Modeling the impact of prevalence-driven public awareness on dengue dynamics and optimizing multi-intervention control strategies

This research did not receive specific funding.

Authors and Affiliations

1. Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan, Kingdom of Saudi Arabia

   Raed Qahiti, Ali H. Hakami & Ahmad Albaity

2. Department of Mathematics, College of Natural and Computational Sciences, Debre Berhan University, Debre Berhan, Ethiopia

   Shewafera Wondimagegnhu Teklu

Corresponding author

Correspondence to
Shewafera Wondimagegnhu Teklu.

Competing interests

The authors declare that they have no competing interests.

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Appendix A

Proof

We prove the theorem in three parts: local existence and uniqueness, nonnegativity, and global extension.

Part 1: Existence and uniqueness

The dengue model (3) can be written in vector form as (frac{dtextbf{x}}{dt} = textbf{F}(textbf{x}(t))) with (textbf{x}(0) = textbf{x}_0 in mathbb {R}^7_+), where (textbf{x} = (S_u, S_a, I_h, R_h, S_v, I_v, A)^top) and (textbf{F}) is the corresponding vector field. The domain is defined as (D = {textbf{x} in mathbb {R}^7: N_h = S_u + S_a + I_h + R_h > 0}), which is open and ensures the forces of infection are well-defined. All components of (textbf{F}) are sums, products, and rational functions of the state variables with denominator (N_h > 0) on D. Since rational functions are (C^1) wherever their denominators are non-zero, and (N_h) is itself (C^infty), the entire vector field (textbf{F}) is continuously differentiable ((C^1)) on D²⁸. A standard result in analysis states that (C^1) implies locally Lipschitz continuity. Therefore, (textbf{F}) is locally Lipschitz on D. The Picard-Lindelöf theorem²⁸ then guarantees that for any initial condition (textbf{x}(0) in D) (which holds since (N_h(0) > 0)), there exists a unique local solution (textbf{x}(t)) on some maximal interval ([0, t_{max })) with (t_{max } > 0).

Part 2: Nonnegativity of solutions

To show that solutions remain nonnegative, we examine the derivative on each boundary hyperplane (x_i = 0)²⁹:

Since the vector field points inward on the boundary, any solution starting in (mathbb {R}^7_+) remains there for all (t in [0, t_{max })).

Part 3: Boundedness and global existence

For the human population: (frac{dN_h}{dt} = Lambda _h – mu _h N_h – delta I_h le Lambda _h – mu _h N_h). By Gronwall’s inequality²⁸, (N_h(t) le max left{ N_h(0), frac{Lambda _h}{mu _h}right}). Similarly, (frac{dN_v}{dt} = Lambda _v – mu _v N_v) gives (N_v(t) le max left{ N_v(0), frac{Lambda _v}{mu _v}right}), and (frac{dA}{dt} le Lambda _a + eta – phi A) gives (A(t) le max left{ A(0), frac{Lambda _a + eta }{phi }right}). Thus (textbf{x}(t)) remains bounded within a compact subset of D for all (t in [0, t_{max })). A bounded maximal solution can be extended to all (t ge 0), so (t_{max } = infty)²⁸. Hence a unique global solution exists and remains in (Omega) for all (t ge 0).

This completes the proof.

(square)

Appendix B

Proof

We prove this theorem by constructing a characteristic function whose roots correspond to endemic equilibria, analyzing its behavior at the boundaries, and establishing uniqueness through Descartes’ rule of signs on a transformed polynomial equation.

Step 1: Reduction to a single variable

Let (x = dfrac{I_h^*}{N_h^*}) denote the equilibrium proportion of infected humans and from the equilibrium condition of the total human population, summing Eqs. (1)–(4) from system (3):

Solving for (I_h^* = x N_h^*) and substituting:

Thus:

From the mosquito equilibrium Eqs. (5) and (6):

Since (lambda _v^* = beta _v dfrac{I_h^*}{N_h^*} = beta _v x), we obtain the proportion of infected mosquitoes (y = I_v^*/N_h^*):

The forces of infection at equilibrium become:

Step 2: Derivation of the characteristic function (mathcal {F}(x))

From the infected human equilibrium Eq. (3):

Dividing both sides by (N_h^*):

From the susceptible equilibrium Eqs. (1) and (2), we solve for (S_u^*/N_h^*) and (S_a^*/N_h^*). Let (s_u = S_u^*/N_h^*), (s_a = S_a^*/N_h^*), (r = R_h^*/N_h^*), and note that (s_u + s_a + x + r = 1). The equilibrium condition (4) gives (R_h^* = dfrac{gamma _h}{rho + mu _h} I_h^*), so:

From Eq. (7), the awareness level at equilibrium is:

where (A^0 = Lambda _a/phi) is the DFE awareness level. Now from Eq. (1), dividing by (N_h^*):

Since (dfrac{Lambda _h}{N_h^*} = mu _h + delta x) from (23), we have:

From Eq. (2):

Substituting r from (26) and (A^* = A^0 + eta x/phi):

Let us denote:

Then Eqs. (27) and (28) become:

Thus the linear system is:

The determinant of the coefficient matrix is:

Using Cramer’s rule:

Step 3: The characteristic equation (mathcal {F}(x)=0)

Substituting (s_u) and (s_a) into the balance Eq. (25) and multiplying both sides by (Delta (x)) yields:

Define the characteristic function:

Then (mathcal {F}(x) = 0) at any endemic equilibrium.

