T2c=0

    (ET0) has as discriminant Δr=(2b21Tc4b0b2)24b21(4b0b2+b21T2c)=16b2b0[b2b0+b21(1Tc)] which is positive for Tc<1, and the equation (ET0) has two solutions T and T+.

    We then have {Tc<T0<1,T0],T[]T+,+[,

    which yields Tc<T0<min(1,T), where max(Tc,T+)<T0<1.

    Remarks 3.1. For the (global) stability of the endemic equilibrium, one could follow the approach in [19] by using a suitable Lyapunov-type functional along the positive flow of the model 5 on a "two domains" subdivision of the phase state R19, under appropriate conditions.

    Note that the disease-free equilibrium is only globally asymptotically stable when T0<ζ<1, so it is possible that if this condition is violated, bistability could occur. That is, for ζ<T0<1, a stable DFE could co-exist with a stable endemic equilibrium, a phenomenon known as backward bifurcation [7,9,14,21,38]. In this case, the condition T0<1, although necessary is insufficient to mitigate the spread of malaria in a community.

    The following figure depicts the backward bifurcation for model system (5), representing the plotting of Ih as a function of T0 for values of the bifurcation parameter m ranging from 0.001 to 0.01, μd=4.54×103 and Tc=0.388975.

    To find the solution to the model (5), following four steps are followed. (i) Description the optimal control (ii) Proof of the existence of optimal control problem (iii) Proof of the uniqueness of the optimal control (iv) Numerically solve the optimal control and show these graphically.

    There are several methods to mitigate the prevalence of malaria in a community by reducing the mosquito density, contact, longevity and competence. Among all these methods, the possession and correct use of insecticide-treated mosquito bednets is the strategy that considers three of the biological elements mentioned above [27]. Therefore, we consider as control the function w representing the fraction of population that owns and properly use a mosquito net.

    Consider the following objective functional

    J(w)=T0[A1Ih+A2(6i=1Eiq+Iq)+Bw2(t)]dt. (25)

    The terms A1Ih and A2(6i=1Eiq+Iq) are the cost of infection while Bw2(t) is a quadratic cost related to the effort of using bednets. Our main goal is to find an optimal control function w such that J(w)=min{J(w)wΓ(T)}, with Γ(T) the set of admissible controls defined as

    Γ(T)={ωω(.) is Lebesgue mesurable on[0,T],
    0ω(t)1,t[0,T]}.

    The aim of this section is to prove the existence of an optimal control for the model system (5) and then derive the optimality system. The existence of optimal control of the system (5) will be considered by applying the following theorem [4,22,23,25,29].

    Theorem 4.1. Consider the objective functional J given by equation (25), with wΓ subject to the constraint state system (5). There exists wΓ(T) such thatJ(w)=min{J(w)wΓ(T)} subject to thecontrol system (5) with initial conditions at t=0.

    Proof 4.1.1. The state and control variables of the system (5) are positive and thecontrol set Γ(T) is closed and convex. Therefore, the integrand of the objectivefunctional J in which it was expressed in the system (5) is a convex function of ω on the control set Γ(T). Since the state solutions are bounded, the Lipschitz property of the state system with respect to the state variables is satisfied. It can also be seen that there exist positive numbers η1 and η2, and a constant ε>1 such that J(ω)η1ωεη2. Therefore, the state variables are bounded and the existence of optimal control of the model system (5) is concluded.

