Drug treatment, snail control, cercariae control, improved sanitation and health education are the effective strategies which are used to control the schistosomiasis. In this paper, we consider a deterministic model for schistosomiasis transmission dynamics in order to explore the role of the several control strategies. The global stability of a schistosomiasis infection model that involves mating structure including male schistosomes, female schistosomes, paired schistosomes and snails is studied by constructing appropriate Lyapunov functions. We derive the basic reproduction number R 0 for the deterministic model, and establish that the global dynamics are completely determined by the values of R 0. We show that the disease can be eradicated when R 0 ≤1; otherwise, the system is persistent. In the case where R 0 >1, we prove the existence, uniqueness and global asymptotic stability of an endemic steady state. Sensitivity analysis and simulations are carried out in order to determine the relative importance of different control strategies for disease transmission and prevalence. Next, optimal control theory is applied to investigate the control strategies for eliminating schistosomiasis using time dependent controls. The characterization of the optimal control is carried out via the Pontryagins Maximum Principle. The simulation results demonstrate that the insecticide is important in the control of schistosomiasis.
Schistosomiasis (also known as bilharzia, bilharziasis or snail fever) is a vector-borne disease caused by infection of the intestinal or urinary venous system by trematode worms of the genus Schistosoma. More than 220.8 million people required preventive treatment worldwide in 2017, out of which more than 102.3 million people were reported to have been treated [
Schistosoma requires the use of two hosts to complete its life cycle: the definitive hosts and the intermediate snail hosts. In definitive hosts, schistosoma has two distinct sexes. Mature male and female worms pair and migrate either to the intestines or the bladder where eggs production occurs. One female worm may lay an average of 200 to 2000 eggs per day for up to twenty years. Most eggs leave the bloodstream and body through the intestines. Some of the eggs are not excreted, however, and can lodge in the tissues. It is the presence of these eggs, rather than the worms themselves, that causes the disease. These eggs pass in urine or feces into fresh water into miracidia which infect the intermediate snail hosts. In snail hosts, parasites undergo further asexual reproduction, ultimately yielding large numbers of the second free-living stage, the cercaria. Free-swimming cercariae leave the snail host and move through the aquatic or marine environment, often using a whip-like tail, though a tremendous diversity of tail morphology is seen. Cercariae are infective to the second host and turn it into single schistosoma, and infection may occur passively (e.g., a fish consumes a cercaria) or actively (the cercaria penetrates the fish) and terminates the life cycle of the parasite.
Many effective strategies are used in the real world, such as: based on preventive treatment, snail control, cercariae control, improved sanitation and health education. The WHO strategy for schistosomiasis control focuses on reducing disease through periodic, targeted treatment with praziquantel. This involves regular treatment of all people in at-risk groups [
However, no work has been done to investigate the global stability of the equilibria which is more in interest. Here, we take this deterministic schistosomiasis model with mating structure [
The model that we consider has been presented in [
· X m ( t ) the male schistosoma population size.
· X f ( t ) the female schistosoma population size.
· X p ( t ) the pair schistosoma population size.
· X s ( t ) the susceptible (uninfected) snail host population size.
· X i ( t ) the infected snail host population size.
The time evolution of the different populations is governed by the following system of equations:
{ d X m d t = k m X i − ( μ m + ϵ m ) X m − ρ X f , d X f d t = k f X i − ( μ f + ϵ f ) X f − ρ X f , d X p d t = ρ X f − ( μ p + ϵ p ) X p , d X s d t = Λ − ( μ s + ϵ s ) X s − β X p X s , d X i d t = β X p X s − ( μ s + ϵ s + α s ) X i . (1)
The different parameters are:
· k m and k f are the recruitment rates of male schistosoma and female schistosoma respectively.
· μ m , μ f , μ p , and μ s denote the natural death rate for male, female, pair and snail hosts respectively. α s is the disease-induced death rate of snail hosts.
· ρ represents the effective mating rate.
· Λ is the recruitment rate of snail hosts.
· β is the transmission rate from pairs parasite to susceptible snails.
· ϵ m , ϵ f , ϵ p and ϵ s are the elimination rates of male shistosoma, female schistosoma, paired schistosoma and snails respectively. These elimination rates represent the control strategies.
