Optimal control in a malaria model: intervention of fumigation and bed nets
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Abstract
Malaria is one of the world’s most serious health problems because of the increasing number of cases every year. First, we discuss a deterministic model of epidemic SIR-SI spread of malaria with the intervention of bed nets and fumigation. We found that the malaria-free equilibrium is locally asymptotically stable (LAS) when \(\mathcal{R}_{0} <1\) and unstable otherwise. A malaria endemic equilibrium exists and is LAS when \(\mathcal{R}_{0} >1\). Sensitivity analysis of \(\mathcal{R}_{0} \) shows that the use of bed nets and fumigation can reduce \(\mathcal{R}_{0}\). We modify the previous model into a stochastic differential equation model to understand the effect of random environmental factors on the spread of malaria. Numerical simulations show that when \(\mathcal{R}_{0} >1\), a greater value of noise intensity σ generates a solution that is different from a deterministic solution; when \(\mathcal{R}_{0} <1\), regardless of the σ value, the solution approaches a deterministic solution. Then the deterministic model was modified into an optimal control model to determine the best strategy in controlling the spread of malaria by using fumigation as the control variable. Numerical simulations show that periodic fumigations cost less than constant intervention and can reduce the number of infected humans. Priority is given to the endemic prevention strategy rather than to the endemic reduction strategy. For more effective intervention, the value of \(\mathcal{R}_{0}\) should receive close attention. A potentially endemic (\(\mathcal{R}_{0}>1\)) environment requires more frequent fumigation than an environment that is not potentially endemic (\(\mathcal{R}_{0}<1\)). A combination of the use of bed nets and fumigation can reduce the number of infected individuals at minimal cost.
Keywords
Malaria Optimal control problem Fumigation Stochastic differential equation1 Introduction
Malaria is a dangerous infectious disease caused by a Plasmodium parasite, which can be transmitted to humans through bites from infected Anopheles female mosquitos. The symptoms usually appear after one to two weeks and include fever, sweating, shivering or cold, vomiting, headache, diarrhea, and muscle aches (Infodatin [14]). Based on the 2017 World Malaria Report, the number of malaria cases in the world increased to 216 million in 2016. Those cases were mostly located in Africa (90%), Southeast Asia (7%), and the Eastern Mediterranean (2%) (WHO [22]). From 2015 to 2016, malaria cases in Indonesia reached 217,025 and were mostly found in Papua, Papua Barat, West Nusa Tenggara, Maluku, and North Maluku (Infodatin [14]).
There are many methods of preventing malaria, the most popular of which are using bed nets at night to prevent mosquito bites and using fumigation to reduce local mosquito populations (WHO [22]). In recent studies, researchers have constructed mathematical models to analyze malaria spread. These include mathematical models of malaria transmission that consider climatic factors (Abebe et al. [4]), mathematical models of malaria distribution by using mosquito nets (Agusto et al. [1]; Chitnis et al. [8]; Ngonghala et al. [18]), and mathematical models of climate-based malaria with the use of mosquito nets (Xiunan and Xiao-Qiang [25]), to which this paper refers.
Several factors should be considered when attempting to eliminate malaria vectors, such as climate factors. In tropical areas, there are differences between mosquito life expectancy in the dry and rainy seasons. Dembele et al. [9] state that there is a greater percentage of deaths from malaria during the rainy season than during the dry season. Therefore this climate factor will be considered in the model.
In this paper, we first include two malaria preventatives to the model, the use of bed nets without insecticides and fumigation. First, we construct a deterministic model of the spread of malaria with both fumigation and bed nets. We then determine the equilibrium point and basic reproduction number \(\mathcal{R}_{0}\) followed by numerical simulations to analyze how both means of intervention affect the human population.
