Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 6 (2016), pp. 4959-4989 © Research India Publications http://www.ripublication.com/gjpam.htm
Optimal Control Analysis of Cryptosporidiosis Disease S.T. Ogunlade African Institute for Mathematical Sciences, Senegal K.O. Okosun1 , R.S. Lebelo, and M. Mukamuri Vaal University of Technology, Vanderbijlpark, South Africa.
Abstract In Sub-Saharan Africa, HIV/AIDS and chronic parasitic diseases (such as trypanosomiasis, schistosomiasis, malaria, etc) have profound effect on the immune system and alter the host’s immune response to infections, thereby paving way for opportunistic parasitic infections (such as cryptosporidiosis, isosporiasis, microsporidiosis, etc) to attack. Hence, mathematical models provide a quantitative and potentially valuable tool for this purpose. This paper therefore is to derive and analyze a deterministic model for the transmission of cryptosporidiosis disease. The basic reproduction number, and the existence and stability of equilibria were investigated. Firstly, we analyzed the model and investigated its stability and bifurcation behaviour. In our findings, model exhibited a backward bifurcation phenomenon. This has epidemiological implication, which means that for effective eradication and control of the disease, bringing R0 less than unity was no longer sufficient, but rather, that R0 < R0c . Furthermore, we modified the model to incorporate susceptible individuals with weakened immune system. We then analyzed the model and investigated its stability property, in order to determine the effect and impact of individuals with weakened immune systems in the disease transmission. From the sensitivity analysis, it was found that increase in the number of susceptible individuals with weak immune system increases the basic reproduction number. AMS subject classification: Keywords:
1
Corresponding Author - E-mail:
[email protected]
2
1.
S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri
Introduction
The study of infectious disease outbreaks (e.g, endemics,epidemics and pandemics) [7, 14, 15], spreads and their control have been major problems over the years. Ever since, scientists, in particular epidemiologists have taken up the quest by formulating different strategies such as formulating and analyzing mathematical models so as to help analyze, minimize and control the disease spread into the population [3, 5, 4, 32]. In time past, following the description of Cryptosporidium in mice by Ernest Edward Tyzzer [16], the genus Cryptosporidium has been studied, and now discovered to contain numerous species and genotypes adapted to parasitic life in almost all classes of vertebrates. Over the years, our knowledge has expanded from microscopic observations of infection and environmental contamination to the knowledge obtained from large application spread of molecular techniques to taxonomy and epidemiology. Although, the medical and veterinary significance of this protozoan was not fully appreciated for an- other 70 years. The interest in Cryptosporidium escalated tremendously over the last two and half decades [38, 37]. It was later recognized as a cause of disease in 1976. As several methods were developed to analyze stool samples, the protozoa was increasingly reported as the cause of human disease [9]. At first, Crypto was categorized as a veterinary problem because, majority of the early cases were diagnosed due to individuals rearing farm animals such as cows. Furthermore, 155 species of animals specifically mammals have been reported to be infected with Cryptosporidium parvum which is also known as C. parvum [42]. Among the 15 named species of Cryptosporidium infectious to non-human vertebrate hosts C. Baileyi, C. canis, C. felis, C. hominis, C meleagridis, C. muris, and C. parvum have been reported to also infect humans. The primary hosts for C. hominis are Humans, except for C. parvum, which is widespread in non-human hosts and is the most frequently reported zoonotic species, the remaining species left have been reported primarily in immunocompromised or immunosuppressed humans [42]. The first Cryptosporidiosis outbreak that was widely known occurred in 1987[Uni] in Carrollton, Georgia. About 13,000 persons became sick as a result of the outbreak the disease. The main cause was traced to a large contaminated water system. In 1993, in Milwaukee area, Wisconsin, a massive outbreak of the disease occurred, causing approximately 400,000 people to fell sick as a result of contaminated drinking water in one of the two treatment plants serving the Milwaukee area [9]. However, there were also evidences of Cryptosporidium cases before the failure of the treatment suggesting that there had been appreciable minor transmission of the disease which occurred in the past thereby resulting in the major outbreak.
2.
Model Formation
We present and analyze a SIR model (1) for Cryptosporidiosis disease. The model stability is investigated. In order to formulate the model (1), the total human population, denoted by N, are sub-divided into sub-populations of susceptible individuals (S), in-
Optimal Control Analysis of Cryptosporidiosis Disease
3
dividuals with Cryptosporidiosis (I ), recovered human (R). So that N = S + I + R. The contaminated environment is denoted by En . We assume a small size of host and environment have contacts and so have chosen a mass action form of infection for the environmental contamination, while a standard form of infection is chosen for between host transmission. The Crypto disease transmission model is shown below. dS dt dI dt dR dt dEn dt
= + σ R − µS − β ∗m S = β ∗m S − (α + µ + ψ)I
(1)
= αI − (µ + σ )R = θI − νEn ,
where β ∗m =
λ I + ρEn S+I +R
Figure 1: Diagram showing the Model Formation
(2)
4
3. 3.1.
S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri
Basic Properties of the Model Positivity Analysis
Lemma 3.1. The closed set 3 D = (S, I, R) ∈ R+ : S + I + R ≤ , (3) µ is positively-invariant for the basic model (i.e. the model does not predict negative values for the state variables at any future time). Proof. Now we let t∗ = sup{t > 0 : S ≥ 0, I ≥ 0, R ≥ 0 ∈ [0, t]} We have that from equations 1, dS (4) + (µ + β ∗m (t) + γ (t))S = , dt −σ R where γ (t) = . S The equation 4 implies that t t d S(t) exp µt + (β(τ ) + γ (τ ))dτ = exp µt + (β(τ ) + γ (τ ))dτ . dt 0 0 (5) Now, integrating 5 from 0 to t∗ , we obtain t∗ (β(τ ) + γ (τ ))dτ − S(0) S(t∗ ) exp µt∗ + 0 ζ t∗ exp µζ + (β(ω) + γ (ω))dω dζ . (6) = 0
Dividing through by exp µt∗ + gives
0
t∗
(β(τ ) + γ (τ ))dτ , and rearranging the expression
0
t∗
S(t∗ ) = S(0) exp −µt∗ − (β(τ ) + γ (τ ))dτ 0 t∗ (β(τ ) + γ (τ ))dτ + exp −µt∗ − 0 ζ t∗ exp µζ + (β(ω) + γ (ω))dω dζ ≥ 0. × 0
0
In the same way, we can show that I ≥ 0, R ≥ 0. This completes the proof.
The lemma and proof above guarantees that the variables of the model are continuously biologically meaningful and therefore the population size can not be negative (that is, greater or equal to zero).
Optimal Control Analysis of Cryptosporidiosis Disease 3.2.