As it has been done in [
μ m + ϵ m = μ m ϵ , μ f + ϵ f = μ f ϵ ,
μ p + ϵ p = μ p ϵ , μ s + ϵ s = μ s ϵ .
In this section, we give some basic results concerning solutions of system (1) that will be subsequently used in the proofs of the stability results.
Proposition 1 The set Γ = { X m ≥ X f ≥ 0 , X p ≥ 0 , X s ≥ 0 , X i ≥ 0 } is a positively invariant set for system (1).
Proof. The vector field given by the right-hand side of system (1) points inward on the boundary of Γ . For example, if X s = 0 , then, X ˙ s = Λ > 0 . In an analogous manner, the same can be shown for the other system components.
Proposition 2 All solutions of system (1) are forward bounded.
Proof. Let us define N X = X m + X f + X p and N Y = X s + X i . Using system (1), we have d N Y d t = Λ − μ s ϵ N Y − α s X i ≤ Λ − μ s ϵ N Y . This implies that the set { N Y ≤ Λ μ s ϵ } is positively invariant and attracts all the solutions of (1).
We also have:
d N X d t = ( k m + k f ) X i − μ m ϵ X m − ( μ f ϵ + ρ ) X f − μ p ϵ X p ≤ ( k m + k f ) Λ μ s ε − min { μ m ϵ , μ f ϵ , μ p ϵ } N X − ρ X f .
Hence, the set { N X ≤ ( k m + k f ) Λ μ s ϵ γ } , where γ = min { μ m ϵ , μ f ϵ , μ p ϵ } , is positively invariant set and attracts all the solutions of (1).
Therefore all feasible solutions of system (1) enter the region
Ω = { ( X m , X f , X p , X s , X i ) ∈ ℝ + 5 : X s + X i ≤ Λ μ s ϵ , X m + X f + X p ≤ ( k m + k f ) Λ μ s ϵ γ } ,
and the set Ω is a compact positively invariant set for system (1). It is then sufficient to consider solutions in Ω .
The disease-free equilibrium of system (1) is
E 0 = ( 0,0,0, X s 0 ,0 ) = ( 0,0,0, Λ μ s ϵ ,0 ) . Using the notations of [
F = ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 β Λ μ s ϵ 0 ) and V = ( − k m μ m ϵ ρ 0 0 ρ + μ f ϵ 0 − k f 0 − ρ μ p ϵ 0 0 0 μ s ϵ + α s )
The basic reproduction number R 0 is equal to the spectral radius of the matrix F V − 1 , a simple computation gives:
R 0 = β ρ k f Λ μ s ϵ μ p ϵ ( μ f ϵ + ρ ) ( μ s ϵ + α s ) = β ρ k f X s 0 μ p ϵ ( μ f ϵ + ρ ) ( μ s ϵ + α s ) .
One can remark that there is a mistake in the formula for R 0 provided in [
The basic reproductive number for system (1) measures the average number of new infections generated by a single infected individual in a completely susceptible population.
As it is well known (see, for instance, [
Hence R 0 determines whether the disease will be prevalent in the given population or will go extinct.
Next, we discuss the global stability of infection-free equilibrium by using suitable Lyapunov function and LaSalle invariance principle for system (1). In recent years, the method of Lyapunov functions has been a popular technique to study global properties of population models. However, it is often difficult to construct suitable Lyapunov functions.
Theorem 3 The disease-free equilibrium E 0 of system (1) is globally asymptotically stable (GAS) on the nonnegative orthant ℝ + 5 whenever R 0 ≤ 1 .
Proof. We shall use the following notations: x = ( X m , X f , X p , X s , X i ) , and X s 0 = Λ μ s ϵ . To show the global stability of infection-free equilibrium of system (1), we use the following candidate Lyapunov function:
V ( x ) = μ s ϵ + α s k f X f + ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ X p + ∫ X s 0 X s X τ − X s 0 X τ d X τ + X i (2)
This function satisfies: V ( x ) ≥ 0 for all x ∈ Ω , and V ( x ) = 0 if and only if x = ( X m , 0 , 0 , X s 0 , 0 ) .