    To derive the necessary conditions that the three controls and corresponding state variables must satisfy, we use Pontryagin's maximum principle [31]. To this end, we define the Hamiltonian function for the system, where λi,i=1,,19 are the adjoint variables or co-state variables

    H=A1Ih+A2(6i=1Eiq+Iq)+Bw2(t)+λ1[Λh+(γminh+w(t)Δγh)Rh(mIQNh(amaxw(t)Δa)+μh)Sh]+λ2[(mIQNh(amaxw(t)Δa))Sh(δh+μh+μd)Ih]+λ3[δhIh(γminh+w(t)Δγh+μh)Rh]+λ4[Λmaxvw(t)ΔΛv((cIhNh+˜cRhNh)((amaxw(t)Δa)+μv+w(t)Δμv)Sq]+λ5[((cIhNh+˜cRhNh)(amaxw(t)Δa))Sq(χ+μv)E1r]+6i=1λi+5[χEir(β+μv+w(t)Δμv)Eiq]+7i=2λi+10[βEi1q(χ+μv)Eir]+λ18[βIq(χ+μv)Ir]+λ19[χ(E7r+Ir)(β+μv+w(t)Δμv)Iq]. (26)

    The following result presents the adjoint system and control characterization.

    Theorem 4.2. Given an optimal control w, and the corresponding state solutions

    Sh,Ih,Rh,Sq,E1r,E1q,E2q,E3q,E4q,E5q,E6q,E2r,E3r,E4r,E5r,E6r,E7r,Ir,Iq

    of the corresponding state system (5), there exists adjoint variables, λi,i=1,,19, satisfying

    {λ1=(amaxΔaw(t))[cIh+˜cRhN2hSq(λ5λ4)                 +mIqNh[ShNh1](λ2λ1)]+μhλ1,λ2=A1mIqN2h(amaxΔaw(t))Sh(λ2λ1)        +cNhcIh˜cRhN2h(amaxΔaw(t))Sq(λ4λ5)        λ3Sh(δh+μh+μd)λ2,λ3=(γminh+w(t)Δγh)(λ3λ1)mIqN2h(amaxΔaw(t))Sh(λ2λ1)        +˜cNhcIh˜cRhN2h(amaxΔaw(t))Sq(λ4λ5)+μhλ3,λ4=(μv+˜μw(t))λ4cIh+˜cRhNh(amaxΔaw(t))(λ5λ4),λ5=(χ+μv)λ5χλ6,λi=(β+Δμvw(t))λiβλi+6A2,  for  i=6,,11,λi=(χ+μv)λiχλi5  for  i=12,,16,λ17=(χ+μv)λ17χλ19,λ18=(χ+μv)λ18χλ19,λ19=(β+μv+w(t)Δμv)λ19βλ18        mNh(amaxΔaw(t))Sh(λ2λ1)A2,λi(T)=0,  for  i=1,,19. (27)

    The control w satisfies the optimality condition.

    w=max{0,min(1,12B[ΔγhRh(λ3λ1)+ΔaShαv(λ2λ1)+(Δaαh(λ5λ4)+Δμvλ4)Sq+λ19ΔμvIq+6i=1λi+5Eiq+ΔΛvλ4])}, (28)

    where αv=mIqNh and αh=cIh+˜cRhNh.

    Proof 4.2.1. The differential equations governing the adjoint variables are obtained by differentiation of the Hamiltonian function, evaluated at the optimal control. Then, the adjoint system can be written as

    λ1(t)=HSh,λ2(t)=HIh,λ3(t)=HRh,λ4(t)=HSq,λ5(t)=HE1r,λi(t)=HEi5q,fori=6,11,λi(t)=HEi10r,fori=12,16,λ17(t)=HE7r,λ18(t)=HIr,λ19(t)=HIq,

    with zero final time conditions (transversality) λi(T)=0. The characterization of the optimal control given by (28) is obtained by solving the equations on the interior of the control set, where 0<w<1. That is,

    Hω=2Bω(ΔγhRh(λ3λ1)+ΔaShαv(λ2λ1)+(Δaαh(λ5λ4)+Δμvλ4)Sq+λ19ΔμvIq+6i=1λi+5Eiq+ΔΛvλ4),

    with Hω=0, where αv=mIqNh and αh=cIh+˜cRhNh.Hence, we obtain

    ω=12B[ΔγhRh(λ3λ1)+ΔaShαv(λ2λ1)+(Δaαh(λ5λ4)+Δμvλ4)Sq+λ19ΔμvIq+6i=1λi+5Eiq+ΔΛvλ4].

