3.1.1 Alcohol addiction-free equilibrium point and its local stability.

Setting the addicted state variables and setting the right-hand side of the sub-model (7) to zero and computing the results yields the alcohol addiction free equilibrium point given by .

Using the next generation operator approach discovered by van den Driesch and Warmouth [43], the effective reproduction number of the alcohol addiction free equilibrium point, represented by ℛ_0A, is used to analyze the local stability of the equilibrium point. Using this technique, we were able to calculate the dissemination matrix F and the transition matrix V, which are provided by and

Afterwards, we have calculated the result provided by using Mathematica as and

Next, the greatest eigenvalue in magnitude of the next generation matrix FV⁻¹ is equal to the effective reproduction number of the alcohol addiction sub-model (7) given by

. Therefore, the average number of secondary alcohol addiction protected and susceptible cases obtained from a typical permanent alcohol addiction or alcohol addiction formed individual during his/her effective addiction time in a susceptible population is described as ℛ_0A in this context. Hence, the threshold result, ℛ_0A, represents the effective reproduction number for the alcohol addiction sub-model.

Theorem 3: The alcohol addiction free equilibrium point of the dynamical system (7) is locally asymptotically stable (LAS) if ℛ_0A<1, and it is unstable if ℛ_0A>1.

Proof: Using the Routh-Hurwitz stability criteria from [44], the local stability of the alcohol addiction free equilibrium point of the alcohol addiction sub-model (7) is examined. At the alcohol addiction free equilibrium point , the Jacobian matrix of the alcohol addiction sub-model (7) is as follows:

Then, the corresponding characteristic equation of the Jacobian matrix is given by

Finally, we have determined the results given by λ = −μ<0, or λ = −(δ+μ)<0, or λ = −μ<0, or (8) where and

Then, using the cubic polynomial Eq (8), we applied the Routh-Hurwitz stability criteria to determine that each eigenvalue is negative if and only if ℛ_0A<1. Since all of the eigenvalues are negative whenever ℛ_0A<1, we can therefore conclude that the model’s alcohol addiction free equilibrium point is locally asymptotically stable whenever ℛ_0A<1. From Lemma 1, biologically speaking, we can conclude that if the initial size of the sub-populations of the alcohol addiction sub-model given in Eq (7) is in the basin of attraction of the alcohol addiction free equilibrium point , then the alcohol addiction dissemination dynamics can be eliminated from the population (whenever ℛ_0A<1).

3.1.2 Existence of alcohol addiction dominance equilibrium point(s).

Let be an arbitrary alcohol addiction dominance equilibrium point of the alcohol addiction sub-model (7) and is determined by making the right hand side of Eq (7) as zero. Then, after a number of steps of computations we have got (9)

Now substitute and given in Eq (9) in to

, then after some simplifications we have (10) where and

It can be seen that, D₁>0 whenever ℛ_0A<1. Thus, the number of possible positive real roots the polynomial (10) can have depends on the signs of D₂, and D₁. Hence, the following results are established.

Theorem 4: The alcohol addiction sub-model (7)

1. Has a unique alcohol addiction dominance equilibrium point if ℛ_0A>1, and D₂<0.
2. Could have a unique alcohol addiction dominance equilibrium point if ℛ_0A>1, and D₂>0.
3. Could have two alcohol addiction dominance equilibrium points if ℛ_0A<1, and D₂<0.

The backward bifurcation in the sub-model (7) is shown in Part (C); that is, if a locally asymptotically stable alcohol addiction dominance equilibrium point and a locally asymptotically stable alcohol addiction free equilibrium point coexist whenever if ℛ_0A<1; instances of this phenomenon’s occurrence in mathematical models as well as its causes can be found in [38–40]. The physical result is that, while required, the traditional physical condition that the effective reproduction number be less than one is insufficient to completely effectively restrict the spread of alcohol addiction in the population. This section of the numerical simulation investigates the backward bifurcation phenomena in the sub-model (7).