Taking the time derivative of the function V (defined by 2), along the solutions of system (1), we obtain
V ˙ = ( 1 − X s 0 X s ) ( Λ − μ s ϵ X s − β X s X p ) + ( β X s X p − ( μ s ϵ + α s ) X i ) + μ s ϵ + α s k f ( k f X i − ( μ f ϵ + ρ ) ) X f + ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ ( ρ X f − μ p ϵ X p )
Using Λ − μ s ϵ X s 0 = 0 , we get
V ˙ = ( 1 − X s 0 X s ) ( − μ s ϵ X s + μ s ϵ X s 0 ) + [ β X s 0 X p − ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ μ p ϵ X p ] = μ s ϵ X s 0 ( 1 − X s 0 X s ) ( 1 − X s X s 0 ) + β Λ μ s ϵ [ 1 − ( μ s ϵ + α s ) ( μ f ϵ + ρ ) μ m ϵ μ p ϵ k f ρ Λ β ] X p = μ s ϵ X s 0 ( 1 − X s 0 X s ) ( 1 − X s X s 0 ) + β Λ μ s ϵ [ 1 − 1 R 0 ] X p = − μ s ϵ X s ( X s 0 − X s ) 2 + β Λ μ s ϵ [ 1 − 1 R 0 ] X p (3)
Hence, V ˙ ≤ 0 if R 0 ≤ 1 , and
Ω ∩ { V ˙ = 0 } = { { x ∈ Ω : x = ( X m , X f ,0, X s 0 , X i ) } if R 0 < 1 { x ∈ Ω : x = ( X m , X f , X p , X s 0 , X i ) } if R 0 = 1
We will show that the largest invariant set L contained in Ω ∩ { V ˙ = 0 } is reduced to the disease-free equilibrium E 0 .
Let x = ( X m , X f , X p , X s , X i ) ∈ L and
x ( t ) = ( X m ( t ) , X f ( t ) , X p ( t ) , X s ( t ) , X i ( t ) ) the solution of (1) issued from this point. By invariance of L , we have X s ( t ) ≡ X s 0 which implies X ˙ s ( t ) = 0 = Λ − μ s X s ( t ) − β X p ( t ) X s ( t ) = Λ − μ s X s 0 − β X p ( t ) X s 0 and hence X p ( t ) = 0 for all t. But, X p ( t ) ≡ 0 implies that X ˙ p ( t ) = 0 for all t which implies, using system (1), that X f ( t ) = 0 for all t. In the same way, it can be proved that X i ( t ) = 0 for all t. Reporting in the first equation of system (1), one obtains that, in L ,
X ˙ m ( t ) = − μ m ϵ X m ( t ) ∀ t
Thus the solution of (1) issued from x = ( X m , X f , X p , X s , X i ) ∈ L is given by x ( t ) = ( X m e − μ m ϵ t ,0,0, X s 0 ,0 ) which clearly leaves Ω and hence L for t < 0 if X m ≠ 0 . Therefore L = { E 0 } and hence E 0 is a globally asymptotically stable equilibrium state for system (1) on the compact set Ω thanks to LaSalle invariance principle [
Biologically speaking, Theorem 3 implies that schistosomiasis may be eliminated from the community if R 0 ≤ 1 . One can remark that R 0 does not depend on μ m ϵ = μ m + ϵ m . Hence it is not helpful to try to control the male schistosoma population and then one can take ϵ m = 0 . Therefore the only way to eliminate schistosomiasis is to increase the killing rates of female schistosoma ( ϵ f ), paired schistosoma ( ϵ p ) and snails ( ϵ s ) in order to have R 0 ≤ 1 .
In the rest of this section, we show that the disease persists when R 0 > 1 . The disease is endemic if the infected fraction of the population persists above a certain positive level. The endemicity of a disease can be well captured and analyzed through the notion of uniform persistence. System (1) is said to be uniformly persistent in Ω if there exists constant c > 0 , independent of initial conditions in Ω ∘ (the interior of Ω ), such that all solutions ( X m ( t ) , X f ( t ) , X p ( t ) , X s ( t ) , X i ( t ) ) of system (1) satysfy
lim inf t → ∞ X m ( t ) ≥ c , lim inf t → ∞ X f ( t ) ≥ c , lim inf t → ∞ X p ( t ) ≥ c ,
lim inf t → ∞ X s ( t ) > c , lim inf t → ∞ X i ( t ) ≥ c ,
provided ( X m ( 0 ) , X f ( 0 ) , X p ( 0 ) , X s ( 0 ) , X i ( 0 ) ) ∈ Ω ∘ , (see [
Theorem 4 System (1) is uniformly persistent in Ω if and only if R 0 > 1 .
