# An Optimal Control Model to Understand the Potential Impact of the New Vaccine and Transmission-Blocking Drugs for Malaria: A Case Study in Papua and West Papua, Indonesia

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## Abstract

**:**

## 1. Introduction

## 2. The Model and Parameter Estimation Result

#### 2.1. The Model

- 1.
- Susceptible without vaccine (${S}_{1}$) consists of a group of individuals susceptible to malaria who have not received a pre-erythrocytic vaccine yet.
- 2.
- Susceptible with a vaccine (${S}_{2}$) consists of a group of individuals who are also susceptible to malaria but have already received a pre-erythrocytic vaccine.
- 3.
- Exposed without vaccine $\left({E}_{1}\right)$ consists of a group of newly infected individuals from ${S}_{1}$ who have not yet gotten the pre-erythrocytic vaccine. We assume that individuals in this compartment are in the leaver stage. Hence, although these individuals do not show any symptoms yet, we assume that they can transmit Plasmodium to susceptible mosquitoes.
- 4.
- Exposed with vaccine $\left({E}_{2}\right)$ consists of a group of newly infected individuals from ${S}_{2}$. Although these individuals have already gotten vaccinated, the description is still the same with ${E}_{1}$.
- 5.
- Infected $\left(I\right)$ consists of a group of individuals who have already gotten infected by malaria and show their symptoms.
- 6.
- Infected individuals undergo transmission-blocking treatment $\left(T\right)$, defined as a group of individuals who already get infected, show symptoms, but get a transmission-blocking treatment. We assume that this treatment can kill the sexual Plasmodium (gametocytes).
- 7.
- Recovered but carrier $\left({R}_{1}\right)$ consists of individuals who recovered from malaria (do not show symptoms anymore) and have a temporal immunity but still have asexual Plasmodium inside their bodies. Hence, this group of individuals can still transmit Plasmodium to the mosquito.
- 8.
- Fully recovered $\left({R}_{2}\right)$ consists of a group of individuals who recovered from malaria and succeeded in the transmission-blocking treatment process. Hence, unlike in ${R}_{1}$, individuals in ${R}_{2}$ lack sexual and asexual Plasmodium in their blood. Therefore, ${R}_{2}$ compartment does not transmit malaria anymore.

- 1.
- The rate of new individuals only came from newborns with a constant rate of ${\mathrm{\Pi}}_{h}$. We ignore migration from our model.
- 2.
- Vertical transmission is neglected [31].
- 3.
- The infected individuals in ${E}_{1},{E}_{2},I,T,$ and ${R}_{1}$ are capable to transmit Plasmodium to mosquitoes with a constant rate ${\beta}_{v}$.
- 4.
- The pre-erythrocytic vaccine is given to ${S}_{1}$ individuals with a constant rate ${u}_{1}$ to give temporal protection from mosquito bites that can lead to malaria infection. Hence, we assume that the transmission rate of ${S}_{2}$ is less than ${S}_{1}$ $({\beta}_{h2}<{\beta}_{h1})$.
- 5.
- The pre-erythrocytic vaccine is not for a lifetime. Hence, after a period of ${\alpha}^{-1}$, individuals in ${S}_{2}$ will return to ${S}_{1}$.
- 6.
- The transmission-blocking treatment is given to individuals in I with a constant rate ${u}_{2}$ to cure malaria and wipe out sexual and asexual Plasmodium from their blood.
- 7.
- The transmission-blocking treatment is not always successful in curing infected individuals.
- 8.
- The description of all parameters is given in Table 1 and assumed to be nonnegative.