3.2.1 Smoking addiction-free equilibrium point and its local stability.

Smoking addiction free equilibrium point of the smoking dissemination sub-model (11) is obtained by making its right-hand side as zero and setting the smoking addiction class and permanent smokers class to zero as E_S = I_S = P_S = T = 0. Then, we have computed the results as , and . Thus, the smoking dissemination sub-model (11) smoking free equilibrium point is given by

Here, we are determining the smoking dissemination sub-model (11) effective reproduction number ℛ_0S by utilizing the van den Driesch and Warmouth next-generation matrix technique [43]. Following lengthy calculations, we now have the dissemination matrix provided by and the transition matrix given by

Then using Mathematica we have computed the result given by and

The characteristic equation of the matrix FV⁻¹ is

Then the spectral radius (effective reproduction number ℛ_0S) of FV⁻¹ of the smoking addiction sub-model (11) is .

Theorem 5: The smoking addiction free equilibrium point of the smoking addiction dissemination sub-model (11) is locally asymptotically stable if ℛ_0S<1, otherwise unstable.

Proof: The local stability of the smoking addiction free equilibrium point of the system (11) at can be studied from its Jacobian matrix and Routh-Hurwitz stability criteria. The Jacobian matrix of the dynamical system at the smoking addiction free equilibrium point is given by

Then the characteristic equation of the above Jacobian matrix is given by

After some steps of calculations we have determined the results as λ₁ = −μ<0 or λ₂ = −(γ₂+μ)<0 or

When ℛ_0S<1, all of the degree three polynomial equation eigenvalues are negative according to the Routh-Hurwiz stability requirements. As a result, the smoking addiction-free equilibrium point of the smoking addiction dissemination sub-model (11) is locally asymptotically stable, similar to [45], as all of the eigenvalues of the characteristics polynomials of the system (11) are negative if ℛ_0S<1.

3.2.2 Smoking persistence equilibrium point (s).

Before checking the global stability of the smoking addiction-free equilibrium point of the smoking addiction dissemination sub-model, we shall find the possible number of smoking addiction persistence equilibrium point(s) of the model (11). Let be the smoking addiction dominance equilibrium point of the sub-model (11) and be the smoking incidence rate at the equilibrium point. To find persistence equilibrium point(s) for which smoking addiction is disseminating in the population, the Eq (11) are solved in terms of at smoking addiction persistence equilibrium point. Now setting the right-hand sides of the equations of the model (11) to zero (at steady state) gives and

Here we have substituted the expressions and T* in the smoking addiction incidence rate given by , then we have computed and simplified the result to determine the second degree polynomial equation with as (12) where

It can be seen from Eq (12) that, b₂>0 whenever ℛ_0S<1. Thus, the number of possible positive real roots the polynomial (12) can have depends on the signs of b₁, and b₀. Hence, the following results are established.

Theorem 6: The smoking addiction sub-model model (11)

1. Has a unique smoking addiction persistence equilibrium point if ℛ_0S>1, and b₁<0.
2. Has a unique smoking addiction persistence equilibrium point if ℛ_0S>1, and b₁>0.
3. Could have two smoking addiction persistence equilibrium points if ℛ_0S<1, and b₁<0.

A locally asymptotically stable smoking addiction persistence equilibrium point coexists with a locally asymptotically stable smoking addiction-free equilibrium point in the model (11) as revealed by Part (C). Examples of the occurrence of backward bifurcation phenomena in mathematical models, as well as their causes, can be found in [38, 39, 45]. The social implication is that, albeit being important, the traditional criterion that the smoking addiction’s effective reproduction number be less than one is insufficient to effectively manage the addiction’s spread across the society. The backward bifurcation curve in the numerical simulation section demonstrates the presence of the backward bifurcation phenomena in sub-model (11).

3.3.1 Dual addiction-free equilibrium point.

The alcohol and smoking dual addiction-free equilibrium point of the complete dynamical system (4) is obtained by taking the state variable and is computed by .

3.3.2 Effective reproduction number of the dual addiction model.

The effective reproduction number of the dynamical system (4) is computed by applying the next generation operator method [43] and is the largest (dominant) eigenvalue (spectral radius) of the matrix: , where ℱ_i is the rate of appearance of new entrance in compartment i, ν_i is the transfer of affected individuals from one compartment i to another, and E_OSA is the co-existence free equilibrium point. Here, we obtained the following matrices: and

Then, the spectral radius of the corresponding next generation matrix, FV⁻¹ is given in terms of smoking and alcohol dual addiction associated effective reproduction number. It is therefore evident that the associated effective reproduction number for the smoking and alcohol dual addiction dissemination model denoted by R_0AS is given by ℛ_0AS = max{ℛ_0A, ℛ_0S}, where ,

Here ℛ_0AS is defined as the average number of secondary smoking and alcohol dual addicted individuals produced by one dual addicted individual who live in a whole individuals who are at risk for dual addiction during his/her addiction period. The threshold result ℛ_0AS is the effective reproduction number for the smoking and alcohol dual addicted individuals. Based on the dual addiction effective reproduction number ℛ_0AS = max{ℛ_0A, ℛ_0S}, where the alcohol only addiction dissemination sub-model (7) effective reproduction number is represented by

and the smoking only addiction dissemination sub-model (11) effective reproduction number is represented by .