Proof. When R 0 ≤ 1 , the infection-free equilibrium E 0 is globally asymptotically stable which precludes any sort of persistence and hence R 0 > 1 is a necessary condition for persistence. In order to show that R 0 > 1 is a sufficient condition for uniform persistence, it suffices to verify conditions (1) and (2) of Theorem 4.1 in [
We use the notations of [
F = μ s ϵ + α s k f X f + ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ X p + X i
The time derivative of F along the solutions of system (1) is given by
F ˙ = β X s X p − ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ μ p ϵ X p = ( β X s − ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ μ p ϵ ) X p = μ p ϵ ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ ( β X s k f ρ μ p ϵ ( μ s ϵ + α s ) ( μ f ϵ + ρ ) − 1 ) X p = μ p ϵ ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ ( R 0 X s X s 0 − 1 ) X p
Since R 0 > 1 , we have F ˙ > 0 for X p > 0 and X s 0 R 0 < X s ≤ X s 0 . Therefore F ˙ > 0 in a neighborhood N of E 0 relative to Ω \ ∂ Ω . This implies that any solution starting in N must leave N at finite time and hence the stable set of M, W s ( M ) is contained in ∂ Ω .
Endemic equilibrium points are steady-state solutions where the disease persists in the population (all state variables are positive).
In this case system (1) has an endemic equilibrium point given by
E h = { X m * = μ p ϵ μ s ϵ ρ ( k m − k f ) + k m μ f ϵ β ρ k f μ m ϵ ( R 0 − 1 ) , X f * = k f Λ ( μ s ϵ + α s ) ( ρ + μ f ϵ ) ( 1 − 1 R 0 ) , X p * = ( R 0 − 1 ) μ s ϵ β , X s * = Λ R 0 μ s ϵ , X i * = Λ μ s ϵ + α s ( 1 − 1 R 0 ) .
This equilibrium has a biological sense only when R 0 > 1 .
Theorem 5 If R 0 > 1 , the unique endemic equilibrium E h is globally asymptotically stable.
Proof. In order to investigate the global stability of the endemic equilibrium, we consider the following function defined on
Ω 1 = { x ∈ Ω : X f > 0, X p > 0, X s > 0 and X s > 0 } :
W ( x ) = ( μ s ϵ + α s ) k f ∫ X f X f * u − X f * u d u + ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ ∫ X p X p * u − X p * u d u + ∫ X s X s * u − X s * u d u + ∫ X i X i * u − X i * u d u
This function satisfies: W ( x ) ≥ 0 for all x ∈ Ω 1 , and W ( x ) = 0 if and only if ( X f , X p , X s , X i ) = ( X f * , X p * , X s * , X i * ) . The time derivative of W with respect to the solutions of system (1) is
W ˙ = ( 1 − X s * X s ) ( Λ − μ s ϵ X s − β X s X p ) + ( 1 − X i * X i ) ( β X s X p − ( μ s ϵ + α s ) X i ) + ( μ s ε + α s ) k f ( 1 − X f * X f ) ( k f X i − ( μ f ϵ + ρ ) X f ) + ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ ( 1 − X p * X p ) ( ρ X f − μ p ϵ X p )
= ( Λ − μ s ϵ X s ) ( 1 − X s * X s ) + β X s * X p − β X s X p − X i * X i β X s X p + ( μ s ϵ + α s ) X i * + β X s X p − ( μ s ϵ + α s ) X i − ( μ s ϵ + α s ) X f * X f X i + ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ X f * − ( μ s ϵ + α s ) X i − ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f X f − ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ μ p ϵ X p + ( μ s ϵ + α s ) X i
− ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f X f − ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ μ p ϵ X p + ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ X p * X p μ p ϵ X p + ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f X f − ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ X p * X p X f
Thus,
W ˙ = ( Λ − μ s ϵ X s ) ( 1 − X s * X s ) + β X s * X p − X i * X i β X s X p + ( μ s ϵ + α s ) X i * − ( μ s ϵ + α s ) X f * X f X i + ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f X f * − ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ μ p ϵ X p − ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f X p * X p X f + ( μ s ϵ + α s ) ( μ f ϵ + ρ ) k f ρ X p * X p μ p ϵ X p .