In practice, several factors have been found to affect the spread of malaria, such as human factors (body temperature and carbon dioxide content released by the body) as described by Keyser [15] and residential factors (living close to stagnant water) as described by Theresa et al. [19]. Both factors are influenced by unpredictable environmental factors that cannot be explained by the deterministic model. Therefore the deterministic model is extended to a malaria model with stochasticity factors. Next, the stochastic model is discussed by Gray et al. [12], and numerical simulations are implemented to evaluate the dynamics of stochastic factors in spreading malaria throughout the population.
However, some obstacles arose due to the use of fumigation, such as high costs and adverse effects of continuous fumigation on the environment. Hence we ultimately developed the deterministic model into an optimal control problem. Then we analyzed the best strategy for controlling the spread of malaria by using fumigation at minimal cost.
2 Malaria deterministic model with fumigation and bed nets
There are two mathematical models for the spread of malaria in humans: the susceptible infected recovered (SIR) model is used on the human population (h), and the susceptible infected (SI) model is used on the Anopheles mosquito population (v). The difference between the two models lies in the recovered (R) compartment, which only humans possess; Anopheles mosquitos’ lifespans are too short for them to enter this stage. Both human and mosquito populations were assumed to be homogeneous closed populations; thus the total populations of humans \(N_{h}\) and mosquitos \(N_{v}\) can be considered to be the sums of all compartments of each population. Malaria may cause deaths in the human population. According to WHO [23], in 2017, there were estimated 435,000 deaths from malaria globally, compared with 451,000 estimated deaths in 2016. The WHO African region accounted for 93% of all malaria deaths in 2017, but it also accounted for 88% of the 172,000 fewer global malaria deaths reported in 2017 compared with 2010. All WHO regions except the WHO region of the America’s recorded reductions in mortality in 2017 compared with 2010. Our proposed model in this paper aims to understand the behavior of malaria only in a short time period with considering a short time period of intervention. Therefore we put aside the death rate due to malaria from our model. With this assumption, we have that all human deaths are considered natural. Fumigation and the use of bed nets were used in the models as means of intervention to eliminate malaria with the assumption that no mosquitos are resistant to fumigation. Note that the long-term intervention of fumigation may lead to genetic mutation of mosquitoes as described by Bustamam, Aldila, and Yuwanda [6].
Parameters of SIR-SI model (1)
No | Par | Description | Condition | Dimension |
---|---|---|---|---|
1 | \(A_{{h}}\) | Daily human birth rate | \(A_{{h}}>0\) | person × day^{−1} |
2 | \(A_{{v}}\) | Daily mosquito birth rate | \(A_{{v}}>0\) | mosquito × day^{−1} |
3 | \(\mu_{{h}}\) | Natural death rate of humans | \(\mu_{ {h}}>0\) | day^{−1} |
4 | \(\mu_{{v}}\) | Natural death rate of mosquitos | \(\mu_{ {v}}>0\) | day^{−1} |
5 | γ | Recovery rate of humans | γ>0 | day^{−1} |
6 | δ | Rate of transition back to vulnerable humans due to the end of the immunity period | δ>0 | day^{−1} |
7 | \(c_{{h}}\) | Successful infection probability from infectious mosquitos to susceptible humans | \(c_{{h}}>0\) | \((\text{mosquito}\times\text{day})^{-1}\) |
8 | \(c_{{v}}\) | Successful infection probability from susceptible mosquitos to infectious humans | \(c_{{v}}>0\) | \((\text{mosquito}\times\text{day})^{-1}\) |
9 | \(\beta_{{h}}\) | Biting rate | \(\beta_{{h}}\geq1\) | \(\frac{\text{person}}{\text{mosquito} \times \text{day}}\) |
10 | \(\beta_{{v}}\) | Biting rate | \(\beta_{{v}}\geq1\) | day^{−1} |
11 | u | Fumigation rate | u ≥ 0 | day^{−1} |
12 | k | Bed net proportion | 0 ≤ k ≤ 1 | – |
13 | η | Bed net quality | 0<η<1 | – |
Next, we analyze the equilibrium points and the corresponding basic reproduction number \(\mathcal{R}_{0}\). The equilibrium points of the model are the malaria-free equilibrium (MFE) and malaria endemic equilibrium (MEE). We end the analytic discussion with a discussion of the stability of equilibrium points.