5
Invariant Region
Lemma 3.2. D is a compact attracting set (i.e. the D limit set of any orbit starting in R3+ lies in D). Proof. By the non-negativity of the model state variables as obtained in the previous lemma we have that, dN = − µN − ψI, (7) dt dN for initial conditions in R3+ and t ≥ 0, we get ≤ − µN. This means that dt d µt Ne ≤ eµt , dt t
0
N(t)eµt
t
dNeµt ≤
eµt dt
0
− N(0) ≤ [eµt − 1] µ ≤ eµt , µ
which implies that N(t)eµt − N(0) ≤
µt e . µ
(8)
. µ
(9)
So for all t ≥ 0, equation (8) becomes N(t) ≤ N(0)e−µt +
Now, if (S ∗ + I ∗ + R ∗ ) is described as a D limit point of an orbit in R3+ then there is a subsequence denoted by ti −→ ∞ such that lim (S(ti ), I (ti ), R(ti )) = (S ∗ , I ∗ , R ∗ ).
i−→∞
Hence, we have
lim (N(ti )) = N ∗ = S ∗ + I ∗ + R ∗ .
i−→∞
Now, from equation (9), evaluating at t = ti and passing to the limit i −→ ∞, it follows that N ∗ ≤ and hence (S ∗ , I ∗ , R ∗ ) ∈ D as required. µ By the above lemmas, for any initial starting point, (S0 , I0 , R0 ) ∈ R3+ , the trajectory lies in D. Thus, in D the basic model (1) is both mathematically and epidemiologically well posed. Hence, it is sufficient to study the dynamics of the model in D.
6 3.3.
S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri Basic Reproduction Number
By the Van den Driessche and Watmough [40], the basic reproduction number R0 of the model is computed by using the Next Generation Matrix Method. It is given by: R0 = r(F V −1 ), where r(.) is the spectral radius. Using the I , R and En classes we have: ρ λ 0 µ F = 0 0 0 0 0 0 α+µ+ψ 0 0 −α µ+σ 0 V = −θ 0 ν
and
Hence, by the maximum eigenvalue, we derived our R0 as R0 = 3.4. 3.4.1
λµν + θρ . µν(α + µ + ψ)
(10)
Analysis of Steady States Disease-free Equilibrium(DFE)
The disease free equilibrium(DFE) of the model (1) is given by: ∗ ∗ ∗ ∗ , 0, 0, 0 . ε 0 = (S , I , R , En ) = µ 3.4.2
Local Stability of DFE
Theorem 3.3. The disease free equilibrium ε0 exists for all R0 and is locally asymptotically stable if R0 < 1 and unstable if R0 > 1. Proof. The Jacobian matrix of the model being evaluated at the disease-free equilibrium is given by: J( ,0,0,0) µ
−µ
= 0 0 0
−λ
σ
−(α + µ + ψ) − λ
0
α θ
−(µ + σ ) 0
µ ρ µ 0 −ν
−ρ
.
Optimal Control Analysis of Cryptosporidiosis Disease
7
Now it suffices that we find the eigenvalues of the Matrix J( ,0,0,0) . Let ϒi , i = 1, 2, 3, 4 µ be the eigenvalues. We have that Det (J( ,0,0,0) − ϒI ) µ −µ − ϒ −λ σ −ρ µ 0 −(α + µ + ψ) + λ − ϒ 0 ρ = 0 . = µ 0 α −(µ + σ ) − ϒ 0 0 θ 0 −ν − ϒ (11) The Characteristic equation of the matrix (11) is given as the quartic equation below: ϒ 4 + K1 ϒ 3 + K2 ϒ 2 + K3 ϒ + K4 = 0,
(12)
where K1 = α − λ + 3µ + ν + σ + ψ,
K2 = ν(α + µ + ψ)[P − R0 ],
K3 = ν(2µ + σ )(α + µ + ψ)[Q − R0 ],
K4 = µν(µ + σ )(α + µ + ψ)[1 − R0 ],
with 3µ(µ + ν) + σ (2µ + ν) − λ(2µ + σ ) + (α + ψ)(2µ + ν + σ ) , ν(α + µ + ψ) 2(µ + σ )(αµ(µ + 2ν) + ασ (µ + ν) + µ(µ2 + 3µν + µσ + 2νσ − λ(µ + σ )) + Gg Q= ν(α + µ + ψ) Gg = (µ(µ + 2ν) + (µ + ν)σ )ψ). P =
(13) Thus, we see that when R0 < 1 provided that R0 < P and R0 < Q, the disease-free equilibrium is locally and asymptotically stable, otherwise it is unstable. The requirement of the real and negative eigenvalues ensuring stability is clearly satisfied by ϒ. Now, for the roots of the quartic equation 12 by which the eigenvalues are obtained, using the Routh-Hurwitz stability criterion. We have that • Firstly, all the coefficients K1 , K2 , K3 , K4 > 0, that is, are positive. • Secondly,for the eigenvalues to have real negative parts, K1 K2 −K3 , K3 K2 −K1 K4 and K1 K2 K3 − K32 − K12 K4 are be positive. Therefore, by the Routh-Hurwitz criterion for stability, we conclude that the disease free equilibrium is locally asymptotically stable whenever R0 < 1.
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S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri
3.5.
Endemic Equilibrium
Next, we investigate the endemic equilibrium points of the basic model that is the equilibria where at least one of the infected components of the model is non-zero, we take the following steps. Let E1 = (S ∗∗ , I ∗∗ , R ∗∗ , En∗∗ ) represents any arbitrary endemic equilibrium of the model. Solving the equations (1) at steady states gives (σ + µ)(µ + α + ψ) S ∗∗ = µ(µ + σ )(α + µ + ψ) + (αµ + (µ + σ )(µ + ψ)β ∗∗ m (σ + µ)β ∗∗ m ∗∗ = I µ(µ + σ )(α + µ + ψ) + (αµ + (µ + σ )(µ + ψ)β ∗∗ m (14) αβ ∗∗ m ∗∗ R = µ(µ + σ )(α + µ + ψ) + (αµ + (µ + σ )(µ + ψ)β ∗∗ m θ(µ + σ )β ∗∗ m ∗∗ E = n µν(µ + σ )(α + µ + ψ) + ν(αµ + (µ + σ )(µ + ψ))β ∗∗ m At this point, we will illustrate the stability of the Existing Endemic equilibrium. Now, by re-writing equation (2), we have β ∗∗ m =
λ I ∗∗ + ρEn∗∗ , S ∗∗ + I ∗∗ + R ∗∗
(15)
be the force of infection of the disease at steady states. We can show that the non-zero equilibria of the basic model (1) satisfy the following polynomial (cubic equation in terms of β ∗∗ m) ∗∗2 ∗∗ ∗∗ ∗∗2 ∗∗ A1 β ∗∗3 m + A2 β m + A3 β m = 0β m (A1 β m + A2 β m + A3 ) = 0
if β ∗∗ m = 0, which is one of the roots of the cubic equation (16), then it corresponds to the disease-free equilibrium. Therefore for the other roots, we have that: ∗∗ A1 β ∗∗2 m + A2 β m + A3 = 0
where,
A1 = ν(α + µ + σ )(αµ + (µ + σ )(µ + ψ)), A2 = µν(µ + σ )(α + µ + ψ)(α + µ + σ )[G − R0 ], A3 = µν(µ + σ )2 (α + µ + ψ)2 [1 − R0 ].