Using the equilibrium relations ( E Q )
E Q = { Λ − μ s ϵ X s * = β X s * X p * , β Λ − μ s ϵ X s * = ( μ s ϵ + α s ) X i * , ρ X f * = μ p ϵ X p * , k f X i * = ( μ s ϵ + ρ ) X f * .
It follows that
W ˙ = ( Λ − μ s ϵ X s ) ( 1 − X s * X s ) + β X s * X p − ( μ s ϵ + α s ) X i * β X s * X p * X i * X i β X s X p + ( μ s ϵ + α s ) X i * − ( μ s ϵ + α s ) X i * X i * X f * X f X i + ( μ s ϵ + α s ) X i * − ( μ s ϵ + α s ) X i * X p * X p − ( μ s ϵ + α s ) X i * X f * X p * X p X f + ( μ s ϵ + α s ) X i * X p X p * X p * X p = ( Λ − μ s ϵ X s ) ( 1 − X s * X s ) + β X s * X p + ( μ s ϵ + α s ) X i * [ X p * X p X f X f * − X p * X p ] − ( μ s ϵ + α s ) X i * [ X i * X i X s X s * X p X p * + X i X i * X f * X f + X p * X p X f X f * − 2 ]
= ( Λ − μ s ϵ X s ) ( 1 − X s * X s ) + ( μ s ϵ + α s ) X i * X p * X p + ( μ s ϵ + α s ) X i * [ X p * X p X p X p * − X p X p * ] − ( μ s ϵ + α s ) X i * [ X s * X s + X i * X i X s X s * X p X p * + X i X i * X f * X f + X p * X f X p X f * − 4 ] + ( μ s ϵ + α s ) X i * X s * X s − 2 ( μ s ϵ + α s ) X i * .
This implies that
W ˙ = ( Λ − μ s ϵ X s − ( μ s ϵ + α s ) X i * ) ( 1 − X s * X s ) + ( μ s ϵ + α s ) X i * [ X p * X p X p X p * − X p X p * + X p X p * − 1 ] − ( μ s ϵ + α s ) X i * [ X s * X s + X i * X i X s X s * X p X p * + X i X i * X f * X f + X p * X f X p X f * − 4 ]
And since ( μ s ϵ + α s ) X i * = Λ − μ s ϵ X s * , it follows that
W ˙ = μ s ϵ X s * ( 1 − X s * X s ) ( 1 − X s X s * ) − ( μ s ϵ + α s ) X i * [ X s * X s + X i * X i X s X s * X p X p * + X i X i * X f * X f + X p * X f X p X f * − 4 ]
From the AM-GM inequality (which says that the algebraic mean is not smaller than the geometric mean), we have
X s * X s + X i * X i X s X s * X p X p * + X i X i * X f * X f + X p * X f X p X f * − 4 ≥ 0
( 1 − X s * X s ) ( 1 − X s X s * ) ≤ 0.
Then, W ˙ ≤ 0 on Ω 1 for R 0 > 1 . Hence, W is a Lyapunov function on Ω 1 . Moreover, W ˙ = 0 if and only if X f = X f * , X p = X p * , X s = X s * , and X i = X i * .
To obtain the largest invariant set L within the region { x ∈ Ω 1 : W ˙ = 0 } , note that the trajectory of X m ( t ) with an initial condition in L must be a solution of:
d X m d t = k m X i * − μ m ϵ X m − ρ X f *
Consequently, we have that
X m ( t ) = k m X i * − ρ X f * μ m ϵ + e − ( μ m ϵ + ϵ m ) t ( X m ( 0 ) − k m X i * − ρ X f * μ m ϵ ) ∀ t
Since X m ( t ) must not leave the domain L for all t, it follows that
X m ( t ) = k m X i * − ρ X f * μ m ϵ ∀ t
Hence, the largest invariant set L contained in { x ∈ Ω 1 : W ˙ = 0 } is reduced to { E h } , and therefore by LaSalle’s principle [
Sensitivity analysis and simulations are important to determine how best we can reduce the effect of schistosomiasis, by studying the relative importance of different factors responsible for its transmission and prevalence. Generally speaking, initial disease transmission is directly related to the basic reproduction number, and the disease prevalence is directly related to the endemic equilibrium state E h , and more specifically to the magnitude of X i * , X m * , X f * , X p * . We perform the analysis by deriving the sensitivity indices of the basic reproduction number to the parameters using both local and global methods.