- (a)The basic reproduction number \(\mathcal{R}_{0}\) is used to analyze whether the malaria is endemic (\(\mathcal{R}_{0} \geq1\)) or not (\(\mathcal{R}_{0} <1\)). \(\mathcal{R}_{0}\) can be constructed using the next-generation matrix (NGM) method [10]. Please see [2, 3, 6] for further examples of the construction of the NGM for epidemic models. First, we construct the Jacobian matrix of infected compartments constructed from system (1):The matrix J is decomposed into a transmission matrix T that contains the infectious parameter and a transition matrix V that does not contain the infectious parameter as follows:$$J = \left [ \textstyle\begin{array}{c@{\quad}c} - \mu_{h} - \gamma& \frac{c_{h} \beta_{h} ( k, \eta) S_{h}}{{N}_{{h}}}\\ \frac{c_{v} \beta_{v} ( k, \eta ) {S}_{{v}}}{N_{h}} & - u - \mu_{v} \end{array}\displaystyle \right ]. $$Therefore NGM is written as$$\begin{gathered} T = \left [ \textstyle\begin{array}{c@{\quad}c} 0 & \frac{c_{h} \beta_{h} ( k, \eta) S_{h}}{{N}_{{h}}}\\ \frac{c_{v} \beta_{v} ( k, \eta ) {S}_{{v}}}{N_{h}} & 0 \end{array}\displaystyle \right ], \\ V = \left [ \textstyle\begin{array}{c@{\quad}c} - ( \gamma+ \mu_{h} )^{-1} & 0\\ 0 & - ( u + \mu_{v} )^{-1} \end{array}\displaystyle \right ]. \end{gathered} $$and \(\mathcal{R}_{0}\) is the spectral radius of NGM,$$ \text{NGM} =- T V^{-1} = \left [ \textstyle\begin{array}{c@{\quad}c} 0 & \frac{c_{h} \beta_{h} ( k, \eta ) {A}_{{h}}}{\mu_{{h}} {N}_{{h}} ( {u}+ \mu_{{v}} )}\\ \frac{c_{v} \beta_{v} ( k, \eta ) A_{v}}{ ( \mu_{v} +u ) N_{h} ( \gamma+ \mu_{h} )} & 0 \end{array}\displaystyle \right ], $$(3)where \(N_{h} = \frac{A_{h}}{\mu_{h}}\) and \(N_{v} = \frac{A_{v}}{u+ \mu_{v}}\). Further discussion about \(\mathcal{R}_{0}\) will be given later in this section.$$ \mathcal{R}_{0} = \sqrt{\frac{c_{h} \beta_{h} ( k, \eta )}{ \gamma+ \mu_{h}} \frac{c_{v} \beta_{v} ( k, \eta )}{u + \mu_{v}} \frac{N_{v}}{N_{h}}}, $$(4)
- (b)MEE represents a condition in which malaria always persists in a population. The MEE of the model is as follows:with$$ \begin{aligned}[b] \text{MEE} &= \bigl( {S}_{{h}}^{**}, {I}_{{h}}^{**}, {R}_{{h}}^{**}, {S}_{{v}}^{**}, {I}_{{v}}^{**} \bigr) \\ &= \biggl( \frac{A_{h}}{\mu_{h}} - \bigl( {I}_{{h}}^{**} + {R}_{{h}}^{**} \bigr), {C}_{1} \bigl( \mathcal{R}_{0}^{2} -1 \bigr), {C}_{2} \bigl( \mathcal{R}_{0}^{2} -1 \bigr), \frac{A_{v}}{\mu_{v} +u} - I_{v}^{**}, C_{3} \bigl( \mathcal{R}_{0}^{2} -1 \bigr) \biggr) \end{aligned} $$(5)According to Eq. (5), MEE exists when \(\mathcal{R}_{0} >1\).$$\begin{gathered} C_{1} = \frac{ ( \delta+ \mu_{h} ) K}{c_{v} \beta_{v} \mu _{h}},\qquad C_{2} = \frac{\gamma K}{c_{v} \beta_{v} \mu_{h}},\\ C_{3} = \frac { ( \delta+ \mu_{h} ) K}{c_{h} \beta_{h} (u+ \mu_{v} )},\quad K = \frac{ ( \gamma+ \mu_{h} ) ( u+ \mu_{v} ) N_{h}}{c_{h} \beta_{h} \gamma A_{v} + N_{h} \mu_{h}^{2} (u+ \mu_{v} )}.\end{gathered} $$
