Optimal Control Model for the Dynamics of Tuberculosis in Mubi North Local Government Adamawa State of Nigeria

References

[1] 0.8

[2] Estimated

[3] 0.3

[4] Estimated

[5] 0.8

[6] Estimated

[7] 0.35

[8] Agusto, 2009

[9] 0.22

[10] Estimated

[11] 0.189

[12] Estimated

[13] 0.03

[14] Nyerere, 2014

[15] 0.204

[16] Estimated

[17] A positive index indicate that the value of increases as the parameter is increased while a negative index means the value of decreases as the parameters is increased.

[18] 3.3.2Sensitivity Index of the Parameters

[19] Table 3.2 Sensitivity indices

[20] Parameter

[21] Sensitivity index

[22] -1.22

[23] -0.06

[24] -0.14

[25] 0.84

[26] 0.16

[27] 0.30

[28] 0.50

[29] -0.64

[30] From table 3.2, the sensitivity indices shows that increasing (or decreasing) the per capital transmission rate from susceptible to expose class, increases (or decreases) the reproduction number by 8.4%. Increasing (or decreasing) quarantine the quarantine individuals goes for treatment, increases (or decreases) the reproduction number by 1.6%. Increasing (or decreasing) birth rate , increases (or decreases) the reproduction number by 3%. Increasing (or decreasing) exposed individuals to infected class , increases (or decreases) the reproduction number by 5%.

[31] The negative sign of the sensitivity index of with respect to, d2, and means an inverse relationship between these parameters and. The relationship implies increase (or decrease) in vaccination at birth () leads to approximately a 12.2% decrease (or increase) in. Which means most people are vaccinated at birth to reduce the probability of being infected. Likewise increase (or decrease) in death in sanitarium (d2) leads to approximately a 0.06% decrease (or increase) in which implies effectiveness of treatment and patients complete their treatment. Similarly increase (or decrease) in () rate of infected individuals goes for treatment and () natural death leads to approximately 1.4% and 6.4% decrease (or increase) in respectively. Therefore, to minimize TB transmission in the population, there is need for the combination of public campaign, case detection, vaccination, quarantine and sanitarium as control strategies to be implemented. This is due to the fact that, public campaign gives education and enlightenment on the transmission of TB, case detection and treatment reduces the progression rate to infectious stage, likewise vaccination and quarantine reduces the likelihood of an individual to get infected.

[32] NUMERICAL SIMULATION AND DISCUSSIONS

[33] In this section, the analytic results of the work was numerically simulated using (MATLAB R 2018a).

[34] 4.1Numerical Simulations of the Optimal Control Model

[35] The numerical simulations conducted in order to investigate the effects of the control strategies on the transmission dynamics of Tuberculosis in Mubi North Local Government to obtained the best control strategies . The simulations are performed using MATLAB and time in years. The estimated initial values of the state variables of the model areandfor the adjoint system and the terminal conditions are , where years. The cost coefficient corresponding to state variables are estimated to beThe quadratic cost coefficient corresponding to control measures is estimated to beand while the values of the remaining parameters are presented in Table 2.4.

[36] The graphs were plotted to show the effects of the control strategies (vaccination, public campaign, case detection, quarantine, and sanitarium) in eradicating tuberculosis infection in Mubi North Local Government Area, Adamawa State, Nigeria.

[37] 4.2 The Control profile of the model

[38] The control profiles are presented in Figure 4.5 to Figure 4.9.

[39] Figure 4.1: The profile of the control variable ()

[40] Figure 4.2: The profile of the control variable ()

[41] Figure 4.3: The profile of the control variable, ()

[42] Figure 4.4: The profile of the control variable, ()

[43] Figure 4.5: The profile of the control variable, ().

[44] 4.3 The analysis and interpretation of the graph of the control values

[45] The controlsand were used to optimize the objective function and the following observations were made:

[46] The optimal control values observed are:

[47] Now the proportion of each control needed to achieve effective control of TB in Mubi North in terms of percentage is computed as follows:

[48] Proportion of

[49] Proportion of

[50] Proportion of

[51] Proportion of

[52] Proportion of

[53] These results imply that for Tuberculosis to be controlled effectively in Mubi North, effort on vaccination of newborn babies and susceptible individuals. effort on public health campaign of susceptible individuals, of effort on case detection, effort on quarantined of suspected is needed and effort on sanitarium of individuals infected with TB are required to be continuously implemented.

[54] CONCLUSION

[55] The optimal analysis of the non-autonomous control model considering five different control strategies (i.e., control effort on vaccination, public health campaign, case detection, quarantine, and sanitarium) is performed. A comparison between an optimal control system and a system without control is presented using Mubi North as a case study. The numerical simulation results further showed that continuous implementation of vaccination, public health campaigns, case detection, quarantine, and sanitariums can bring the TB outbreak under effective control in Mubi North.

[56] The numerical simulation results of the optimal control model have shown that to minimize the number of cases of TB transmission in Mubi North, there is a need for continuous implementation of the five control strategies, i.e., vaccination, public health campaigns, case detection, quarantine, and sanitarium.

[57] RECOMMENDATION

[58] Proper education and sensitization on TB should be given to the general public by the government and non-governmental organizations (NGO).

[59] Government at national, state, local level should create an awareness program for early diagnoses of TB disease.

[60] Health care workers should ensure that patients have completed their treatments.

[61] Government should implement a 35.69% control effort on vaccination of new births and susceptible individuals, a 35.69% effort on a public health campaign for susceptible individuals, a 0.0642% effort on TB case detection, a 28.55% effort on the quarantine of suspected TB cases, and a 0.0021% control effort on the sanitarium to eradicate tuberculosis infection.

[62] REFERENCES

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@article{1711482,
      author = {Njida James Andest, Lazarus Gayus Ndatuwong, Paul Inuwa Dalatu},
      title = {Optimal Control Model for the Dynamics of Tuberculosis in Mubi North Local Government Adamawa State of Nigeria},
      journal = {Iconic Research And Engineering Journals},
      year = {2025},
      volume = {9},
      number = {4},
      pages = {1187-1201},
      issn = {2456-8880},
      url = {https://www.irejournals.com/formatedpaper/1711482.pdf},
      abstract = {The optimal control model with five different control strategies; the control effort on vaccination (u1), the control effort on public health campaigns (u2), the control effort on tuberculosis case detection (u3), the control effort on quarantine (u4) and the control effort on sanitarium (u5) is formulated and analyzed using Pontrayagin?s maximum principle. The numerical simulations of the optimal control model have been applied to dynamics of TB in Mubi North Local Governt Area of Adamawa State and the results have shown that tuberculosis infection can be effectively controlled in Mubi Northn provided that 35.69% control effort on vaccination (u1), 35.69% control effort on public health campaign (u2), 0.0642% control effort on tuberculosis case detection (u3), 28.55% control effort on quarantine (u4), and 0.0021% control effort on sanitarium (u5) are continuously implemented.},
      keywords = {Tuberculosis, Optimal Control, Public Health Campaign, Quarantine, Sanitarium.},
      month = {October},
      doi = {https://doi.org/10.64388/IREV9I4-1711482-9100}
  }