We calculate the sensitivity indices of the reproductive number, R 0 , and the endemic equilibrium point, E h , to the parameters in the model. we can derive an analytical expression for its sensitivity to each parameter using the normalized forward sensitivity index as described by Chitnis et al. [
The normalized forward sensitivity index of a variable to a parameter is a ratio of the relative change in the variable to the relative change in the parameter. When a variable is a differentiable function of the parameter, the sensitivity index may be alternatively defined using partial derivatives.
Definition 1 The normalised forward sensitivity index of a variable p that depends differentiable on a parameter q is defined as:
ϒ q p = ∂ p ∂ q × q p
Sensitivity analysis is commonly used to determine the robustness of model predictions to parameter values (since there are usually errors in data collection and presumed parameter values). Here we use it to discover parameters that have a high impact on R 0 , and E h , and should be targeted by intervention strategies.
The sensitivity analysis of R 0 has already been done in [
ϒ μ s ϵ R 0 = ∂ R 0 ∂ μ s ϵ × μ s ϵ R 0 = − 2 μ s ϵ + α s μ s ϵ + α s
ϒ μ f ϵ R 0 = ∂ R 0 ∂ μ f ϵ × μ f ϵ R 0 = − μ f ϵ μ f ϵ + ρ
ϒ μ p ϵ R 0 = ∂ R 0 ∂ μ p ϵ × μ p ϵ R 0 = − 1
However the conclusions are not affected: the authors remarked that | ϒ μ f ϵ R 0 | < 1 = | ϒ μ p ϵ R 0 | < | ϒ μ s ϵ R 0 | , and so the most sensitive parameter most sensitive parameter is μ s ϵ the death rate of snails, followed by μ p ϵ the death rate of pair schistosoma. The least sensitive parameter is μ f ϵ the death rate of female single schistosoma. Therefore the most efficient way to reduce the value of R 0 is to reduce the snail host population.
Sensitivity analysis of E h
Since in general it is not easy to reduce the value of R 0 to be less than one and hence to eradicate the disease, one of the control strategy goal could be to reduce the disease prevalence. To this end, we perform a sensitivity analysis of the endemic equilibrium state. Sensitivity analysis of the endemic equilibrium has usually been used to determine the relative importance of different parameters responsible for equilibrium disease prevalence. Equilibrium disease prevalence is related to the magnitude of ( X m * , X f * , X p * , X i * ) , and specifically to the magnitude of X i * .
The sensitivity indices of X i * , to the parameters, μ s ϵ , μ f ϵ and μ p ϵ are given by
ϒ μ s ϵ X i * = ∂ X i * ∂ μ s ϵ × μ s ϵ X i * = μ s ϵ ( β ρ k f Λ s + μ p ϵ ( μ f ϵ + ρ ) ( α s + μ s ϵ ) 2 ) ( α s + μ s ϵ ) ( μ p ϵ μ s ϵ ( μ f ϵ + ρ ) ( α s + μ s ϵ ) − β ρ k f Λ s )
ϒ μ f ϵ X i * = ∂ X i * ∂ μ f ϵ × μ f ϵ X i * = μ s ϵ ( β ρ k f Λ s + μ p ϵ ( μ f ϵ + ρ ) ( α s + μ s ϵ ) 2 ) ( α s + μ s ϵ ) ( μ p ϵ μ s ϵ ( μ f ϵ + ρ ) ( α s + μ s ϵ ) − β ρ k f Λ s )
ϒ μ p ϵ X i * = ∂ X i * ∂ μ p ϵ × μ p ϵ X i * = μ p ϵ μ s ϵ ( μ f ϵ + ρ ) ( α s + μ s ϵ ) μ p ϵ μ s ϵ ( μ f ϵ + ρ ) ( α s + μ s ϵ ) − β ρ k f Λ s
It follows that
ϒ μ s ϵ X i * ϒ μ f ϵ X i * = β ρ k f Λ s μ f ϵ μ p ϵ ( α s + μ s ϵ ) 2 + μ f ϵ + ρ μ f ϵ
ϒ μ f ϵ X i * ϒ μ p ϵ X i * = μ f ϵ μ f ϵ + ρ
ϒ μ s ϵ X i * ϒ μ p ϵ X i * = β ρ k f Λ s μ p ϵ ( μ f ϵ + ρ ) ( α s + μ s ϵ ) 2 + 1
This implies that
| ϒ μ f ϵ X i * | < | ϒ μ p ϵ X i * | < | ϒ μ s ϵ X i * |
We note that the most sensitive parameter for X i * is μ s ϵ the death rate of host snails followed by μ p ϵ the death rate of pair parasites and μ f ϵ the death rate of female parasites.