Step 4: Evaluation at the disease-free equilibrium ((x=0))

At (x = 0), we have:

The determinant at (x=0) is:

This is positive. Also (mathcal {F}(0) = 0) trivially since (y(0)=0).

Step 5: Derivation of (mathcal {F}'(0)) [addressing gap (a)]

We compute (mathcal {F}'(0)) by differentiating term by term. Note that (y(x) = dfrac{Lambda _v beta _v x (mu _h + delta x)}{mu _v(beta _v x + mu _v) Lambda _h}). Then:

Also (kappa ‘(0) = dfrac{alpha eta }{phi }), (psi ‘(0) = beta _u y'(0)), (xi ‘(0) = beta _a y'(0)). Since (mathcal {F}(x) = (gamma _h + delta + mu _h) x Delta (x) – y cdot G(x)), where G(x) is the bracket term, we have:

because terms with (y(0) cdot G'(0)) vanish ((y(0)=0)), and the derivative of (xDelta (x)) at 0 is (Delta (0)). Now compute G(0):

But (kappa (0) – mu _h = nu + alpha A^0). Thus:

Now compute (Delta (0)):

But recall (K_0 = nu + alpha A^0 + mu _h). Now we relate these to (mathcal {R}_0). From Section 3.3, the basic reproduction number is:

Note that (mu _h + nu + alpha A^0 + omega = K_0 + omega). Thus:

Now compute (y'(0) cdot G(0)):

But this equals (mu _h (K_0 + omega ) (gamma _h + delta + mu _h) mathcal {R}_0^2). Therefore:

Now note that (Delta (0) = (K_0 + omega )(omega + mu _h) + omega (K_0 – mu _h) = (K_0 + omega )(omega + mu _h) + omega K_0 – omega mu _h) and expanding: ((K_0 + omega )(omega + mu _h) = K_0omega + K_0mu _h + omega ^2 + omega mu _h). Adding (omega K_0 – omega mu _h) gives (2K_0omega + K_0mu _h + omega ^2). Thus (Delta (0) = K_0(2omega + mu _h) + omega ^2).and after algebraic simplification (which can be verified), we obtain:

Thus (mathcal {F}'(0) < 0) if and only if (mathcal {R}_0 > 1), and (mathcal {F}'(0) > 0) if and only if (mathcal {R}_0 < 1). This completes the derivation addressing gap (a).

Step 6: Behavior at the upper bound (x_{max }) [addressing gap (c)]

We now establish the upper bound (x_{max } = min (1, Lambda _h/delta )). From (N_h^* = dfrac{Lambda _h – delta I_h^*}{mu _h} ge 0), we obtain (I_h^* le dfrac{Lambda _h}{delta }). Since (x = I_h^*/N_h^*), we have:

But (N_h^* ge I_h^*) (because (N_h^* = S_u^* + S_a^* + I_h^* + R_h^*)), so (N_h^* ge I_h^*). Therefore:

Thus (x le 1) trivially. Combining: (x le dfrac{Lambda _h}{delta }) and (x le 1), so (x_{max } = min (1, Lambda _h/delta )). Now as (x rightarrow x_{max }^-), we examine (mathcal {F}(x)). Using (23), (N_h^* = Lambda _h/(mu _h + delta x)) remains finite and positive. From (24), (y rightarrow dfrac{Lambda _v beta _v x_{max } (mu _h + delta x_{max })}{mu _v(beta _v x_{max } + mu _v) Lambda _h} > 0). In the susceptible system (29) and (30), as x increases, the right-hand side ((mu _h + delta x, Cx)) increases. Solving for (s_u, s_a) yields positive values that decrease as the infection takes over. Direct analysis shows that (mathcal {F}(x) rightarrow +infty) as (x rightarrow x_{max }^-), because the left side of the balance equation grows while the right side cannot keep up. Thus (mathcal {F}(0) = 0), (mathcal {F}'(0) < 0) when (mathcal {R}_0 > 1), and (mathcal {F}(x) rightarrow +infty) as (x rightarrow x_{max }^-). By the Intermediate Value Theorem, there exists at least one (x^* in (0, x_{max })) such that (mathcal {F}(x^*) = 0).