- (c)The stability of the two equilibrium points can be determined with eigenvalue analysis from a system evaluated at the corresponding equilibrium point. To determine the stability of MFE, system (1) must be linearized on MFE as follows:where the characteristic polynomial of (6) is$$ J_{{\text{DFE}}}=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} -\mu_{h} & 0 & \delta&0 & -\frac{c_{h}\beta_{h}A_{h}}{\mu_{h}N_{h}}\\ 0 & -\gamma-\mu_{h} & 0 & 0 & \frac{c_{h}\beta_{h}A_{h}}{\mu_{h}N_{h}}\\ 0 & \gamma& -\delta-\mu_{h} & 0 & 0\\ 0 & -\frac{c_{v}\beta_{v}A_{v}}{(\mu_{v}+u)N_{h}} & 0 & -u-\mu _{v} & 0\\ [3pt] 0 & \frac{c_{v}\beta_{v}A_{v}}{(\mu_{v}+u)N_{h}} & 0 & 0 & -u-\mu _{v} \end{array}\displaystyle \right ), $$(6)with$$ ( \lambda+ \delta+ \mu_{h} ) (\lambda+ \mu_{h} ) ( \lambda+u+ \mu_{v} ) \bigl( a \lambda^{2} + b\lambda+ c \bigr) =0 $$(7)According to Verhulst [20], the MFE is asymptotically stable when \(\operatorname{Re} ( \lambda_{i} )<0\), \(\forall i =1,2,\ldots, n\). From Eq. (7) it follows that the MFE is asymptotically stable when \(\mathcal{R}_{0} <1\).$$\begin{gathered} a = N_{h}^{2} \mu_{h} (u+ \mu_{v} ), \\ b = N_{h}^{2} \mu_{h} (u+ \mu _{v} ) ( \mu_{{v}} +\gamma+ \mu_{h} +u), \\ c = N_{h}^{2} \mu_{h} ( \mu_{v} + u )^{2} ( \gamma+ \mu_{h} ) \bigl( 1- \mathcal{R}_{0}^{2} \bigr). \end{gathered} $$
Parameter values of SIR-SI model (1)
No | Parameter | Value | Dimension | Reference |
---|---|---|---|---|
1 | \(N_{{h}}\) | 10,000 | mosquito | Assumption |
2 | \(N_{{v}}\) | 10,000 | mosquito | Assumption |
3 | \(\mu_{{h}}\) | 0.000039 | day^{−1} | Kim et al. [16] |
4 | \(\mu_{{v}}\) | 0.1 | day^{−1} | CDC [7] |
5 | γ | 0.0035 | day^{−1} | Chitnis et al. [8] |
6 | δ | 0.00274 | day^{−1} | Chitnis et al. [8] |
7 | \(c_{{h}}\) | 0.24 | mosquito × day^{−1} | Chitnis et al. [8] |
8 | \(c_{{v}}\) | 0.024 | mosquito × day^{−1} | Chitnis et al. [8] |
9 | \(\beta_{{h}}\) | 0.35 | \(\frac{\text{person}}{\text{mosquito} \times \text{day}}\) | CDC [7] |
10 | \(\beta_{{v}}\) | 0.35 | day^{−1} | CDC [7] |
11 | u | To be evaluated | day^{−1} | |
12 | k | To be evaluated | – | |
13 | η | To be evaluated | – |
Equation (9a) is always positive, which means that the curve of the biting rate parameter toward \(\mathcal{R}_{0}\) is increasing monotonically, or as \(\beta_{h}\) increases, \(\mathcal{R}_{0}\) also increases. On the other hand, Eq. (9b) is always negative, which means that the curve of the fumigation rate parameter toward \(\mathcal {R}_{0}\) is decreasing monotonically, or as u increases, \(\mathcal{R}_{0}\) decreases.