Whenever ℛ_0AS<1, the dual addiction dissemination model exhibit the phenomenon of backward bifurcation, i.e. the dual addiction may not be eliminated in the near future without implementing further control efforts, but if ℛ_0AS>1, the dual addiction will disseminate in the community. In order to minimize the dual addiction model effective reproduction number ℛ_0AS, we can vary the model parameters incorporated in ℛ_0AS. Since ℛ_0AS is dependent on the model parameters β₁,α₁, μ, α₂, π, ε₁, d_A,d_PA, δ, ρ₁, β₂, γ₁, ε₂, γ₂, d_S, δ, δ₁, and d_PS. From the sensitivity indices results illustrated in Table 3 and the sensitivity indices diagram given by Fig 2 the dual addiction model effective reproduction number is directly proportional to some of the model parameters such as β₁,α₁, μ, α₂, ρ₁, β₂, γ₁, ε₂, γ₂, and δ₁ and also inversely proportional to some of the model parameters like π, ε₁, d_A, d_PA, δ, ε₂, d_S, and d_PS.

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Fig 2. Model parameters sensitivity indices whenever max{ℛ_0A, ℛ_0S} = ℛ_0A = 1.74>1.

https://doi.org/10.1371/journal.pone.0309356.g002

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Table 3. Sensitivity indices whenever ℛ_0AS = max{ℛ_0A, ℛ_0S} = ℛ_0A.

https://doi.org/10.1371/journal.pone.0309356.t003

3.3.3 Locally asymptotic stability of the dual addiction-free equilibrium.

Theorem 7: The smoking and alcohol dual addiction-free equilibrium point E_0AS of the complete dynamical system (4) is locally asymptotically stable whenever ℛ_0AS<1, otherwise unstable.

Proof: The local stability of the smoking and alcohol dual addiction free equilibrium point of the complete dynamical system (4) can be studied from its Jacobian matrix at the smoking and alcohol dual addiction-free equilibrium point E_0AS and Routh-Hurwitz stability criteria [41]. Then, the Jacobian matrix of the dynamical system (4) at is given by where, , , and A₁₂ = μ.

Then, the characteristic equation of the above Jacobian matrix is given by

After some steps of computations we have got λ = −μ, λ = −μ<0, λ = −A₄<0, λ = −A₁₀<0, λ = −A9<0, λ = −A₈<0,λ = −A₁₁<0μ, λ = −A₁₂<0, or (13)

Now by applying the Routh-Hurwiz stability criteria stated in [41], we can reveal that each of the eigenvalue of the polynomial is negative. Therefore, since all the eigenvalues of the characteristics polynomials of the system (4) are negative whenever ℛ_0AS<1, the smoking and alcohol dual addiction-free equilibrium point of the smoking and alcohol dual addiction dissemination model (4) is locally asymptotically stable.

3.3.4 Smoking and alcohol dual addiction persistence equilibrium point.

Persistence equilibrium points are steady-state solutions where both smoking addiction and alcohol addiction persists in the population (all state variables are positive). Let the smoking and alcohol dual addiction persistence equilibrium of the complete model system (4) is denoted by . It is obtained by setting the right hand side of each equation of the dual addiction dissemination model (4) equal to zero, and computed as

3.3.5 Global dynamics of the dual addiction model at ℛ_0AS = 1.

Theorem 8: The smoking and alcoholism dual addiction (4) exhibits the phenomenon of backward bifurcation at ℛ_0AS = 1 whenever the inequality K₁>K₂ holds.

Proof: Let us use the Centre Manifold criteria and change of dual addiction model state variables by S = z₁,P = z₂, E_S =z₃,I_S =z₄, P_S = z₅, E_A =z₆, I_A = z₇, P_A = z₈, C_SA = z₉, P_SA = z₁₀ and T = z₁₁.