- 1.
**Cost to implement pre-erythtocytic vaccine.**The total cost to implement the pre-erythrocytic vaccine is given by$${\int}_{0}^{{t}_{f}}\left(\right)open="("\; close=")">{\omega}_{1}{u}_{1}^{2}$$- 2.
**Cost to implement transmission-blocking treatment.**Similar to ${u}_{1}$, we also use a quadratic term for ${u}_{2}$. Hence, the total cost of transmission blocking is given by$${\int}_{0}^{{t}_{f}}\left(\right)open="("\; close=")">{\omega}_{2}{u}_{2}^{2}$$- 3.
**Cost related to the high number of infected individuals.**Except the cost related to the implementation of pre-erythrocytic and transmission-blocking, the cost for malaria control strategy also comes from the cost related to the high number of infected individuals who were not treated (I and ${R}_{1}$). This cost is given by$${\int}_{0}^{{t}_{f}}\left(\right)open="("\; close=")">{\omega}_{3}I+{\omega}_{4}{R}_{1}$$

#### 2.2. Parameter Estimation

- 1.
- Papua is more malaria-endemic than West Papua given the non-controlled basic reproduction number (${\mathcal{R}}_{0}{|}_{\mathrm{Papua}}=1.75$ and ${\mathcal{R}}_{0}{|}_{\mathrm{West}\phantom{\rule{4.pt}{0ex}}\mathrm{Papua}}=1.53$). See the formula of non-controlled basic reproduction number (${\mathcal{R}}_{0}$) in (12).
- 2.
- Based on the value of p and q, we conclude that individuals in Papua have a slightly bigger chance of reaching the malaria elimination target if there is a continuous implementation of treatment efforts compared to West Papua.
- 3.
- The infection rates from mosquitoes to humans $({\beta}_{h1},$ and ${\beta}_{h2})$ in West Papua is higher than in Papua. Hence, it is important to develop a media campaign that targets reducing the contact between humans and mosquitoes in West Papua than in Papua. Such campaigns may include the use of bed nets or mosquito repellent.

## 3. Dynamical Analysis of the Model

#### 3.1. Preliminary Results on the Positiveness and Boundedness of the Solution

**Theorem**

**1.**

**Proof.**

#### 3.2. The Malaria-Free Equilibrium and the Controlled Reproduction Number

- 1.
- ${\mathcal{R}}_{0v}^{{E}_{1}}$ shows the path of transmission for a new case in mosquito due to a bite from susceptible mosquitoes to an exposed human in compartment ${E}_{1}$.
- 2.
- ${\mathcal{R}}_{0h1}^{{E}_{1}}$ shows the transmission path which gives a new infection in humans without pre-erythrocytic vaccine $\left({E}_{1}\right)$ due to a bite from infected mosquitoes.
- 3.
- ${\mathcal{R}}_{0v}^{{E}_{2}}$ presents the transmission path for a new infection in the mosquito population after they bite exposed humans in ${E}_{2}$.
- 4.
- ${\mathcal{R}}_{0h2}^{{E}_{2}}$ presents the transmission path for a new infection in vaccinated human ${E}_{2}$ due to a bite from infected mosquitoes.
- 5.
- ${\mathcal{R}}_{0v}^{I}$ shows a transmission for a new infection in the mosquitoes population after biting an infected individual in I.
- 6.
- ${\mathcal{R}}_{0h}^{I}$ presents a transmission path for a new cases in I due to progression of ${E}_{1}$ and ${E}_{2}$.
- 7.
- ${\mathcal{R}}_{0v}^{T}$ presents a transmission path for a new case in the mosquitoes population after biting an infected individual who is undergoing treatment $\left(T\right)$.
- 8.
- ${\mathcal{R}}_{0h}^{T}$ presents a transmission path for a new case in treated human population $\left(T\right)$ due to the treatment rate from I.
- 9.
- ${\mathcal{R}}_{0v}^{{R}_{1}}$ presents a transmission path for a new case in the mosquito population after biting individuals in ${R}_{1}$.
- 10.
- ${\mathcal{R}}_{0h}^{{R}_{1}}$ presents a transmission path for new cases in the compartment of humans who partially succeed in conducting transmission-blocking treatment.