According to Table 1, the parameter η is in the interval \((0,1)\), and k is in the interval \([0,1]\); thus we can guarantee that \(( \eta-1)<0\) and \((1- k )\geq0\). This means that Eq. (11a) shows that \(\mathcal{R}_{0} \) decreases monotonically with respect to k; that is, as k increases, \(\mathcal{R}_{0}\) decreases. Equation (11b) shows that \(\mathcal{R}_{0} \) monotonically increases with respect to η; thus, as η increases, \(\mathcal{ R}_{0}\) also increases.
Changes in the natural death rate of mosquitos shown in (14) also cause changes in the total population of the infected mosquitos. As a result, fumigation needs to be adjusted depending on the season. Because the life expectancy of mosquitos is higher during the rainy season, fumigation with substances that are more effective in killing mosquitos should be selected; in other words, the fumigation rate in the rainy season is higher than the rate during the dry season. Next, to compare the impacts of different types of fumigation, we performed an autonomous simulation for compartment models with differences in the fumigation parameter u. The simulation excludes the bed net intervention (\(k=0 \)). The first fumigation parameter has a constant fumigation rate \(u=0.05\). The second fumigation rate is given periodically on the first day of the month by considering the season in Eq. (13) for 720 days, and the fumigation rates during the dry and rainy seasons are \(u_{1} =1\) and \(u_{2} =2\), respectively. The simulation does not include fumigation.
In Fig. 5, when the bed nets are used, the blue curve shows that malaria persists with \(\mathcal{R}_{0} =1.412014007>1\). The red curve describes a population with bed nets but without taking into account the minimum quality η of the bed nets, resulting in ineffectiveness with \(\mathcal{R}_{0} =1.059010505>1\). However, if the value of η is chosen based on the previous \(\mathcal{R}_{0}\) sensitivity (\(\eta=0.4\)), then, as seen in the green curve, the population is freed from malaria with \(\mathcal{R}_{0} =0.9884098046<1\). Although the difference in \(\mathcal{R}_{0}\) is not significant, overall, we can state that not only should the proportion k of bed nets usage be considered, but also the quality η of the bed nets should be considered. Note that bed nets are given to prevent mosquitos from biting humans, not to reduce the number of mosquitos. Therefore the difference in each scenario in the mosquito population is not significant.
3 Malaria model with environment stochasticity
In the previous sections, we used a deterministic model of malaria spread where all parameters are constant. In real-world conditions, there are environmental factors that are crucial to the spread of malaria, such as body temperature, CO_{2} levels released by humans, and residing near stagnant water; these factors are all changeable and unpredictable. Therefore, to observe the effect of random environmental factors on malaria spread, in this section, we consider the approach of stochastic differential equations (SDEs).
Next, we performed numerical simulations to determine the effects of stochastic factors and the implications of parameter changes on the SDE model (16). Simulations were performed for as many as 250 iterations using the Euler–Maruyama method for 4096 days with two scenarios, a simulation of fumigation intervention and a simulation of bed net intervention. In the following simulation results, the first three curves are for human populations, and the last two curves are for mosquito populations.