(16)
Where G=
µ(2α + 2µ)ν(α + µ + σ ) + ν((− λ + 3µ)(µ + σ ) + α(3µ + σ ))ρ + ν(µ + σ )ρ 2 . µν(α + µ + ψ)(α + µ + σ ) (17)
Optimal Control Analysis of Cryptosporidiosis Disease
9
Theorem 3.4. 1. If G ≥ 1 then system (1) exhibits a transcritical bifurcation. 2. If G < 1 then system (1) exhibits a backward bifurcation. Proof. 1. For G ≥ 1 we obtain when R0 > 1 that A3 < 0. This implies that the system (1) has a unique endemic steady state. If R0 ≤ 1, then A3 ≥ 0 and A2 ≥ 0. In this case system (1) has no endemic steady states. 2. For G < 1 we discuss the following cases: i. R0 > 1, in this case A3 < 0 and system (1) has a unique endemic steady state. ii. R0 ≤ G, in this case both A2 and A3 are positive implying that system (1) has no endemic steady states. √ iii. G < R0 < 1, here A3 > 0 and A2 < 0 while the discriminant of (1), (R0 ) := A22 − 4A1 A3 , can be either positive or negative. We have (1) = A22 > 0 and (G) = −4A1 A3 < 0, then there exists R0c such that (R0c ) = 0, (R0 ) < 0 for G < R0 < R0c and (R0 ) > 0 for R0c < R0 . This together with the signs of A2 and A3 imply that system (1) has no endemic steady states when G < R0 < R0c , one endemic steady state when R0 = R0c and two endemic steady states when R0c < R0 < 1.
4.
Sensitivity Analysis of Cryptosporidiosis Model
In order to investigate the above model (1) robustness, due to uncertainties associated with the estimation of certain parameter values, it is important and useful to carry out a sensitivity analysis to investigate how sensitive the basic reproduction number is with respect to these parameters [29]. To carry out this analysis, we compute the normalized forward sensitivity index of the reproduction number 10 with respect to these parameters [26, 36]. The normalized forward sensitivity index of a variable h, that depends differentially on a parameter m, is defined as: ∂h m × . (18) ϒ h := ∂m h
10 4.1.
S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri Sensitivity indices of R0
We derive the sensitivity of R0 (10) corresponding to the following parameters: ∂R0 θρ × = ϒθh = ϒρh , = ∂ R0 λµν + θρ λµν = ϒlh λ, ϒ h := λµν + θρ θρ , ϒνh := − λµν + θρ α , ϒαh := − α+µ+ψ ψ , ϒψh := − α+µ+ψ µ θµρ − . ϒµh := − λµ2 ν + θµρ α + µ + ψ h := ϒ
(19)
Table 1: Sensitivity analysis of R0 Parameters Descriptions µ Natural death rate ν Rate at which Crypto leaves the environment Human Recruitment θ Average contribution of each Crypto infected persons to the environment ρ Modification parameter due to environmental treatment α Recovery rate from the disease ψ Crypto related death Human contact rate λ Transmission Probability
Sensitivity −1.8061 −0.9999 0.9996 0.9996 0.9996 −0.1660 −0.0277 0.0001 0.0001
Using parameter values from table (4.1), (it should be stated that these parameters are chosen for illustrative purpose only, and may not necessarily be realistic in terms of epidemiological interpretations), we calculate the sensitivity indices of R0 based on the following parameters µ, , θ, ρ, ν, , λ, α, ψ. The parameters are therefore, arranged from the most sensitive to least. The most sensitive parameter is proportion of the natural death rate ϒµh = −1.8061. While the least of the sensitivity parameters is the Transmission probability ϒλh = −0.0001. An increase (or decrease) in the natural death rate of Humans µ by 10% decreases (or increases) the R0 by 18.06%. Similarly increasing (or decreasing) the rate at which Crypto leaves the environment ν by 10% decreases (or increases) the R0 by 9.999%. In other words, increasing (or decreasing) the Human Recruitment rate , Average contribution of each Crypto infected individuals into the environment θ and the modification parameter due to environmental treatment ρ by 10% each would increase (or decrease) R0 by 9.996%. From the sensitivity analysis, it is clear that control efforts should be targeted towards the rate at which Crypto leaves the environment (ν) through treatment of water e.g, water
Optimal Control Analysis of Cryptosporidiosis Disease
11
plants and environment. Since it is not epidemiologically reasonable to increase the mortality rate as a means of controlling the disease [29, 26].
5.
Modified Cryptosporidiosis Model
In this section, we modified the Cryptosporidiosis model (1) to include Susceptible individuals with weakened immune system, that is, in order to investigate the effect of such individuals in the disease dynamics, we thereby sub-dividing the total human population at time t, denoted by N, into the following sub-populations of two susceptible individuals; susceptible individuals with strong immunity S1 , susceptible individuals with weakened immunity S2 , infected strong immune individuals I1 , infected weakened immune individuals I2 , those who are treated T , the recovered R and together with the environmental contamination En . We have chosen a mass action form of infection here in order to reduce the model analysis complexity. Therefore, we have, where N = S1 + S2 + I1 + I2 + T + R, therefore we have the following corresponding system of differential equations dS1 dt dS2 dt dI1 dt dI2 dt dT dt dR dt dEn dt where
= (1 − a) + σ κR − bβ ∗m S1 − µS1 = a + (1 − κ)σ R − β ∗m S2 − µS2 = bβ ∗m S1 − (α + µ + ψ)I1 = β ∗m S2 − (δ + µ + ψ)I2
(20)
= δI2 − (η + µ)T = αI1 + ηT − (σ + µ)R = (I1 + I2 )θ − νEn , β ∗m = λ (I1 + I2 ) + ρEn .
(21)
5.1.