#### 3.3. The Malaria-Endemic Equilibrium

## 4. Sensitivity Analysis

#### 4.1. Global Sensitivity Analysis on the Basic Reproduction Number

#### 4.2. Local Sensitivity Analysis on the Model Variables

## 5. Existence of Solution and Characterization of the Optimal Control Problem

#### 5.1. Existence of the Optimal Solution

**Theorem**

**2.**

- 1.
- The admissible control parameters and each model state variable are non-empty.
- 2.
- Control $\mathcal{H}$ is convex and bounded.
- 3.
- Right hand side of our system is continuous and bounded above by a linear function in the state variables and the control parameters.
- 4.
- The integrand of $\mathcal{J}({u}_{1}\left(t\right),{u}_{2}\left(t\right))$ is concave on $\mathcal{H}.$
- 5.
- There exists positive constants ${b}_{1},{b}_{2}>0$ and $\tau >1$ such that $\mathcal{J}\left({u}_{k}\left(t\right)\right)$ satisfies$$\mathcal{J}({u}_{1}\left(t\right),{u}_{2}\left(t\right))\le {b}_{1}+{b}_{2}{\left(\right|{u}_{1}{|}^{2}+\left|{u}_{2}{|}^{2}\right)}^{{\displaystyle \frac{\tau}{2}}}.$$

**Proof.**

- 1.
- Applying the methodology of Theorem 9.2.1 on page 182 of [57], the first criteria is fulfilled as the solutions of our model system exist and are bounded in $\omega $ as also shown by the existence of the super-solutions.
- 2.
- Using the definition of $\mathcal{H},$ we have that set as bounded and closed.
- 3.
- 4.
- Suppose the objective function$$\mathcal{L}={\omega}_{1}{u}_{1}^{2}+{\omega}_{2}{u}_{2}^{2}+{\omega}_{3}I+{\omega}_{4}{R}_{1}$$$$\begin{array}{cc}\hfill \mathcal{L}(tp+(1-t\left)q\right)& ={\omega}_{3}I+{\omega}_{4}{R}_{1}+{\omega}_{1}{(t{u}_{11}+(1-t){u}_{12})}^{2}+{\omega}_{2}{(t{u}_{21}+(1-t){u}_{22})}^{2}\hfill \end{array}$$$$\begin{array}{cc}\hfill t\mathcal{L}\left(p\right)+(1-t)\mathcal{L}\left(q\right)& ={\omega}_{3}I+{\omega}_{4}{R}_{1}+{\omega}_{1}(t{u}_{11}^{2}+(1-t){u}_{12}^{2})+{\omega}_{2}(t{u}_{21}^{2}+(1-t){u}_{22}^{2}).\hfill \end{array}$$From (2), we observe that$${(t{u}_{11}+(1-t){u}_{12})}^{2}\ge (t{u}_{11}^{2}+(1-t){u}_{12}^{2}),$$$${(t{u}_{21}+(1-t){u}_{22})}^{2}\ge (t{u}_{21}^{2}+(1-t){u}_{22}^{2}).$$Following this observation, we can then generalize that given any $(p,q)\in {\theta}^{2}$ for $0\le t\le 1$ we obtain $\mathcal{L}(tp+(1-t\left)q\right)\ge t\mathcal{L}\left(p\right)+(1-t)\mathcal{L}\left(q\right).$ Hence, the integrand of $\mathcal{J}$ is concave.
- 5.
- Following the fact that the model state variable in the objective function, that is, I and ${R}_{1}$ are bounded as shown in 3. Above, there exits positive constants ${\phi}_{1},{\phi}_{2}>0$ such that the sum of $(I+{R}_{1})\le {\phi}_{2}.$ If we set ${\phi}_{2}={\mathrm{max}}_{n}=3,4\cdots ,$ we have that$$\mathcal{L}(I,{R}_{1},{u}_{1},{u}_{2})\le {\eta}_{1}+{\eta}_{2}+{\eta}_{3}{\left(\right)}^{|}{\displaystyle \frac{2}{\tau}}$$$$\mathcal{L}(I,{R}_{1},{u}_{1},{u}_{2})\le {\phi}_{1}+{\phi}_{2}{\left(\right)}^{|}{\displaystyle \frac{2}{\sigma}}$$

#### 5.2. Characterization of the Optimal Control Problem

## 6. Numerical Simulation of the Optimal Control Problem

#### 6.1. Different Combination of Interventions

#### 6.2. Different Initial Condition of Population

#### 6.3. Different Initial Basic Reproduction Number

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Transmission diagram of model (1).