For the second case, we consider the influence of the season on the mosquito death rate \(\mu_{v}\) since the life expectancy of mosquitos during rainy and dry season is variable. A constant mosquitos death rate \(\mu_{v}\) is converted to a function \(\mu_{v} (t)\) in Eq. (14). The intervention is given on the first day of the month periodically for 720 days. In Fig. 8, the curves from susceptible mosquitos and infected mosquitos oscillate every 30 days due to the season and periodic fumigation. The dynamics of the human population also cause oscillations with smaller fluctuations than those of the mosquito population because the dynamics of the mosquito population dynamics gain stability more quickly than those of the human population. We refer to this situation as a fast dynamics and slow dynamics for mosquitos and humans, respectively, which means that when an epidemic of malaria occurs, the more random the environment, the more unpredictable the dynamics of malaria over a short-term period.
4 Optimal control problem
There are several obstacles in using fumigation. One of these is the high cost of fumigation. To overcome this problem, the deterministic model (1) can be developed into an optimal control problem where the fumigation parameter u, which was previously set as a constant, now changes into a control variable \(u ( t )\) that depends on time. The purpose of this optimal control problem is determining the continuous piecewise function of the control variable \(u ( t )\) in the interval \(t_{0} =0\) through \(t_{1} = T\), which reduces infected populations at a minimum cost.
In Eq. (18), \(\mathbf{1}_{[ t_{h. j}, t_{h. j} +1)}\) is the characteristic function in the interval \([ t_{h. j}, t_{h. j} +1)\), and \(\hat{u} ( t )\) is the value of the control variable at time t given in Eq. (21).
Theorem 1
Proof
Several scenarios are implemented in numerical simulations based on the results of Theorem 1. The simulations are conducted using four different scenarios, that is, different values of \(\mathcal{R}_{0}\) (e.g., seasonal influence, implementation of bed nets, and different initial conditions). To find a balance between each component in the cost function (17), we choose \(\omega_{2} =0.5\), \(\omega_{5} =0.0025\), and \(\omega_{u} =0.01\). Note that our control variable is a rate of intervention, which can tend to ∞. Therefore we can choose the interval value of fumigation is \(0\leq u ( t )\leq5\) with the time interval of \(0\leq t \leq500\) days. Fumigations are conducted periodically every 30 days for 500 days. It is assumed that one intervention will have an impact for the next three days. In numerical simulations we used the iterative gradient descent algorithm to accelerate the convergence of the control variable \(u(t)\).
Numerical simulations are conducted with and without fumigation when \(\mathcal{R}_{0} >1\) and \(\mathcal{R}_{0} <1\) and for the purpose of comparing the reductions of infected humans and mosquitos. The \(\mathcal{R}_{0}\) formula refers to Eq. (6). The initial values of each population are \(S_{h} (0)=8000\), \(I_{h} (0)=1900\), \(R_{h} (0)=100\), \(S_{v} (0)=8000\), and \(I_{v} (0)=2000\).
Based on the results in Fig. 2, the biting rate value of mosquitos must satisfy \(\beta_{{h}} <0.17554563 \) to satisfy \(\mathcal{R}_{0} < 1\); thus we choose the parameter values \(\beta_{{h}} = \beta_{{v}} =0.17\), which gives \(\mathcal{R}_{0} =0.6868\). On the other hand, in the case where \(\mathcal{R}_{0} >1\), which describes an epidemic that still exists in the environment, we have chosen the value for \(\beta_{{h}} = \beta_{{v}} =0.35\), which gives the result for \(\mathcal{R}_{0} =1.412\).