    We numerically solve the optimal transmission parameter control for the malaria model. The optimal control is obtained by solving the optimality system, consisting of 19 non-linear ordinary differential equations from the state and adjoint equations. An iterative scheme is used for solving the optimality system [24]. For the simulations, we consider the initial (and arbitrary) number of individuals at time t=0: Sh(0)=100000,Ih(0)=100,Rh(0)=1000,Sq(0)=100000, E1r(0)=10, E2r(0)=9,E3r(0)=8, E4r(0)=7,E5r(0)=6,E6r(0)=5,E7r(0)=4, E1q(0)=3,E2q(0)=3,E3q(0)=3,E4q(0)=3, E5q(0)=3, E6q(0)=2,Ir(0)=35,Iq(0)=800.

    For the cost weight in the objective functional J, we take B=4.5USD (for three years) which represents what the state of Cameroon spends on the purchase of an insecticide-treated mosquito net for two individuals [32]. This is somehow comparable to B=3.95USD for the average cost for a household (with about 5.5 individuals) per month (for the first two largest cities of Cameroon in terms of population - Douala and Yaounde [28]). The main practical problem is the difficulty to provide bednets to everybody in the household as well as individuals complain of feeling excessive heat when sleeping under a bednet [28], the latter being a potential reason why some individuals use other vector control measures. We could consider the cost per household2 for T=3years, and finally discuss the impact of the optimal controls associated to B1=4.55.52 and B2=3.9536. Clearly, from the economic stand point, the other vector control strategies are more expensive than the bednet control. Thus, we focus on the bednet control strategies. The simulations are carried out with T = 3 years, the duration of one LLIN (Long-lasting insecticide-treated bednet) efficacy [32]. We define the uniform control uunif(t)=k and the multi-intervals ("stage") one ustages(t)={u1fflt[0;365.25 days ],u2fflt]365.25;730.5 months ],u3fflt]730.5;1080.75 months ], over three years with k,u1,u2,u3R+ and 11080.751080.750ustages(t)dt=k as the mean value. uoptimal is an optimal control for our optimal problem in Theorem 4.1. uforced is an administrative control of distribution of the bednets over three years (it is imposed or "forced"); it is either uunif or ustages. We also define the following in percentage:

    1. tuoptimals(uforced)=100×Total of susceptible humans on [ 0;T] under uforced Total of susceptible humans on [ 0;T] under uoptimal;

    2. tuoptimalIh(uforced)=100×Total of infectious humans on [ 0;T] under uoptimal Total of susceptible humans on [ 0;T] under uforced;

    3. tuoptimalRh(uforced)=100×Total of recovered humans on [ 0;T] under uoptimalTotal of recovered humans on [ 0;T] under uforced.

    Definition 4.1. Let Γ(T) be the set of admissible controls relative to a dynamical system D(u(.)), u(.)Γ(T). An optimal control, mathematically admissible, is biologically admissible if tuoptimals(uforced)100, tuoptimalIh(uforced)100 and tuoptimalRh(uforced)100.

    It is easy (even numerically) to study the biological admissibility to an (mathematically) admissible control uforced. But (for all uforced), the biological admissibility is a challenge related to the choice of the objective function.

    Numerically, for T=1080.75 days, uunif(t)=0.65 and ustages(t)={u1=0.9t[0;365.25 days ],u2=0.6t]365.25;730.5 days ],u3=0.45t]730.5;1080.75 days ]. For all our numerical simulations, graphs related to optimal control are in black solid lines, while those linked to the "forced" control are in solid green lines. The effects of the "uniform" control are graphically represented in Figures 4-9 while the "stage" control effects are shown in Figures 10-15.