Thus, N(t) = z₁(t)+z₂(t)+z₃(t)+z₄(t)+z₅(t)+z₆(t)+z₇(t)+z₈(t)+z₉(t)+z₁₀(t)+z₁₁(t).

Moreover, based on the vector notation represented by

Z = (z₁,z₂,z₃,z₄,z₅,z₆,z₇,z₈,z₉,z₁₀,z₁₁)^T, the smoking and alcoholism dual-addiction model (4) can be re-written in the form where H = (h₁,h₂,h₃,h₄,h₅,h₆,h₇,h₈,h₉,h₁₀,h₁₁)^T, such that (14) with initial conditions represented by , , , , , , , , and , and where θ₁≥1 and θ₂≥1.

The Jacobian matrix of the Eq (14) at the dual addiction-free equilibrium point E_0AS is the same as that of Eq (13). Now without loss of generality, let us consider the case when ℛ_0S>ℛ_0A and ℛ_0AS = 1, so that ℛ_0S = 1. Moreover, let β₂ = β* is chosen as a parameter that will cause bifurcation phenomenon, then solving for β₂ from ℛ_0S = 1 we have determined that .

Now at β* = β₂, the characteristics equation illustrated in Eq (13) has a single zero eigenvalue because and each of the remaining eigenvalues has negative real part. Now let us apply the Center Manifold Criteria on the new dynamical system (14) near to β* = β₂. Then, the right and left eigenvectors u = (u₁,u₂,u₃,u₄,u₅,u₆,u₇,u₈,u₉,u₁₀,u₁₁)^Tand v = (v₁,v₂,v₃,v₄,v₅,v₆,v₇,v₈,v₉,v₁₀,v₁₁)^T with respect to the zero eigenvalue of the Jacobian matrix J(E_0AS) evaluated at β* = β are respectively computed and represented by and

We found the non-zero partial derivatives with respect to the bifurcation parameter and the non-zero second-order mixed derivatives of h with respect to variables for the dynamical system (14) at the dual addiction-free equilibrium point E_0AS in order to derive the bifurcation coefficients a and b. The following formulae are used to compute the coefficients:

The only non-zero second order partial derivatives at E_0AS are given by

Thus, where

Similarly, we compute the result represented by

Finally, since the bifurcation coefficients whenever K₁>K₂ and b>0 the smoking and alcoholism dual addiction free equilibrium point E_0AS of the model (4) is not globally asymptotically stable at ℛ_0AS = max{ℛ_0A, ℛ_0S} =1. Thus, the dual addiction model (4) shows bifurcation in the backward direction whenever ℛ_0AS = max{ℛ_0A, ℛ_0S} = 1.

3.3.6 Sensitivity analysis.

We conducted a sensitivity analysis of the smoking and alcohol dual addiction model parameters included in the effective reproduction numbers in this subsection since our study takes optimal control theory into consideration. It is vital to look at the most important model parameters that might change the threshold quantity (also known as the dual addiction dissemination model effective reproduction number, or ℛ_0AS) between drinking and smoking. Reducing the prevalence of drinking and smoking combined with dual addiction in the community requires identifying significant model parameters that have an influence on the dual addiction model. We must do a sensitivity analysis of the model parameters using the values listed in Table 4 below, together with the dual addiction effective reproduction number, represented by ℛ_0AS, using the following widely recognized approach.

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Table 4. The dual addiction model parameters values used for numerical simulation.

https://doi.org/10.1371/journal.pone.0309356.t004

Definition: Let a be an arbitrary model parameter incorporated in the dual addiction model effective reproduction number ℛ_0AS, and then the forward sensitivity index formula is defined by [23, 33, 45].

Applying the dual addiction dissemination model parameters stated in Table 4, we have demined that ℛ_0A = 1.74>1 and ℛ_0S = 1.53>1 and also we have computed the values ℛ_0AS = max{ℛ_0A, ℛ_0S} = ℛ_0A = 1.74>1. And also we computed the sensitivity indices as:

Based on the model parameter values described in Table 4 below we have determined the sensitivty index values represented by Table 3 below.

Based on the sensitivity indices described in Table 3 aboe we have the following diagrams that show the graphical representations of the values represented in Table 3 above.

Case 1 (Implementation of three control measures).

Strategy A: Implementing protection strategies (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0) simultaneously.