**Figure 2.**Estimation result on fitting the accumulated cases in Papua with $c\left(t\right)$ in Equation (3). The best-fit initial conditions for ${S}_{1},$ ${E}_{1},$ $I,$ $T,$ ${R}_{1},$ ${R}_{2},$ $V,$W are 3,435,430, 2000, 2000, 2000, 1000, 21, 6,870,860, 3000, respectively.

**Figure 3.**Estimation result of fitting the accumulated cases in West Papua with $c\left(t\right)$ in Equation (3). The best-fit initial conditions for ${S}_{1},$ ${E}_{1},$ $I,$ $T,$ ${R}_{1},$ ${R}_{2},$ $V,$ W are 974,300, 400, 400, 51, 21, 125, 1,789,099, 202, respectively.

**Figure 6.**Local sensitivity analysis with full–normalization technique of all variables with respect to all parameters (

**a**) and without ${\delta}_{1}$ (

**b**). We use incidence data from Papua and the following initial conditions: ${S}_{1}\left(0\right)=\mathrm{3,208,839},$ ${S}_{2}\left(0\right)=0,\phantom{\rule{0.277778em}{0ex}}{E}_{1}\left(0\right)=2000,\phantom{\rule{0.277778em}{0ex}}{E}_{2}\left(0\right)=0,\phantom{\rule{0.277778em}{0ex}}I\left(0\right)=2000,\phantom{\rule{0.277778em}{0ex}}T\left(0\right)=1000,\phantom{\rule{0.277778em}{0ex}}{R}_{1}\left(0\right)=1000,\phantom{\rule{0.277778em}{0ex}}{R}_{2}\left(0\right)=317,\phantom{\rule{0.277778em}{0ex}}V\left(0\right)=\mathrm{6,664,102},\phantom{\rule{0.277778em}{0ex}}W\left(0\right)=2781$ to run the simulation.

**Figure 7.**Local sensitivity analysis with full–normalization technique of all variables with respect to all parameters (

**a**) and without ${\delta}_{1}$ (

**b**). We use incidence data from West-Papua and the following initial conditions: ${S}_{1}\left(0\right)=\mathrm{918,167},\phantom{\rule{0.277778em}{0ex}}{S}_{2}\left(0\right)=0,\phantom{\rule{0.277778em}{0ex}}{E}_{1}\left(0\right)=399,\phantom{\rule{0.277778em}{0ex}}{E}_{2}\left(0\right)=0,\phantom{\rule{0.277778em}{0ex}}I\left(0\right)=242,\phantom{\rule{0.277778em}{0ex}}T\left(0\right)=51,\phantom{\rule{0.277778em}{0ex}}{R}_{1}\left(0\right)=22,\phantom{\rule{0.277778em}{0ex}}{R}_{2}\left(0\right)=88,\phantom{\rule{0.277778em}{0ex}}V\left(0\right)=\mathrm{1,915,967},\phantom{\rule{0.277778em}{0ex}}W\left(0\right)=641$ to run the simulation.

**Figure 8.**Dynamics of human (

**a**–

**h**), mosquitoes (

**i**,

**j**), and controls (

**k**,

**l**) under the scenario of pre-erythrocytic vaccine intervention only applies. Blue and red curve represents implementation strategies with and without controls.

**Figure 9.**Dynamics of human (

**a**–

**h**), mosquitoes (

**i**,

**j**), and controls (

**k**,

**l**) under the scenario of transmission-blocking treatment intervention only applies. Blue and red curve represents implementation strategies with and without controls.

**Figure 10.**Dynamics of human (

**a**–

**h**), mosquitoes (

**i**,

**j**), and controls (

**k**,

**l**) under the scenario of both intervention applies. Red and blue curve present a condition of without and with implementation of control.

**Figure 11.**Dynamics of total susceptible human (

**a**) total of infected human (

**b**) for four different strategies (no intervention (red), ${u}_{1}$ and ${u}_{2}$ (blue), ${u}_{1}$ only (black), and ${u}_{2}$ only (magenta)).

**Figure 12.**Dynamics of human (

**a**–

**h**), mosquitoes (

**i**,

**j**), and controls (

**k**,

**l**) under the scenario of both intervention applies, but with a higher number of infected individuals at time $t=0$. Red and blue curve present a condition of without and with implementation of control.