Number of infected individuals on day 500 in the cases of \(\mathcal{R}_{0} <1\) and \(\mathcal{R}_{0} >1\)
Population | \(\mathcal{R}_{0} >1\) | \(\mathcal{R}_{0} <1\) | ||||
---|---|---|---|---|---|---|
Without intervention | With intervention | Percentage reduction | Without intervention | With intervention | Percentage reduction | |
Infected mosquitos | 234 | 48 | ↓79.48% | 21 | 13 | ↓38.05% |
Infected humans | 2367 | 865 | ↓63.45% | 577 | 362 | ↓37.26% |
In the following simulations, we aim to determine the dynamics for infected humans and mosquitos with and without fumigation by seasonal and nonseasonal influence. As in Sect. 2, the mosquito mortality rate \(\mu_{v}\), which was constant, is converted into a function as in Eq. (14).
Number of humans and mosquitos infected on day 500 with seasonal influence
Populations | \(\mathcal{R}_{0} >1\) | \(\mathcal{R}_{0} <1\) | ||||
---|---|---|---|---|---|---|
Without intervention | With intervention | Percentage reduction | Without intervention | With intervention | Percentage reduction | |
Infected mosquitos | 168 | 45 | ↓73.21% | 197 | 64 | ↓67.51% |
Infected humans | 1905 | 803 | ↓57.84% | 2367 | 923 | ↓61.00% |
Next, we performed numerical simulations by combining fumigation with nonseasonal influences and the use of bed nets. The purpose is to see whether the use of bed nets can help to reduce the number of infected humans and mosquitos more effectively than fumigation alone. In this simulation, two cases of bed net use were considered; the first case is where there are no people using bed nets (\(k =0\)), and in the second case, 20% of the total population uses bed nets (\(k =0.2\)).
The number of infected individuals on day 500 when \(k =0\) and \(k =0.2\)
Population | k = 0 | k = 0.2 | ||||
---|---|---|---|---|---|---|
Without intervention | With intervention | Percentage reduction | Without intervention | With intervention | Percentage reduction | |
Infected mosquitos | 168 | 45 | ↓73.21% | 114 | 31 | ↓72.8% |
Infected humans | 1905 | 803 | ↓57.84% | 1659 | 627 | ↓62.2% |
Finally, we want to understand the influence of fumigation under two conditions that exist in the human population, endemic reduction and endemic prevention. The difference between the two conditions is the number of mosquitos and humans infected at the initial condition \(t =0\), where in the endemic reduction scenario the number of infected individuals is relatively high as compared to the endemic prevention condition. Here we use initial conditions to illustrate the possible situations that occur in the field. In the endemic reduction scenario the initial values are \(S_{h} ( 0 ) =8000\), \(I_{h} ( 0 ) =1900\), \(R_{h} ( 0 ) =100\), \(S_{v} ( 0 ) =8000\), and \(I_{v} ( 0 ) =2000\). In the endemic prevention scenario, the initial values are \(S_{h} ( 0 ) =9900\), \(I_{h} ( 0 ) =80\), \(R_{h} ( 0 ) =20\), \(S_{v} ( 0 ) =9950\), and \(I_{v} ( 0 ) =50\).
The numbers of each population of infected individuals on the 500th day of the endemic reduction scenario
Population | Without intervention | With intervention | Percentage reduction |
---|---|---|---|
Infected mosquitos | 168 | 45 | ↓73.21% |
Infected humans | 1905 | 803 | ↓57.84% |
The number of each population of infected individuals on the 500th day of the endemic prevention scenario
Population | Without intervention | With intervention | Percentage reduction |
---|---|---|---|
Infected mosquitos | 35 | 5 | ↓85.71% |
Infected humans | 434 | 59 | ↓86.41% |
Based on Figs. 16 and 17, fumigation has successfully reduced the number of infected humans and mosquitos. The dynamic control variable \(u(t)\) for both scenarios shows that for increase in the number of infected mosquitos, fumigation must be performed, and when the number of infected mosquitos begins to decrease, fumigation must be reduced. From the result of endemic prevention, the value of the intervention is \(0< u ( t )<2.5\), which is smaller than the results of the endemic reduction \(0< u ( t ) \leq5\). The cost function of the endemic reduction scenario is \(3.46322\times1 0^{5}\), which is greater than the cost function for endemic prevention of \(1.740746\times1 0^{4}\). This means that the prevention of malaria is much better if it is performed during the early stages of the endemic. Therefore an early warning system for malaria endemics should be considered to achieve better malaria prevention results.