    Figure 3.  Backward bifurcation diagram, showing the co-existence of a stable DFE and two branches of endemic equilibria (a stable and an unstable branch).
    Figure 4.  Number of susceptible humans Sh: optimal versus uniform controls.
    Figure 5.  Number of infectious humans Ih: optimal versus uniform controls.
    Figure 6.  Number of recovered individuals Rh: optimal versus uniform controls.
    Figure 7.  The optimal control uoptimal compared to the uniform control uunif: optimal versus uniform controls.
    Figure 8.  Total number of latent questing E.q and latent resting E.r mosquitoes: optimal versus uniform controls.
    Figure 9.  Number of questing Iq and resting Ir infectious mosquitoes: optimal versus uniform controls.
    Figure 10.  Number of susceptible humans Sh: optimal versus "stage" controls.
    Figure 11.  Number of infectious humans Ih: optimal versus "stage" controls.
    Figure 12.  Number of recovered individuals Rh: optimal versus "stage" controls.
    Figure 13.  Optimal control uoptimal compared to the "stage" control ustage.
    Figure 14.  Total number of latent questing E.q and latent resting E.r mosquitoes: optimal versus "stage" controls.
    Figure 15.  Number of questing Iq and resting Ir infectious mosquitoes: optimal versus "stage" controls.

    Practically, the common strategies uforced in malaria affected countries are decreasing functions of time (as trends), due to the difficulty to maintain a constant or high (>90%) level of possession and use of bednets throughout a 3-year campaign of LLINs distribution.

    Table 4 suggests that, even if the "stage" and uniform controls have the same mean, it is better to use a "stage" control with emphasis on the first few months of the 3 years. Clearly, the effort should be done to cover the gap (between the results of the system state following uforced and uoptimal) in Table 4. This approach, based on the reality of the malaria programs in each country, could support health policies and decision-makers in order to obtain an accurate threshold in the percentage ϵuforced of the "administrative/public planners controls" uforced (ustage or uunif) applications compared to optimal effects, such that 100tuoptimals(uforced)ϵuforced, 100tuoptimalIh(uforced)ϵuforced and 100tuoptimalRhϵuforced. This allows us to introduce the definitions of the ϵuforced-approximate weak or strong "sub-optimal" controls.

    Table 4.  Results in percentage.
    tuoptimals(uforced) tuoptimalIh(uforced) tuoptimalRh(uforced)
    uunif 99.769 43.530 52.164
    ustages 99.929 68.500 77.009

     | Show Table
    DownLoad: CSV

    Definition 4.2. (Approximate controlability) Let Γ(T) be the set of admissible controls relative to a dynamical system D(u(.)), u(.)Γ(T), for T>0. For Vuoptimaluforced=(tuoptimals(uforced),tuoptimalIh(uforced),tuoptimalRh(uforced)), let define

    Normstrong(Vuoptimaluforced):=max{100tuoptimals(uforced),100tuoptimalIh(uforced),100tuoptimalRh(uforced)},

    also written as

    Normstrong(Vuoptimaluforced):=100min{tuoptimals(uforced),tuoptimalIh(uforced),tuoptimalRh(uforced)},

    and

    Normweak(Vuoptimaluforced):=13(100tuoptimals(uforced))+(100tuoptimalIh(uforced))+(100tuoptimalRh),

    That is,

    Normweak(Vuoptimaluforced):= 10013{tuoptimals(uforced)+tuoptimalIh(uforced)+tuoptimalRh},

    A biologically admissible control uforced is ϵuforced-approximate weak "sub-optimal" if

    Normweak(Voptimal(uforced))ϵuforced.

    A biologically admissible control uforced is ϵuforced-approximate strong "sub-optimal" if

    Normstrong(Vuoptimaluforced)ϵuforced.