Strategy B: Implementing treatment strategies (w₄ ≠ 0, w₅ ≠ 0, w₆ ≠ 0) simultaneously.

Case 2 (Implementation of four control measures).

Strategy C: Implementing protection strategies (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0) and treatment strategy (w₄ ≠ 0).

Strategy D: Implementing protection strategies (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0) and treatment strategy (w₅ ≠ 0).

Strategy E: Implementing protection strategies (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0) and treatment strategy (w₆ ≠ 0).

Case 3 (Implementation of five control measures).

Strategy F: Implementing protection strategies (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0) and treatment strategies (w₄ ≠ 0, w₅ ≠ 0).

Strategy G: Implementing protection strategies (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0) and treatment strategies (w₄ ≠ 0, w₆ ≠ 0).

Strategy H: Implementing protection strategies (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0) and treatment strategies (w₅ ≠ 0, w₆ ≠ 0).

Case 4 (Implementation of all control measures).

Strategy I: Implementing all proposed strategies (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0, w₄ ≠ 0, w₅ ≠ 0, w₆ ≠ 0) simultaneously.

5.3.1 Effect of the triple control measures on the dual addiction.

The numerical simulation results of the optimal control system (15) when the strategies A and B are presented in Fig 6 above respectively. It can be seen from Fig 6(A) that implementing Strategy A tends to minimize the number of smoking and alcoholism dual addiction greatly. However, the number of smoking and alcoholism dual addicted individuals is high whenever no intervention strategies put in place against the dual addiction dissemination. The result shown us the impact of implementing the smoking, alcoholism, and smoking and alcoholism dual addiction educations protection measures simultaneously (implementing w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0, w₄ = w₅ = w₆ = 0 simultaneously) on the dual addiction dissemination dynamics in the community. Thus, implementing the education protection measures simultaneously (i.e., implementing Strategy A) has a significance role to reduce the dual addiction dissemination in the community as compared with the simulation result without implementing any proposed control strategies. Similarly, the simulation curve given by Fig 6(B) reveals the possible effects (impacts) of smoking addiction, alcoholism addiction, and smoking and alcoholism dual addiction improvement (rehabilitation) measures simultaneously, i.e., w₁ = w₂ = w₃ = 0, w₄ ≠ 0, w₅ ≠ 0, w₆ ≠ 0 denoted by Strategy B on the dual addiction dissemination dynamics in the community. The result of the simulation given by Fig 6(B) shows that implementing the improvement (rehabilitation) control measures simultaneously (i.e., implementing Strategy B) has a significance role to reduce the dual addiction dissemination dynamics in the community as compared with the simulation result without implementing any proposed control strategies. From the simulation results explained implementing the Strategy A (i.e., implementing protection measures simultaneously is more effective to reduce the dual addiction dissemination than implementing Strategy B (i.e., implementing treatment measures simultaneously).

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Fig 6.

Impacts of Strategy A and Strategy B on the dual addiction dissemination respectively.

https://doi.org/10.1371/journal.pone.0309356.g006

5.3.2 Effect of four control measures on the dual addiction.

The potential repercussions of applying all four control measures at once are shown by the simulation curves illustrated in Fig 7. Fig 7(C) demonstrated the effects of concurrently implementing the smoking improvement (rehabilitation) control measure and the alcoholism addiction and smoking addiction protection measures, as described by Strategy C, on the dual addiction dissemination dynamics in the community. As compared to the simulation cure without implementing any suggested strategies, the simulation result shown in Fig 7(C) indicates that implementing protection control measures and smoking improvement (rehabilitation) control measures simultaneously (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0, w₄ ≠ 0 or Strategy C) has a significant role to reduce the dual addiction dissemination dynamics in the community. Additionally, as shown by Fig 7(D) in the simulation curve, the potential impacts (effects) of alcoholism addiction, smoking addiction, and protection measures against both alcoholism and smoking addiction as well as alcohol addiction improvement (rehabilitation) control measures simultaneously (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0, w₅ ≠ 0 or Strategy D) on the dual addiction dissemination problem in the community are revealed. As compared to the simulation cure without implementing any suggested strategies, the simulation result shown in Fig 7(D) indicates that implementing protection and alcoholism addiction improvement (rehabilitation) control measures simultaneously (Strategy D) has a significant role to reduce the dual addiction dissemination in the community. In a similar vein, the simulation curve depicted in Fig 7(E) shows the potential impacts (effects) of alcoholism addiction, smoking addiction, and simultaneous implementation of protection and improvement (rehabilitation) control measures for both alcoholism and smoking addiction (i.e., implementing Strategy E) on the dual addiction dissemination problem in the community. The simulation result shown in Fig 7(E) demonstrates that, in comparison to the simulation result without implementing any suggested control strategies, implementing all protection measures and dual addiction improvement (rehabilitation) control measures simultaneously (i.e., implementing Strategy E) has a significant role to reduce the dual addiction dissemination in the community. Ultimately, if we do not take into account the cost-effectiveness analysis of each of these control strategies, we compare the results of Fig 7C–7E and find that the most effective strategy to reduce the prevalence of alcoholism and smoking dual addiction in the community is to implement Strategy E, which entails implementing all of the protection measures and dual addiction rehabilitation measures simultaneously (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0, w₆ ≠ 0).