**Figure 13.**Time series dynamics for; humans (

**a**–

**h**), mosquitoes (

**i**,

**j**), and controls (

**k**,

**l**), but different initial ${\mathcal{R}}_{0}$ when no control applied. The color in the legend are explained in Table 5.

**Table 1.**The parameters of malaria model in (1). Some parameter values are taken from parameter estimation results in Section 2.2 and the rest are from assumptions.

Symbols | Biological Definitions | Papua | West Papua | Sources |
---|---|---|---|---|

${\mathrm{\Pi}}_{h}$ | Natural birth rate of humans | $\frac{3,435,430}{71.5\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}12}$ | $\frac{981,822}{71.5\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}12}$ | [32,33,34] |

${\mathrm{\Pi}}_{v}$ | Natural birth rate of mosquitos | $\frac{2\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}3,435,430}{\frac{21}{30}}$ | $\frac{2\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}981,822}{\frac{21}{30}}$ | [32,33,35] |

b | The average number of mosquitoes bite per unit time | $9.07523$ | $5.94285$ | estimated |

$\overline{{\beta}_{h1}}$ | The successful transmission rate of susceptible humans per bite | $0.89999$ | $0.87753$ | estimated |

$\overline{{\beta}_{h2}}$ | The successful transmission rate of susceptible humans who receive pre-erythrocytic vaccine per bite | $0.42299$ | $0.41244$ | estimated |

${\beta}_{h1}=\frac{b\overline{{\beta}_{h1}}}{N}$ | Average infection rate to humans per unit time per mosquito | $2.37749\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | $5.31162\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | estimated |

${\beta}_{h2}=\frac{b\overline{{\beta}_{h2}}}{N}$ | Average infection rate to humans who receive pre-erythrocytic vaccine per unit time per mosquito | $1.11742\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | $2.49646\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | estimated |

$\overline{{\beta}_{v}}$ | Average of successful transmission rate of susceptible mosquitos per bite | $0.00572$ | $0.01054$ | estimated |

${\beta}_{v}=\frac{b\overline{{\beta}_{v}}}{N}$ | Average infection rate to mosquitos per unit time per human | $1.51190\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $6.38376\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | estimated |

$\kappa $ | Waning rate of temporal immunity from recovered carriers | $0.33333$ | $0.33333$ | estimated |

$\vartheta $ | Waning rate of temporal immunity from fully recovered class | $0.05555$ | $0.05555$ | estimated |

${u}_{1}$ | Vaccination rate with pre-erythrocytic vaccine | $0.001$ | $0.001$ | assumption |

${u}_{2}$ | Rate of treatment with transmission-blocking drugs | $0.499999$ | $0.499997$ | estimated |

$1-\xi $ | pre-erythrocytic vaccine efficacy level | $0.53$ | $0.53$ | [36] |

$\alpha $ | Waning rate of vaccine efficacy | $\frac{1}{6}$ | $\frac{1}{6}$ | [37] |

${\delta}_{1}$ | Progression rate from exposed without vaccine class | $6.08333$ | $6.08333$ | estimated |

${\delta}_{2}$ | Progression rate from exposed with vaccine class | 2 | 2 | [36,38] |

${\gamma}_{1}$ | Natural recovery rate of infected humans by immune response | $0.19999$ | $0.19999$ | estimated |

${\gamma}_{2}^{-1}$ | Duration of treatment with transmission-blocking drugs | $\frac{1}{1.08631}$ | $\frac{1}{1.08631}$ | estimated |

p | Proportion of people in treatment who managed to get protection | $0.59999$ | $0.59964$ | estimated |

q | Proportion of people in treatment who fail to receive protection and then recover naturally | $0.29999$ | $0.29994$ | estimated |

$1-p-q$ | Proportion of people in treatment who fail to receive protection and then return to the infected class | $0.100000004$ | $0.10040$ | estimated |

${\zeta}_{e1}$ | Correction factor for infection rate in mosquitoes by exposed humans | $0.04999$ | $0.04999$ | estimated |

${\zeta}_{e2}$ | Correction factor for infection rate in mosquitoes by exposed humans with vaccine | $0.03999$ | $0.03999$ | assumption |