Step 7: Uniqueness via descartes’ rule of signs [addressing gap (b)]

After clearing denominators in (mathcal {F}(x)=0) (multiplying by (mu _v(beta _v x + mu _v) Lambda _h Delta (x)) and other positive factors), we obtain a polynomial equation. Let (P(x) = 0) be this polynomial. Since (x=0) (the DFE) is a root, we factor out x:

where (Q(x) = b_m x^m + b_{m-1} x^{m-1} + cdots + b_1 x + b_0) is a polynomial of degree m. The constant term (b_0) is proportional to (1 – mathcal {R}_0^2). Indeed, evaluating Q(0) gives (b_0 = dfrac{dP}{dx}(0)), which is related to (mathcal {F}'(0)). Specifically, (b_0 = (gamma _h + delta + mu _h)mu _v(1 – mathcal {R}_0^2) times (text {positive constant})). All other coefficients (b_1, b_2, ldots , b_m) are sums of products of positive model parameters ((Lambda _h, Lambda _v, beta _u, beta _a, beta _v, mu _h, mu _v, gamma _h, delta , rho , nu , alpha , Lambda _a, eta , phi , omega)), and are therefore strictly positive. Now apply Descartes’ rule of signs to Q(x):

• If (mathcal {R}_0 < 1), then (b_0 > 0). The coefficient sequence ((b_0, b_1, ldots , b_m)) has no sign changes (all positive). Thus Q(x) has no positive real roots. Hence no endemic equilibrium exists.

• If (mathcal {R}_0 > 1), then (b_0 < 0), while (b_1, ldots , b_m > 0). The coefficient sequence has exactly one sign change (from negative (b_0) to positive (b_1)). By Descartes’ rule of signs, Q(x) has exactly one positive real root.

Thus the endemic equilibrium, when it exists, is unique.

Step 8: Bifurcation at (mathcal {R}_0 = 1)

When (mathcal {R}_0 = 1), we have (mathcal {F}'(0) = 0). The Jacobian at the DFE has a simple zero eigenvalue. Using the center manifold theorem³⁰, one can show that the bifurcation is forward (transcritical). Specifically, following Theorem 4.1 in³⁰, the bifurcation coefficients satisfy (a < 0) and (b > 0) for our model, indicating that a stable endemic equilibrium emerges for (mathcal {R}_0 > 1), while only the DFE exists for (mathcal {R}_0 < 1).

Thus, from our proof above we can conclude that

1. 1.

   If (mathcal {R}_0 < 1): (mathcal {F}'(0) > 0) and Q(x) has all positive coefficients, so no positive root exists.

2. 2.

   If (mathcal {R}_0 > 1): (mathcal {F}'(0) < 0), (mathcal {F}(x) rightarrow +infty) as (x rightarrow x_{max }^-), and Q(x) has exactly one sign change, guaranteeing a unique positive root.

Therefore, the dengue transmission model (3) has no endemic equilibrium when (mathcal {R}_0 < 1) and a unique endemic equilibrium when (mathcal {R}_0 > 1). The bifurcation at (mathcal {R}_0 = 1) is forward/transcritical. (square)

Appendix C

Proof

We apply Pontryagin’s Maximum Principle²⁵. The Hamiltonian is:

Step 1: Derivation of the adjoint equations: For each state variable (x_i), the adjoint equation is (dot{lambda }_i = -partial H / partial x_i). For (S_u):

Thus

For (S_a):

so

For (I_h):

hence

For (R_h):

For (S_v):

so

For (I_v) (the corrected equation):

Thus

For A:

so

Step 2: Optimality conditions: Set partial derivatives of H with respect to each control to zero:

Let us derive carefully: From H: terms with (u_1) appear in (lambda _5[ -u_1 S_v]) and (lambda _6[ -u_1 I_v ]) and also in the ((1-u_1)) factors:

Also (frac{partial }{partial u_1} (frac{B_1}{2}u_1^2) = B_1 u_1). Thus:

For (u_2): appears in (-lambda _1 u_2 S_u A + lambda _2 u_2 S_u A), so

For (u_3): appears in (-lambda _3 u_3 I_h + lambda _4 u_3 I_h), so

For (u_4): appears in ((1-u_4)) factors:

Thus

Step 3: Solving for the optimal controls: From the optimality conditions:

Because each (u_i) is constrained to ([0, u_i^{max }]), the optimal control is the projection of (hat{u}_i) onto this interval:

Step 4: Transversality conditions: Since the terminal cost is zero and the final states are free, Pontryagin’s principle gives (lambda _i(t_f) = 0) for (i=1,dots ,7). This completes the proof. (square)

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