Basic Properties of the Cryptosporidiosis Modified model
5.1.1
Positivity Analysis
From the above system of differential equations (20) lim N(t) ≤ t→∞
. In other words, µ
by the dynamics described in equation 20, the region defined by 6 = (S1 , S2 , I1 , I2 , T , R) ∈ R+ |S1 + S2 + I1 + I2 + T + R ≤ . µ
(22)
12
S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri
Figure 2: Diagram showing the modified Cryptosporidiosis model
By lemmas (3.1) and (3.2), we deduce thet the model (20) is positively invariant and for any initial starting point, (S10 , S20 , I10 , I20 , T0 , R0 ) ∈ R6+ , the trajectory lies in . Thus, in the basic model (20) is both mathematically and epidemiologically well posed. Hence, it is sufficient to study the dynamics of the model in . By the Van den Driessche and Watmough [40], the basic reproduction number R0 of the modified Cryptosporidiosis model is given by
R0 =
( λν + ρθ)(b(δ + µ + ψ) + a(α + µ + ψ − b(δ + µ + ψ))) . µν(α + µ + ψ)(δ + µ + ψ)
(23)
Optimal Control Analysis of Cryptosporidiosis Disease
13
Table 2: Modified Cryptosporidiosis Model Notations Parameters S1 S2 I1 I2 T R En N κ a b β ∗m λ α δ η θ σ ρ µ ψ ν
5.2. 5.2.1
Definition Dimension The number of Susceptible individuals. H umans The number of Susceptible individuals with weakened immune system. H umans The number of Infected individuals. H umans The number of Infected individuals with weakened immune system at time t. H umans The number of treated infected individuals with weakened immune system. H umans H umans The number of Recovered individuals. Environmental contamination class. Cont The total number of human population. H umans The rate of ingestion of Cryptosporidiosis disease. time−1 The proportion of the Humans with weakened immunity. H umans × time−1 The modification parameter of the infection rate. time−1 Human recruitment rate. H umans × time−1 Human contact rate. (H umans × time)−1/2 The force of infection. time−1 (H umans × time)−1/2 The transmission probability. time−1 The recovery rate from the disease. The treatment rate of the infected individuals with weakened immune system. time−1 The recovery rate of the treated individuals with weakened immune system from the disease. time−1 Average contribution of each cryptosporidiosis infected individual to the environment. time−1 Immunity waning rate. time−1 The modification parameter due to environmental treatment. (time × Cont)−1 The Human mortality rate. time−1 The Cryptosporidiosis related death. time−1 The rate at which cryptos leaves the environment. time−1
Analysis of Steady States of the Modified Model Stability Disease-Free Equilibrium
The Disease-free equilibrium of the modified model (20) is given by: (1 − a) a 0 0 0 0 0 0 0 , , 0, 0, 0, 0, 0 . ε1 = (S1 , S2 , I1 , I2 , T , R , En ) = µ µ 5.2.2
(24)
Local Stability of the Diseases-Free Equilibrium
Theorem 5.1. The disease-free equilibrium point ε0 , is locally asymptotically stable if R0 < 1 and unstable if R0 > 1. Proof. Evaluating the Jacobian matrix of disease-free equilibrium we have that: Let the Jacobian be defined by J11 J12 J13 J21 J22 J23 J31 J32 J33 J = J41 J42 J43 J51 J52 J53 J61 J62 J63 J71 J72 J73
the modified cryptosporidiosis model at the
J14 J24 J34 J44 J54 J64 J74
J15 J25 J35 J45 J55 J65 J75
J16 J26 J36 J46 J56 J66 J76
J17 J27 J37 J47 J57 J67 J77
.
14
S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri
Therefore, the Jacobian matrix of the model (20) evaluated at DFE is gven as:
Jε
0
=
J11 0 0 J22 0 0 0 0 0 0 0 0 0 0
J13 J23 J33 J43 0 J63 J73
J14 0 J16 J24 0 J26 J34 0 0 J44 0 0 J54 J55 0 0 J65 J66 J74 0 0
J17 J27 J37 J47 0 0 J77
,
(25)
where J11 = −µ, J13 = −(1 − a)bλ J17 = −(1 − a)bρ
, J14 = −(1 − a)bλ , J16 = σ κ, µ µ
, J22 = −µ, µ
, J24 = −aλ , J26 = (1 − κ)σ , J27 = −aρ , µ µ µ = −(α + µ + ψ) + (1 − a)bλ , µ
J23 = −aλ J33
, J37 = (1 − a)bρ , J43 = aλ , µ µ µ = −(δ + µ + ψ) + aλ , J47 = aρ , µ µ
J34 = (1 − a)bλ J44
J54 = δ, J55 = −(η + µ), J63 = α, J65 = η, J66 = −(µ + σ ), J73 = θ, J74 = θ, J77 = −ν.
(26)
From the above Jacobian matrix (25) we see that the first and the second columns contain only the diagonal terms which form the two negative eigenvalues that is J11 and J22 . To obtain the other eigenvalues, the matrix (25) can be reduced to a sub-matrix (27). Jε is 0 formed by removing the first and the second rows and columns of the matrix (25). We now obtain: J33 J34 0 0 J37 J43 J44 0 0 J47 0 (27) Jε = 0 J54 J55 0 . 0 J63 0 J65 J66 0 J73 J74 0 0 J77
Optimal Control Analysis of Cryptosporidiosis Disease
15
Therefore, we have:
Jε 0
=
−(α + µn + ψ) + (1 − a)bλ aλ
µ
µ
(1 − a)bλ
µ
−(δ + µ + ψ) + aλ
0 α θ
µ
δ 0 θ
0
0
0
0
µ aρ µ , 0 0 −ν
(1 − a)bρ
−(η + µ) 0 η −(µ + σ ) 0 0
(28) we let Ni , i = 1, 2, 3, 4, 5, be the eigenvalues of the above matrix (28), then we have
Det (Jε 0
− NI ) =
−(α + µ + ψ) + (1 − a)bλ aλ 0 α θ
µ
−N µ
(1 − a)bλ
µ
−(δ + µ + ψ) + aλ
−N µ
δ 0 θ
0
0
0
0
−(η + µ) − N 0 η −(µ + σ ) − N 0 0
µ aρ µ =0 0 0 −ν − N
(1 − a)bρ
.