${\zeta}_{r}$ | Correction factor for infection rate in mosquitoes by recovered humans but carrier | $0.06$ | $0.06$ | estimated |

${\zeta}_{t}$ | Correction factor for infection rate in mosquitoes by humans in treatment | $0.08$ | $0.08$ | estimated |

${\mu}_{h}$ | Natural death rate of humans | $\frac{1}{71.5\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}12}$ | $\frac{1}{71.5\times 12}$ | [34] |

${\mu}_{v}$ | Natural death rate of mosquitos | $\frac{30}{21}$ | $\frac{30}{21}$ | [35] |

Parameters | Interval Values | Sources |
---|---|---|

${\mathrm{\Pi}}_{h}$ | $\frac{3,435,430}{71.5\times 12}$ or $\frac{981,822}{71.5\times 12}$ | [32,33,34] |

${\mathrm{\Pi}}_{v}$ | $2\times 3,435,430\times \frac{30}{21}$ or $2\times 981,822\times \frac{30}{21}$ | [32,33,35] |

b | $\left(\right)$ | [27] |

$\overline{{\beta}_{h1}}$ | $\left(\right)$ | [16,27] |

$\overline{{\beta}_{h2}}$ | $\left(\right)$ | [16,27,36] |

${\beta}_{h1}=\frac{b\overline{{\beta}_{h1}}}{N}$ | $\left(\right)$ or $\left(\right)$ | [27,32,33] |

${\beta}_{h2}=\frac{b\overline{{\beta}_{h2}}}{N}$ | $\left(\right)$ or $\left(\right)$ | [27,32,33,36] |

$\overline{{\beta}_{v}}$ | $\left(\right)$ | [27] |

${\beta}_{v}=\frac{b\overline{{\beta}_{v}}}{N}$ | $\left(\right)$ or $\left(\right)$ | [27,32,33] |

$\kappa $ | $\left(\right)$ | [38] |

$\vartheta $ | $\left(\right)$ | [38] |

${u}_{1}$ | $\left(\right)$ | varied |

${u}_{2}$ | $\left(\right)$ | varied |

$1-\xi $ | $0.53$ | [36] |

$\alpha $ | $\left(\right)$ | [37] |

${\delta}_{1}$ | $\left(\right)$ | [38] |

${\delta}_{2}$ | $\left(\right)$ | [36,38] |

${\gamma}_{1}$ | $\left(\right)$ | [46] |

${\gamma}_{2}$ | $\left(\right)$ | [47] |

p | $\left(\right)$ | varied |

q | $\left(\right)$ | varied |

$1-p-q$ | $\left(\right)$ | varied |

${\zeta}_{e1}$ | $\left(\right)$ | [16] |

${\zeta}_{e2}$ | $\left(\right)$ | varied |

${\zeta}_{r}$ | $\left(\right)$ | [16] |

${\zeta}_{t}$ | $\left(\right)$ | [16] |

${\mu}_{h}$ | $\frac{1}{71.5\times 12}$ | [34] |

${\mu}_{v}$ | $\frac{30}{21}$ | [35] |

Parameter | Value | Parameter | Value |
---|---|---|---|

$\kappa $ | $0.333333327466586$ | ${\gamma}_{1}$ | $0.199999993947735$ |

$\vartheta $ | $0.0555555854358188$ | ${\gamma}_{2}$ | $1.08631070229205$ |

$\alpha $ | $\frac{1}{6}$ | ${\beta}_{v}$ | $1.51190029406934\times {10}^{-8}$ |

${\beta}_{h1}$ | $2.37749246426963\times {10}^{-6}$ | ${\zeta}_{e1}$ | $0.0499997440960968$ |

${u}_{1}$ | $0.001$ | ${\zeta}_{e2}$ | $0.0399997440960968$ |

${\beta}_{h2}$ | $1.11742145820672\times {10}^{-6}$ | ${\zeta}_{t}$ | $0.080000001876738$ |

${\delta}_{1}$ | $6.08333998105411$ | ${\zeta}_{r}$ | $0.060000000761188$ |

${\delta}_{2}$ | 2 | ${\mu}_{h}$ | $1.165501166\times {10}^{-3}$ |

p | $0.599999998192909$ | ${\mathrm{\Pi}}_{h}$ | $4003.99766899767$ |

q | $0.299999997668467$ | ${\mu}_{v}$ | $1.428571429$ |

${u}_{2}$ | $0.499999999825239$ | ${\mathrm{\Pi}}_{v}$ | $\mathrm{9,815,514.28571429}$ |