In this subsection we propose the global sensitivity analysis of the model parameter to determine how much the parameters affect the output of the model. Global sensitivity analysis is a collection of more robust procedures, modifying groups of parameters simultaneously, with a specific goal to recognize the impacts of interactions between various parameters. LHS is at present the most productive and refined statistical techniques [
| Parameter | Description | Sample value | Range |
|---|---|---|---|
| Λ | recruitment rate of snail hosts | 150 per year | 100 - 200 |
| k m | recruitment rate of single male | 145 per year | - |
| k f | recruitment rate of single female | 100 per year | - |
| μ m | elimination rate of single male | 0.1 per year | 0.01 - 0.2 |
| μ p | elimination rate of single pair | 0.02 per year | 0.001 - 0.05 |
| μ f | elimination rate of single female | 0.2 per year | 0.1 - 0.5 |
| μ s | elimination rate of snail hosts | 0.1 per year | 0.01 - 0.2 |
| α s | disease-induced death rate of snail hosts | 0.5 per year | 0.1 - 0.9 |
| β | transmission rate from pairs to susceptible snails | 1.8 × 10−4 per year | 10 × 10−4 - 25 × 10−4 |
| ρ | the effective mating rate 0.467 per year | 0.467 per year | 0.1 - 0.5 |
In this section, we aim to place the system (1) thereof in an optimal control setting, in order to be able to calculate the optimal intervention strategies. The optimal control represents the most effective way of controlling the disease that can be adopted by authorities in response to its outbreak. We now modify our model (1) with time-dependent treatment effort u ( t ) as control for the system. The variable u ( t ) represents the amounts of insecticide that is continuously applied during a considered period, as a measure to fight the disease:
u ( t ) ≡ level of insecticide campaigns at time t
Our model with snails treatment can be described with the following differential equations:
{ d X m d t = k m X i − ( μ m + ϵ m ) X m − ρ X f , d X f d t = k f X i − ( μ f + ϵ f ) X f − ρ X f , (4)
{ d X p d t = ρ X f − ( μ p + ϵ p ) X p , d X s d t = Λ − ( μ s + ϵ s ) X s − β X p X s , d X i d t = β X p X s − ( μ s + ϵ s + α s + u ( t ) ) X i .
The control variable u ( t ) is a bounded, Lebesgue integrable function that is considered in relative terms, varying from 0 to 1. The goal is to maximize the following objective function
J ( u ) = min u ∫ 0 T ( c s X s ( t ) + c i X i ( t ) + 1 2 c u u ( t ) 2 ) d t
subject to the system differential Equations (4), where c s , c i and c u are the positive balancing constants. We seek to find an optimal control u * such that
J ( u * ) = min u { J ( u ) }
where the control set is defined as
U = { u : [ 0, T ] → [ 0,1 ] , u is Lebesgue measurable } . Here, the running costs of susceptible snails are given by c s X s ( t ) , while term c i X i ( t ) determines the costs of infected snails. Notice that 1 2 c u u ( t ) 2 is the cost of eliminating a fraction N Y = ( X s ( t ) + X i ( t ) ) of the snails population. The choice of the cost function as linear in the number of susceptible and infected and quadratic in the control is as generally done [
This system satisfies standard conditions for the existence of an optimal control and thus by using Pontryagins Maximum Principle as stated in [
H = c s X s ( t ) + c i X i ( t ) + 1 2 c u u ( t ) 2 + λ 1 ( t ) d X m d t + λ 2 ( t ) d X f d t + λ 3 ( t ) d X p d t + λ 4 ( t ) d X s d t + λ 5 ( t ) d X i d t
The adjoint variables λ i ( i = 1 , 2 , 3 , 4 , 5 ) are the solution of the following system:
{ d λ 1 d t = μ m λ 1 ( t ) , d λ 2 d t = − λ 2 ( t ) ( − μ f − ρ ) + ρ λ 1 ( t ) − ρ λ 3 ( t ) , d λ 3 d t = μ p λ 3 ( t ) − β λ 4 ( t ) X s ( t ) − β λ 5 ( t ) X s ( t ) , d λ 4 d t = − c s − β λ 4 ( t ) X p ( t ) − β λ 5 ( t ) X p ( t ) , d λ 5 d t = − c i − k f λ 2 ( t ) − λ 4 ( t ) ( − α s − μ s − u ( t ) ) − λ 5 ( t ) ( − α s − μ s − u ( t ) ) . (5)
with the boundary conditions λ i ( T ) = 0 .