(29) The eigenvalues of the above characteristic equation (29) are the zeros which satisfies the following equation below N 5 + N 4 F1 + N 3 F3 + N 2 F3 + NF4 + F5 = 0,
(30)
= µ(α + δ + η + 4µ + ν + σ + 2ψ) + (1 − a)bλ − aλ , = (f21 + f22 + f23 + µ(f24 + f25 )), = (f31 + µ(f32 + f33 ) + ψf34 + f35 + f36 ), = f41 + (a − 1)bλ f42 + µ(f43 + α(f44 )) + (f45 + µf46 )ψ + f47 , = µν(µ + η)(µ + σ )(α + µ + ψ)(δ + µ + ψ)[1 − R0 ],
(31)
where F1 F2 F3 F4 F5 with f21 = (aρ + bρ − abρ)(−θ), f22 = aλ (−(α + η + 3µ + ν + σ + ψ)), f23 = (a − 1)bλ (δ + η + 3µ + ν + σ + ψ), f24 = 3ηµ + 6µ2 + ην + 4µν + ησ + 3µσ + νσ + 2(η + 3µ + ν + σ )ψ + ψ 2 , f25 = αδ + (α + δ)(η + 3µ + ν + σ + ψ), f31 = (a − 1)bρθ(δ + η + 3µ + σ ) − aρθ(α + η + 3µ + σ ), f32 = µ(3η(µ + ν) + 2µ(2µ + 3ν)) + (3µ(µ + ν) + η(2µ + ν))σ , f33 = δ(3µ(µ + ν) + (2µ + ν)σ + η(2µ + ν + σ )) + α(3µ(µ + ν) + (2µ + ν)σ + η(2µ + ν + σ ) + δ(η + 2µ + ν + σ )),
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S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri
f34 = µ(α(η + 2µ + ν + σ ) + δ(η + 2µ + ν + σ ) + 2(3µ(µ + ν) + (2µ + ν)σ + η(2µ + ν + σ ))) − (aρ + bρ − abρ)θ, f35 = (−aλ )(3µ2 + 3µν + 2µσ + νσ + α(η + 2µ + ν + σ ) + (2µ + ν + σ )ψ + η(2µ + ν + σ + ψ)) + µ(η + 2µ + ν + σ )ψ 2 , f36 = (a − 1)bλ (3µ2 + 3µν + 2µσ + νσ + δ(η + 2µ + ν + σ ) + (2µ + ν + σ )ψ + η(2µ + ν + σ + ψ)) f41 = (−aλ )(ηµ(µ + 2ν) + η(µ + ν)σ + µ(µ2 + 3µν + µσ + 2νσ ) + α(µ(µ + 2ν) + (µ + ν)σ + η(µ + ν + σ ))), f42 = ηµ(µ + 2ν) + η(µ + ν)σ + µ(µ2 + 3µν + µσ + 2νσ ) + δ(µ(µ + 2ν) + (µ + ν)σ + η(µ + ν + σ )), f43 = µ(µ(δ + µ)(η + µ) + (2δη + 3(δ + η)µ + 4µ2 )ν) + (µ(δ + µ)(η + µ) + (δη + 2(δ + η)µ + 3µ2 )ν)σ , f44 = (ηµ(µ + 2ν) + η(µ + ν)σ + µ(µ2 + 3µν + µσ + 2νσ ) + δ(µ(µ + 2ν) + (µ + ν)σ + η(µ + ν + σ ))), f45 = (a − 1)bλ (µ(µ + 2ν) + (µ + ν)σ + η(µ + ν + σ )) − aλ (µ(µ + 2ν) + (µ + ν)σ + η(µ + ν + σ )), f46 = 2µ(η(µ + 2ν) + µ(µ + 3ν)) + 2(η(µ + ν) + µ(µ + 2ν))σ + α(µ(µ + 2ν) + (µ + ν)σ + η(µ + ν + σ )) + δ(µ(µ + 2ν) + (µ + ν)σ + η(µ + ν + σ )), f47 = µ(µ(µ + 2ν) + (µ + ν)σ + η(µ + ν + σ ))ψ 2 − aρθ(3µ2 + 2µσ + α(η + 2µ + σ ) + 2µψ + σ ψ + η(2µ + σ + ψ)) + (a − 1)bρθ(3µ2 + 2µσ + δ(η + 2µ + σ ) + 2µψ + σ ψ + η(2µ + σ + ψ)).
(32)
Clearly, if R0 < 1, then the DFE is locally asymptotically stable, otherwise it is unstable. 5.3.
Endemic Equilibrium
In order to obtain the endemic equilibria of the modified model that is the equilibria where at least one of the infected components of the model is non-zero, we let Eme = (S1e , S2e , I1e , I2e , T e , R e , Ene ) be the endemic equilibrium of the model (20). By solving
Optimal Control Analysis of Cryptosporidiosis Disease
17
the system of equations (20) at steady states[26], we obtain: (1 − a)(µ + σ )(η + µ) + κσ ((η + µ)αI1e + ηδI2e ) e S1 = (µ + bβ ∗m )(µ + σ )(η + µ) a(µ + σ )(η + µ) + (1 − κ)σ ((η + µ)αI1e + ηδI2e ) e = S 2 (µ + bβ ∗m )(µ + σ )(η + µ) b((1 − a)(µ + σ )(η + µ) + κσ ηδI2e )β ∗m e I1 = (η + µ)(µ(µ + σ )(α + µ + ψ) + b((µ + σ )(µ + ψ) + α(µ + σ + κσ ))β ∗m e (a(µ + σ )(η + µ) + (1 − κ)σ (α(η + µ)I1e + δηI2e ))β ∗m I1 = µ(η + µ)(µ + σ )(δ + µ + ψ) + ((µ + σ )(µ + η)(δ + µ + ψ) − (1 − κ)σ ηδ)β ∗m δI2e e T = (η + µ) (η + µ)αI1e + ηδI2e e = R (η + µ)(σ + µ) (I1e + I2e )θ e = E n ν (33) In order to analyse the stability of the exixting endemic equilibrium, we rewrite equation (21) as follows: e e e (34) β ∗e m − λ (I1 + I2 ) − ρEn = 0. On substituting equations (33) into equation (34), then we can observe that the endemic value of β ∗e m satisfies the following cubic polynomial given as ∗e2 ∗e M1 β∗e3 m + M2 β m + M3 β m = 0 ∗e2 ∗e β ∗e m M1 β m + M2 β m + M3 = 0,
(35)
if β ∗e m = 0, which is one of the zeros of the cubic equation (35), then it corresponds to the disease-free equilibrium. Therefore to obtain the other roots, we have that: ∗e M1 β ∗e2 m + M2 β m + M3 = 0,
(36)
18
S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri
where M1 = bν(m11 + η(m12 + αm13 )), M2 = (µ + σ )(m21 + ab( λν + ρθ)m22 + b(η(m23 + m24 + m25 + m26 ) + ν(m27 + ψm28 + m29 + δm20 ))), M3 = µ2 ν(η + µ)(µ + σ )2 (α + µ + ψ)(δ + µ + ψ)(1 − R0 ), with m11 = µ(µ + σ )(δ + µψ)(α(µ + σ − κσ ) + (µn + σ )(µ + ψ)), m12 = (µ + σ )(µ + ψ)(δ(µ + κσ ) + (µ + σ )(µ + ψ)), m13 = δ(µ2 + µσ + 2κσ 2 − 2κ 2 σ 2 ) + (µ + σ )(µ + σ − κσ )(µn + ψ), m21 = µν(α + µ + ψ)(µ(µ + σ )(δ + µ + ψ) + η(δ(µ + κσ ) + (µ + σ )(µ + ψ))), m22 = (−η(δµ + α(µ − 2(−1 + k)σ ) + 2(µ + σ )(µ + ψ)) + µ(α(µ − 2(−1 + κ)σ ) + (µ + σ )(δ + 2(µ + ψ)))), m23 = αµ3 ν + µ4 ν + θµ2 ρ + αµ2 νσ − καµ2 νσ + µ3 νσ + αθρσ − καθ ρσ + θµρσ , m24 = ((2µ2 ν + θρ)(µ + σ ) + αµν(µ + σ − κσ ))ψ + µν(µ + σ )ψ 2 , m25 = δ(µ3 ν + θµρ + µ2 νσ + kθρσ + αµν(µ + σ − κσ ) + λν(µ + κσ ) + µν(µ + σ )ψ), m26 = λν(−(−1 + k)ασ + (µ + σ )(µ + ψ)), m27 = αµ3 ν + µ4 ν + θµ2 ρ + αµ2 νσ − καµ2 νσ + µ3 νσ + αθρσ − καθ ρσ + θµρσ , m28 = ((2µ2 ν + θρ)(µ + σ ) + αµν(µ + σ − κσ )), m29 = µν(µ + σ )ψ 2 + λν(−(−1 + k)ασ + (µ + σ )(µ + ψ)), m20 = ( λν(µ + σ ) + αµν(µ + σ − κσ ) + (µ + σ )(θρ + µν(µ + ψ))). (37) Clearly, if R0 > 1, we have an unstable DFE which means that there exists an Endemic Equilibrium which is locally assymptotically stable.