**Table 4.**PRCC parameter values and p-values for the global sensitivity analysis against ${\mathcal{R}}_{0}$.

Parameter | PRCC Values | p-Values | Significant? |
---|---|---|---|

${\mathrm{\Pi}}_{h}$ | $0.271036046$ | 0 | TRUE |

${\beta}_{h1}$ | $0.631568463$ | 0 | TRUE |

${\mu}_{h}$ | $-0.656060428$ | 0 | TRUE |

${\mu}_{v}$ | $-0.856948705$ | 0 | TRUE |

${\beta}_{v}$ | $0.643359162$ | 0 | TRUE |

${\xi}_{e1}$ | $0.038340724$ | $2.282\times {10}^{-1}$ | FALSE |

${\mathrm{\Pi}}_{v}$ | $0.276430743$ | 0 | TRUE |

${\delta}_{1}$ | $0.003574822$ | $9.106\times {10}^{-1}$ | FALSE |

${\gamma}_{1}$ | $-0.162743705$ | $2.486\times {10}^{-7}$ | TRUE |

$\kappa $ | $-0.079425150$ | $1.240\times {10}^{-2}$ | FALSE |

${\xi}_{r}$ | $0.183579610$ | $5.425\times {10}^{-9}$ | TRUE |

**Table 5.**Numerical results for various initial value of the non-controlled basic reproduction number.

Case | Scenario for ${\mathit{\beta}}_{\mathit{h}1}$, ${\mathit{\beta}}_{\mathit{h}2}$ and ${\mathit{\beta}}_{\mathit{v}}$ | ${\mathcal{R}}_{0}$ | Inverted Case | Cost | Colour |
---|---|---|---|---|---|

1 | Reduced 50% from case 3 | 1.005 | 52,787 | $7.038\times {10}^{11}$ | Red |

2 | Reduced 25% from case 3 | 1.508 | 129,702 | $1.101\times {10}^{12}$ | Green |

3 | As in Table 1 for Papua data | 2.011 | 170,680 | $1.334\times {10}^{12}$ | Blue |

4 | Increased 50% from case 3 | 3.016 | 185,168 | $1.857\times {10}^{12}$ | Cyan |

5 | Increased 100% from case 3 | 4.022 | 183,584 | $2.67\times {10}^{12}$ | Magenta |

6 | Increased 150% from case 3 | 5.027 | 184,294 | $3.573\times {10}^{12}$ | Yellow |

7 | Increased 200% from case 3 | 6.033 | 185,505 | $4.562\times {10}^{12}$ | Black |

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**MDPI and ACS Style**

Handari, B.D.; Ramadhani, R.A.; Chukwu, C.W.; Khoshnaw, S.H.A.; Aldila, D.
An Optimal Control Model to Understand the Potential Impact of the New Vaccine and Transmission-Blocking Drugs for Malaria: A Case Study in Papua and West Papua, Indonesia. *Vaccines* **2022**, *10*, 1174.
https://doi.org/10.3390/vaccines10081174

**AMA Style**

Handari BD, Ramadhani RA, Chukwu CW, Khoshnaw SHA, Aldila D.
An Optimal Control Model to Understand the Potential Impact of the New Vaccine and Transmission-Blocking Drugs for Malaria: A Case Study in Papua and West Papua, Indonesia. *Vaccines*. 2022; 10(8):1174.
https://doi.org/10.3390/vaccines10081174

**Chicago/Turabian Style**

Handari, Bevina D., Rossi A. Ramadhani, Chidozie W. Chukwu, Sarbaz H. A. Khoshnaw, and Dipo Aldila.
2022. "An Optimal Control Model to Understand the Potential Impact of the New Vaccine and Transmission-Blocking Drugs for Malaria: A Case Study in Papua and West Papua, Indonesia" *Vaccines* 10, no. 8: 1174.
https://doi.org/10.3390/vaccines10081174