    Remarks 4.1. These definitions in 4.2 improve on the efficiency index [1]. It is possible to consider the reduction of noise Nmosq (similar to Norm. for mosquitoes) produced by mosquitoes as the percentage of mosquitoes with optimal control compared to the states with forced control: then the new index would be Normα,β.:=αNorm.+βNmosq such that α+β=1. The coefficients α and β relate respectively the importance of the humans' group and mosquitoes' group. Herein, we focus on the optimal impact on humans and consider α=1.

    Straightforward computations lead to the following proposition 4.1.

    Proposition 4.1. There is an equivalence between Normweak and Normweak:

    NormweakNormstrong3.Normweak

    In Table 4,

    Normweak(Vuoptimaluunif)=34.845926,
    Normweak(Vuoptimalustage)=18.186947,
    Normstrong(Vuoptimaluunif)=56.470157,

    and

    Normstrong(Vuoptimalustage)=31.499672.

    We see that for ϵuforced=35%, the Vuoptimaluunif is 35%-approximate weak "sub-optimal" like Vuoptimalustage. But only Vuoptimalustage is 35%-approximate strong "sub-optimal" and not Vuoptimaluunif. Another interesting point is the fact that Normweak(Vuoptimalustage)=18.186947, and this comes from the fact that the "weak" deviation from the optimal strategy is only about 18.187% in total (with the collective effort/contribution of Sh,Ih and Rh to reach the optimal strategy). By the way, Normstrong(Vuoptimalustage)=31.410, and this corroborates the fact that the "weak" deviation from the optimal strategy is only about 31.410% following the individual efforts/contributions of Sh,Ih and Rh to reach the optimal objective. We observe that the disadvantage (loss of immunity) in the use of bednets is compensated by the benefit of the number of susceptible/infected individuals excluded from the malaria disease dynamics. Then, it is possible to get closer to the optimal results with a realistic strategy.

    We formulated and rigorously analyzed a vector multi-stage malaria model with the use of mosquito treated bednets as preventive measure. The proposed model is biologically meaningful and mathematical well-posed. We investigated the local and global stability of equilibria. The analytical results reveal the possibility of bistability when T0<ζ<1 (see subsection 3.1.1 with additional mortality δh less than 105 see discussion in [6]). That is, the model could exhibit the phenomenon of backward bifurcation, an epidemiological situation which although necessary, having the basic reproduction number less than unity is no longer sufficient to mitigate the malaria transmission dynamics [38]. Thus, a low level of additional (disease-induced) mortality could lead to the existence of an endemic equilibrium even if the basic reproduction number is less than one.

    Next, an optimal control strategy is investigated with the proper usage of LLINs (during three years compared to a "forced" control) as the control parameter. We observe that the disadvantage (loss of immunity) in the use of bednets is compensated by the benefit of the number of susceptible/infected individuals excluded from the malaria disease dynamics. Moreover, it is possible to get close to the optimal results with a realistic strategy.

    Results from this study could help inform health policy and decision-makers on the potential optimum strategies to mitigate malaria transmission dynamics in affected communities by designing reachable malaria program implementation objectives "close" to the optimal strategies ϵu.% by the "weak" collective contribution or the "strong" individual effort to achieve the optimal objective. The notions of ϵuforced-approximate strong/weak "sub-optimal" control are more practical than the theoretical optimal control which remains a daunting task to health officials. The upper bound ϵuforced of the gap, from the "sub-optimal" results to the optimal ones, is of great interest practically since it delineates the acceptable error one could essentially make if we apply uforced instead of uoptimal.

    The proposed model is not exhaustive. One could consider splitting the human population into adults and juveniles since malaria disproportionately affects children under five years of age. The model could then be fitted with real data from an affected region/country with most model parameter values estimated for the purpose. Also, because of uncertainties in most parameter values, a detailed sensitivity and uncertainty analysis could be carried out to understand the dependence of the basic reproduction number and model state variables on their components [30].

    The authors declare that they have no conflicts of interest to this work.



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