6.
Sensitivity Analysis of the Modified Model
Following the investigation of model (1) we also check for the robustness of model (20), due to uncertainties associated with the estimation of certain parameter values that were included in the model modification. As this will reveal the slight and vast changes with respect to the parameters that affect the Cryptosporidiosis disease transmission, in terms of the R0 and hence give the special control measure that is most effective.
Optimal Control Analysis of Cryptosporidiosis Disease 6.1.
19
Sensitivity indices of R0
We derive the sensitivity of R0 (23) corresponding to the following parameters: h ϒ :=
ϒθh := ϒ h := ϒνh := ϒλh := ϒρh := ϒah := ϒbh := ϒαh :=
∂R0 ∂ ∂R0 ∂θ ∂R0 ∂ ∂R0 ∂ν ∂R0 ∂λ ∂R0 ∂ρ ∂R0 ∂a ∂R0 ∂b ∂R0 ∂α
× × × × × × × × ×
R0 θ R0 R0 ν R0 λ R0 ρ R0 a R0 b R0 α R0
= 1, = = = = = = = =
ρθ , λν + ρθ λν , λν + ρθ −ρθ , λν + ρθ λν , λν + ρθ ρθ , λν + ρθ a(α + µ + ψ − b(δ + µ + ψ)) , b(δ + µ + ψ) + a(α + µ + ψ − b(δ + µ + ψ)) b(δ + µ + ψ − a(δ + µ + ψ)) , b(δ + µ + ψ) + a(α + µ + ψ − b(δ + µ + ψ)) α(a − 1)b(δ + µ + ψ) , (α + µ + ψ)(b(δ + µ + ψ) + a(α + µ + ψ − b(δ + µ + ψ)))
∂R0 aδ(α + µ + ψ) δ = × , ∂δ R0 (δ + + ψ))) µ + ψ)(b(δ +2 µ +ψ)2 + a(α + µ + ψ − b(δ + µ b(δ + µ + ψ) + a α + 2α(µ + ψ) + (µ + ψ)2 − b(δ + µ + ψ)2 ψ ψ ∂R0 ϒψh := × =− ∂ψ R0 (α + µ + ψ)(δ + µ + ψ)(b(δ + µ + ψ) + a(α + µ + ψ − b(δ + µ + ψ))) −1 −1 a(b − 1) − b h ϒµ = µ + − − 1, α+µ+ψ δ+µ+ψ b(δ + µ + ψ) + a(α + µ + ψ − b(δ + µ + ψ)) ϒσh = ϒηh = ϒκh = 0. (38) ϒδh :=
From the table (3) below, we outlined some values of parameters and calculated the sensitivity of R0 with respect to such parameters. In the Table the parameters are arranged in the order of the most sensitive to the least. The most sensitive parameter here is the natural death rate ϒµh = −1.8650. Increasing (or decreasing) the natural death rate of Humans µ by 10% decreases (or increases) the R0 by 18.65%. Similarly increasing (or decreasing) the rate at which Crypto leaves the environment ν by 10% decreases (or increases) the R0 by 9.901%. In the same way, increasing (or decreasing) the Human Recruitment rate , Average contribution of each Crypto infected individuals into the environment θ and the modification parameter due to environmental treatment ρ by 10% each would increase (or decrease) R0 by 9.9%. Also, increasing(of decreasing) the Infection rate modification parameter b and Proportion of Humans with weakened immunity a by 10% each would increase (or decrease) the R0 by 5.467% and 2.189% respectively. From our results, it is clear that the natural death rate has the most sensitivity but it is biologically and epidemiologically unreasonable to increase the death rate as a control measure. Therefore, control efforts is aimed at reducing the θ, the average contribution of each Cryptosporidiosis infected persons to the environment, ρ, the modification parameter due to the environmental treatment and increasing ν, the rate at which Crypto
20
S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri
Table 3: Sensitivity analysis of R0 Parameters Descriptions Sensitivity µ Natural death rate −1.8650 Human Recruitment 1.0000 ν Rate at which Crypto leaves the environment −0.9901 θ 0.9900 Average contribution of each Crypto infected persons to the environment Modification parameter due to environmental treatment 0.9900 ρ b Infection rate modification parameter 0.5467 a 0.2189 Proportion of Humans with weakened immunity α −0.0908 Recovery rate from the disease ψ −0.0297 Crypto related death δ Treatment rate of infected persons with −0.0146 weakened immunity Human contact rate 0.0099 λ Transmission Probability 0.0099
leaves the environment would go a long way in reducing the disease burden.