By using Pontryagins Maximum Principle and the existence result for the optimal control from Fleming and Rishel [
Theorem 6 There exists an optimal strategy u * ∈ U such that
J ( u * ) = min u ∈ U { J ( u ) } ,
given by
u * = min { 1 , max { 0 , λ 4 X i + λ 5 X i c u } }
where λ i ( i = 1 , 2 , 3 , 4 , 5 ) are the solutions of (5).
Proof. Here the control and the state variables are nonnegative values. The necessary convexity of the objective functional in u is satisfied for this minimising problem. The control variable set u ∈ U is also convex and closed by definition. In addition, the integrand of J ( u ) with respect to control variables u * is convex and it is easy to verify the Lipschitz property of the state system with respect to the state variables. Together with a priori boundedness of the state solutions, the existence of an optimal control has been given by in [
System (5) is obtained by differentiating the Hamiltonian function. Furthermore, by equating to zero the derivatives of the Hamiltonian with respect to the control, we obtain
∂ H ∂ t = c u ( t ) − λ 4 ( t ) X i ( t ) − λ 5 X i ( t ) = 0
⇒ u * ( t ) = λ 4 ( t ) X i ( t ) + λ 5 X i ( t ) c u
Using the property of the control space, we obtain
u * ( t ) = { 0 , if l 4 ( t ) X i ( t ) + l 5 X i ( t ) c u ≤ 0 λ 4 ( t ) X i ( t ) + λ 5 X i ( t ) c u , if l 4 ( t ) X i ( t ) + l 5 X i ( t ) c u ∈ ( 0 , 1 ) 1 , if l 4 ( t ) X i ( t ) + l 5 X i ( t ) c u ≥ 1.
Those can be rewritten in compact notation
u * = min { 1 , max { 0 , λ 4 X i + λ 5 X i c u } }
The numerical simulations are completed utilizing Matlab and making use of parameter values in [
The control is updated by using a convex combination of the previous control. The iteration is stopped when the values of the unknowns at the previous iteration are very close to the ones at the present iteration. For more details see, e.g., [
We represent the solution curves of the five state variables both in the presence and absence of the control. When viewing the graphs, remember that each of the individuals with control is marked by dashed blue lines. The individuals without control are marked by red lines. It is observed that the application of optimal control reduces a quite larger number of schistosoma (male, female, pair) and snails in the absence of the control. This is occurring as the application of pesticide control reduces the snails population significantly as seen in
We have considered the schistosomiasis infection in an endemic population (when R 0 ). In Figures 4-6, we observe that the fraction of schistosoma (male, female, pair) is lower when control is considered. More precisely, at the end of 15 years, the total number of male, female and pair schistosoma is 105, 0.1 × 10 5
and 2 × 10 5 respectively when control is considered, and 3.5 × 10 5 , 0.5 × 10 5 and 8 × 10 5 respectively without control. The schistosoma can survive for very long periods in a dry state, often for more than a few years, that’s why in the stage without control evolution of the number of schistosoma (male, female, pair) take higher levels, once controlled, the number of schistosoma doesn’t develop as shown in Figures 4-6.
In
In
This research was carried out with financial support of CEA-MITIC for postdoc project in Université Gaston Berger de SAINT-LOUIS.
The authors declare no conflicts of interest regarding the publication of this paper.
Diaby, M., Sène, M. and Sène, A. (2019) Global Transmission Dynamics of a Schistosomiasis Model and Its Optimal Control. Applied Mathematics, 10, 397-418. https://doi.org/10.4236/am.2019.106029