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Fig 7.

Impacts of Strategy C, Strategy D, and Strategy E on the dual addiction dissemination respectively.

https://doi.org/10.1371/journal.pone.0309356.g007

5.3.3 Effect of five control measures on the dual addiction.

The potential impacts of implementing protection measures and smoking and alcoholism single addictions improvement (rehabilitation) control measures simultaneously (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0, w₄ ≠ 0, w₅ ≠ 0 described by strategy F) on the dual addiction dissemination problem in the community are shown by the simulation curves depicted in Fig 8. The simulation result shown in Fig 8(F) demonstrates that, when compared to the simulation cure without implementing any suggested control measures, implementing protection measures and smoking and alcoholism single addictions improvement (rehabilitation) control measures simultaneously (i.e., implementing Strategy F) has a significant role to reduce the dual addiction dissemination in the community. Additionally, the numerical simulation curve depicted in Fig 8(G) shows the potential outcomes (impacts) of putting protective measures in place as well as the improvement of multiple addictions and smoking-only addiction (rehabilitation). The simulation result shown in Fig 8(G) demonstrates that, in comparison to the simulation result without implementing any suggested control measures, implementing protection measures and smoking only addiction and dual addictions improvement (rehabilitation) control measures simultaneously (i.e., implementing Strategy G) has a significant role to reduce the dual addiction dissemination in the community. Similarly, the simulation curve shown in Fig 8(H) shows the potential impacts (effects) of concurrently implementing alcoholism only and dual addictions improvement (rehabilitation) control measures (i.e., implementing Strategy H) on the problem of dual addiction dissemination in the community. The simulation result presented in Fig 8(H) demonstrates that putting protective measures in place concurrently with alcoholism-only and multiple addictions improvement (rehabilitation) management measures (i.e., applying Strategy H) has a significant influence in reducing the spread of dual addictions. Lastly, let’s compare the numerical simulation results shown in Fig 8F–8H without doing the cost-effectiveness analysis. We find that Strategy H is the most successful tactic used by the stakeholders to lessen the problem of alcoholism and smoking dual addiction spreading throughout the community.

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Fig 8.

Impacts of Strategy F, Strategy G, and Strategy H on the dual addiction dissemination respectively.

https://doi.org/10.1371/journal.pone.0309356.g008

5.3.4 Effect of all control measures on the dual addiction.

Here, the numerical simulation result described by Fig 9 reveals the effects (impacts) of implementing all the smoking addiction, alcoholism addiction, and smoking and alcoholism dual addiction protection and improvement (rehabilitation) control measures (w₁ ≠ 0, w₂ ≠ 0, w₃ ≠ 0, w₄ ≠ 0, w₅ ≠ 0, w₆ ≠ 0) simultaneously (i.e., implementing the Strategy I) on the dual addiction dissemination problem in the community. From the result of the simulation illustrated by Fig 9 one can observe that implementing Strategy I has a fundamental impact to reduce the dual addiction dissemination problem in the community as compared with all the simulation cures from Figs 4–8 discussed previously and the simulation results obtained without implementing any proposed control measures. Therefore, without considering the cost-effectiveness analysis of the nine proposed control measures discussed above, Strategy I, is the most influential strategy to reduce the dual addiction dissemination problem in the community. Thus, the public stake holders shall put their effort to implement this strategy to reduce the dual addiction dissemination in the community.

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Fig 9. Impact of Strategy I on the dual addiction dissemination.

https://doi.org/10.1371/journal.pone.0309356.g009