7. Analysis of optimal control In the this section, we apply optimal control method using Pontryagin’s Maximum Principle to determine the necessary conditions for the optimal control of the disease. We incorporate time dependent controls into the model (20) to determine the optimal strategy for controlling the disease. Hence we have,
dS1 = (1 − a)u1 + σ κR − (1 − u2 )bβ ∗m S1 − µS1 dt dS2 = au1 + (1 − κ)σ R − (1 − u2 )β ∗m S2 − µS2 dt dI1 = (1 − u2 )bβ ∗m S1 − (u3 mα + µ + ψ)I1 dt dI2 = (1 − u2 )β ∗m S2 − (u3 δ + µ + ψ)I2 dt dT = δI2 − (η + µ)T dt dR = u3 mαI1 + ηT − (σ + µ)R dt dEn = (1 − u2 )(I1 + I2 )θ − νEn , dt
(39)
Optimal Control Analysis of Cryptosporidiosis Disease
21
For this, we consider the objective functional J (u1 , u2 , u3 ) = 0
tf
[z1 I1 + z2 I2 + z3 En + Au21 + Bu22 + Cu23 ]dt
(40)
The control functions, u1 (t), u2 (t) and u3 (t) are bounded, Lebesgue integrable functions. Our choice control functions here, agrees with what is in other literature on control of epidemic [2, 22, 24, 30]. The control u1 (t) and u2 (t) represents the efforts on preventing infection. The control efforts on treatment of infected individuals u3 (t) satisfies 0 u3 g2 , where g2 is the drug efficacy use for treatment of infected individuals. Where tf is the final time and the coefficients, z1 , z2 , z3 , A, B, C are the balancing cost factors due to scales and importance of the eight parts of the objective function. We seek to find an optimal control, u∗1 , u∗2 and u∗3 , such that J (u∗1 , u∗2 , u∗3 ) = min{J (u1 , u2 , u3 )|u1 , u2 , u3 ∈ U}
(41)
where U = {(u1 , u2 , u3 ) such that u1 , u2 , u3 are measurable with 0 ≤ u1 ≤ 1, 0 ≤ u2 ≤ 1 and 0 ≤ u3 ≤ g2 for t ∈ [0, tf ]} is the control set. The necessary conditions that an optimal solution must satisfy come from the Pontryagin et al. [31] Maximum Principle. This principle converts (39)–(40) into a problem of minimizing pointwise a Hamiltonian H , with respect to u1 , u2 , u3 , u4 and u5 H = z1 I1 + z2 I2 + z3 En + Au21 + Bu22 + Cu23 +MS1 (1 − a)u1 + σ κR − (1 − u2 )bβ ∗m S1 − µS1 +MS2 au1 + (1 − κ)σ R − (1 − u2 )β ∗m S2 − µS2 +MI1 (1 − u2 )bβ ∗m S1 − (u3 mα + µ + ψ)I1 +MI2 (1 − u2 )β ∗m S2 − (u3 δ + µ + ψ)I2
(42)
+MT {δI2 − (η + µ)T } +MR {u3 mαI1 + ηT − (σ + µ)R} +MEn {(1 − u2 )(I1 + I2 )θ − νEn .} where the MS1 , MS2 , MI1 , MI2 , MT , MR and MEn are the adjoint variables or co-state variables. The system of equations is found by taking the appropriate partial derivatives of the Hamiltonian (42) with respect to the associated state variable. Theorem 7.1. Given optimal control u∗1 , u∗2 , u∗3 and solutions S1 , S2 , I1 , I2 , T , R, En of the corresponding state system (39)- (40) that minimize J (u1 , u2 , u3 ) over U . Then there exists adjoint variables MS1 , MS2 , MI1 , MI2 , MT , MR , MEn satisfying ∂H −dMi = dt ∂i
(43)
22
S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri
where i = S1 , S2 , I1 , I2 , T , R, En and with transversality conditions MS1 (tf ) = MS2 (tf ) = MI1 (tf ) = MI2 (tf ) = MT (tf ) = MR (tf ) = MEn (tf ) = 0 (44) and u∗1
u∗2
aMS2 − (1 − a)MS1 = min 1, max 0, 2A
,
(45)
(MI1 − MS1 )bβ m S1 + (MI2 − MS2 )β m S2 + θMEn (I1 + I2 ) , = min 1, max 0, 2B (46) αm(MI1 − MR )I1 + δ(MI2 − MT )I2 u∗3 = min 1, max 0, . (47) 2C
Proof. Corollary 4.1 of Fleming and Rishel [17] gives the existence of an optimal control due to the convexity of the integrand of J with respect to u1 , u2 and u3 , a priori boundedness of the state solutions, and the Lipschitz property of the state system with respect to the state variables. The differential equations governing the adjoint variables are obtained by differentiation of the Hamiltonian function, evaluated at the optimal control. Then the adjoint equations can be written as dMS1 = (MS1 − MI 1 )(1 − u2 )bβ m + µMS1 dt dMS2 − = (MS2 − MI2 )(1 − u2 )β m + µMS2 dt dMI1 = −z1 + (1 − u2 )(MI1 − MS1 )λ bS1 + (1 − u2 )(MI2 − MS2 )λ S2 − dt +(u3 mα + µ + ψ)MI1 + u3 mαMR + (1 − u2 )θMEn
−
−
dMI2 = dt
−z2 + (1 − u2)(MI1 − MS1 )bλ S1 + (1 − u2)(MI2 − MS2 )λ S2 +u3 δMT + (u3 δ + µ + ψ)MS2 + (1 − u2 )θMEn
dMT = −(η + µ)MT + ηMR dt dMR = σ κMS1 + (1 − κ)σ MS2 − (σ + µ)MR − dt dMEn = −z3 − νMEn − dt −
Solving for
u∗1 , u∗2 , u∗3 , u∗4
and
u∗5
(48) subject to the constraints, the characterization (45–47)
Optimal Control Analysis of Cryptosporidiosis Disease
23
can be derived and we have ∂H = 2Au1 − aMS2 + (1 − a)MS1 ∂u1 ∂H 0= = 2Bu2 + (MS1 − MI1 )bβ m S1 + (MS2 − MI2 )β m S2 − θMEn (I1 + I2 ) ∂u2 ∂H 0= = 2Cu3 + αm(MR − MI1 )I1 + δ(MT − MI2 )I2 ∂u3 (49) Hence, we obtain (see Lenhart and Workman (2007)) 0=
aMS2 − (1 − a)MS1 2A (MI1 − MS1 )bβ m S1 + (MI2 − MS2 )β m S2 + θMEn (I1 + I2 ) u∗2 = 2B αm(MI1 − MR )I1 + δ(MI2 − MT )I2 u∗3 = 2C
u∗1 =
(50)
By standard control arguments involving the bounds on the controls, we conclude 0 If ξ ∗i ≤ 0 ∗ ∗ u∗i = ξ i If 0 < ξ i < 1 1 If ξ ∗i ≥ 1 for i ∈ 1, 2, 3 and where ξ ∗1 =
aMS2 − (1 − a)MS1 2A
(MI1 − MS1 )bβ m S1 + (MI2 − MS2 )β m S2 + θMEn (I1 + I2 ) 2B αm(MI1 − MR )I1 + δ(MI2 − MT )I2 ξ ∗3 = 2C Next, we discuss the numerical solutions of the optimality system and the corresponding results of varying the optimal controls u1 , u2 , u3 , u4 and u5 , the parameter choices, and the interpretations from various cases. ξ ∗2 =
8.
Numerical results and discussions
In this section, we investigate numerically the effect of the following itemized optimal control strategies listed below on the spread of malaria-cholera co-infection in a population. The optimal control solution is obtained by solving the optimality system, which
24
S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri
consists of the state system and the adjoint system. An iterative scheme is used for solving the optimality system. We start by solving the state equations with a guess for the controls over the simulated time using the fourth order Runge-Kutta scheme. Because of the transversality conditions (44), the adjoint equations are solved by the backward fourth order Runge-Kutta scheme using the current iterations solutions of the state equations. Then the controls are updated by using a convex combination of the previous controls and the value from the characterizations (45)–(48). This process is repeated and the iterations are stopped if the values of the unknowns at the previous iterations are very close to the ones at the present iteration. 8.1.
Prevention u1 and u2 only
The prevention controls u1 and u2 are used to optimize the objective function J while setting the other interventions (u3 ) to zero. We observed in Figure 3(a-d) that due to the control strategies, the number of infected individuals I1 and infected individuals with weak immunity I2 decreases significantly and while the environmental contamination En show signifcant decrease too. This strategy clear suggest that with 75% maximum starting efforts on control u2 and 20% maximum starting efforts on control u1 the disease can be effectively controlled, see Figure 3(d). 8.2.
Prevention u1 and treatment u3 only
The prevention control u1 and treatment control u3 are used to optimize the objective function J while setting the other interventions (u2 ) to zero. We observed in Figure 4(a - d) that due to the control strategies, the number of infected individuals I1 and infected individuals with weak immunity I2 shows no significant difference between cases with control and cases without control. This strategy clear suggest that with 75% efforts on control u2 and 20% efforts on control u1 the disease can be effectively controlled, see Figure 4(d). 8.3.
Prevention u2 and u3 only
The prevention control u2 and treatment control u3 are used to optimize the objective function J while setting the other interventions (u1 ) to zero. We observed in Figure 5(a - d) that due to the control strategies, the number of infected individuals I1 and infected individuals with weak immunity I2 decreases significantly and while the environmental contamination En show signifcant decrease too. This strategy clear suggest that with 75% maximum starting efforts on control u2 and 32% maximum starting efforts on control u3 the disease can be effectively controlled, see Figure 5(d). 8.4.
Preventions with treatment (u1 , u2 , u3 )
All control mechanism (u1 , u2 , u3 ) are used to optimize the objective function J . We observed in Figure 6(a-d) that due to the control strategies, the number of infected individuals I1 and infected individuals with weak immunity I2 decreases significantly.
Optimal Control Analysis of Cryptosporidiosis Disease
300
25
1.8
u1 = u2 = u3 =0 u1≠ 0, u2≠ 0, u3=0
u1 = u2 = u3 =0 u1≠ 0, u2≠ 0, u3= 0
1.6
250
Infected with weakened immunity
1.4
Infected Individuals
200
150
100
1.2
1
0.8
0.6
0.4 50 0.2
0
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4
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(a)
3 Time (years)
4
5
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(b)
20 u = u = u =0 1
2
3
u1
0.7
u1≠ 0, u2≠ 0, u3= 0
u2 u3
0.6
15
Controls
Environment
0.5
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0.3 10 0.2
0.1
5 0
1
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3 Time (years)
(c)
4
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0
0
1
2
3 Time (years)
4
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(d)
Figure 3: Simulations of the malaria-cholera model showing the effect of malaria prevention and treatment only on transmission This strategy suggest that optimal preventions and treatment regime against the disease in a community would be a very effective approach to effectively control the disease.
9.
Conclusion
In this paper, we derived and analyzed a deterministic model (1) together with its modified version (20) which includes Susceptible and Infected individuals with weakened immune system and Treatment for the Infected persons with weakened immune system for the
26
S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri
300 u1 = u2 = u3 =0
1.8
u1≠ 0, u2= 0, u3≠0 1.6
Infected with weakened immunity
250
Infected Individuals
200
150
100
1.4 1.2 1 0.8
u = u = u =0 1
2
3
u ≠ 0, u = 0, u ≠ 0 1
0.6
2
3
0.4 50 0.2 0
0
1
2
3 Time (years)
4
5
0
6
0
1
2
(a)
3 Time (years)
4
5
6
(b)
20 u1 = u2 = u3 =0
0.7
u ≠ 0, u = 0, u ≠ 0
19
1
2
u1
3
u
2
0.6
u3
18 0.5
Controls
Environment
17
16
15
0.4
0.3
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0.2
13 0.1 12 0
1
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3 Time (years)
(c)
4
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0
0
1
2
3 Time (years)
4
5
6
(d)
Figure 4: Simulations of the showing the effect of prevention and treatment only on transmission
transmission of Cryptosporidiosis disease. We analyzed the two models for the existence of diseases-free and endemic equilibrium points. We also carried out the sensitivity analysis for the reproduction numbers R0 and R0 for the models (1) and (20) respectively. Analyzing the models so as to gain insights into the qualitative dynamics, the following results listed below were obtained. (i) The basic model (1) and its modified version (20) (with Susceptible individuals with weakened immune system) are locally asymptotically stable at their disease-free
Optimal Control Analysis of Cryptosporidiosis Disease
300
27
1.8
u1 = u2 = u3 =0
u = u = u =0 1
u1= 0, u2≠ 0, u3≠0
2
3
u1= 0, u2≠ 0, u3≠ 0
1.6
250
Infected with weakened immunity
1.4
Infected Individuals
200
150
100
1.2
1
0.8
0.6
0.4 50 0.2
0
0
1
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3 Time (years)
4
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0
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0
1
2
(a)
3 Time (years)
4
5
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(b)
20 u = u = u =0 1
2
0.7
3
u = 0, u ≠ 0, u ≠ 0 1
2
u
1
3
u2 0.6
15
u3
Controls
Environment
0.5
0.4
0.3 10 0.2
0.1 5 0
1
2
3 Time (years)
(c)
4
5
6
0
0
1
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3 Time (years)
4
5
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(d)
Figure 5: Simulations of the model showing the effect of prevention and treatment only on transmission equilibrium points whenever the associated reproduction numbers are less than one. (ii) The basic model, with the formulation of the standard incidence form and its modified version, does not undergo the phenomenon of backward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium [18]. (iii) The two models (both the basic and its modified version) are locally asymptotically
28
S.T. Ogunlade, K.O. Okosun, R.S. Lebelo, and M. Mukamuri
300
1.8
u1 = u2 = u3 =0
u = u = u =0 1
u1≠ 0, u2≠ 0, u3≠0
2
3
u1≠ 0, u2≠ 0, u3≠ 0
1.6
250
Infected with weakened immunity
1.4
Infected Individuals
200
150
100
1.2
1
0.8
0.6
0.4 50 0.2
0
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3 Time (years)
4
5
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0
1
2
(a)
3 Time (years)
4
5
6
(b)
20 u = u = u =0 1
2
1
2
u1
0.7
3
u ≠ 0, u ≠ 0, u ≠ 0
u2
3
u3
0.6
15
Controls
Environment
0.5
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0.3 10 0.2
0.1 5 0
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(c)
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4
5
6
(d)
Figure 6: Simulations of the model showing the effect of prevention and treatment only on transmission stable at their endemic equilibrium points whenever the associated reproduction numbers are greater than one. (iv) Based on the Sensitivity analysis of the model (1), Control strategy should be aimed at increasing the rate at which Crypto leaves the environment by treatment of water and environment to avoid contamination. And for the model (20), Control efforts should be targeted at reducing the θ, the average contribution of each Cryptosporidiosis infected persons to the environment by providing basic health facilities for treatment, sensitization on proper hygiene and cleanliness, ρ, the
Optimal Control Analysis of Cryptosporidiosis Disease
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modification parameter due to the environmental treatment i.e, by ensuring proper sanitation and purification of water, and increasing ν, the rate at which Crypto leaves the environment would go a long way in reducing the disease burden.
10.
Concluding Remarks
In this study, we derived and analyzed a deterministic model which includes Susceptible and Infected individuals with weakened immune system and treatment for the infected persons with weakened immune system for the transmission of cryptosporidiosis disease. We analyzed the model for the existence of diseases-free and endemic equilibrium points. We also carried out the optimal control analysis for the model.
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