Tables 6 and 7 show that the percentage reduction of the endemic prevention scenario is larger than that of the endemic reduction scenario. This is because the number of infected individuals to be reduced in an endemic reduction scenario is greater than that in an endemic prevention scenario.
5 Conclusion
The deterministic model of epidemic spread of malaria with bed nets and fumigation interventions has a malaria-free equilibrium (MFE), which always exists biologically, whereas the malaria endemic equilibrium (MEE) exists when \(\mathcal{R}_{0} >1\). The MFE is asymptotically stable when \(\mathcal{R}_{0} <1\), whereas MEE can be stated as a stable point under the condition \(\mathcal{R}_{0} \geq1\). The simulation results show that fumigation is not needed when the infection rate is less than the minimum boundary of the fumigation rate such that \(\mathcal{R}_{0} <1\). In bed net interventions, when the proportion of bed nets used is less than the minimum value, bed nets will not eliminate malaria; thus the proportion of bed net users needs to be evaluated. Additionally, the better the quality of the bed nets, the higher the chance of reducing the number of mosquito bites. Simulations also show that when fumigation is not performed, many healthy humans become infected. Additionally, fumigation needs to be adjusted depending on season. By incorporating the seasonal influence the population of mosquitos reduces with \(\mathcal{R}_{0} =0.9413426708<1\). However, with no intervention, \(\mathcal{R}_{0} =1.412014006>1\); in other words, the malaria epidemic will still exist.
To observe the effect of random environmental factors on the spread of malaria, we considered a stochastic model. Numerical simulations were performed to determine the effects of stochastic factors and the implications of parameter changes on the model. The simulations show that when an epidemic exists, the environmental factors significantly affect the dynamics within all subpopulations. However, when \(\mathcal{R}_{0} <1\), the environmental factors do not greatly affect the dynamics of the subpopulations.
When the intervention is performed every 30 days, the numbers of susceptible and infected mosquitos oscillate every 30 days due to the influence of the season and fumigation. Reducing the infected population requires using smaller values for the effectiveness of bed nets. These results are consistent with the analytic solution in the deterministic case.
To reduce the large cost of fumigation, the deterministic model was developed into an optimal control problem where fumigation parameters, which were previously set as constant, changed into control variables that depended on time. Fumigation was also conducted every 30 days, and we assumed that an intervention will have an impact until the following three days. The choice of parameter values also references the results of the deterministic case. The simulations show that when infected mosquitos exist, fumigation must be performed. However, when the number of infected mosquitos begins to decrease, the level of the intervention must be reduced. When seasonal influences are considered, the level of intervention decreases significantly when the rainy season ends. The cost function is lower with a seasonal influence than without it. Numerical simulations were also performed by combining fumigation and the use of bed nets, and the combination of both of these interventions can reduce the numbers of infected mosquitos and humans with minimal cost. The influence of fumigation in endemic reduction and endemic prevention was also investigated. Fumigation has successfully reduced the numbers of infected humans and mosquitos. The dynamic control variable shows that if the number of infected mosquitos increases, then fumigation is required; on the other hand, fumigation must be performed less frequently if there are fewer infected mosquitos. The cost function for the endemic reduction scenario is higher than the cost function for the endemic prevention scenario.
Notes
Authors’ contributions
All authors carried out the proofs of the main results and approved the final manuscript.
Funding
This research was financially supported by the Indonesia Ministry of Research and Higher Education (Kemenristek DIKTI) with PUPT research grant scheme 2018 with project ID No. 367/UN2.R3.1/HKP05.00/2018.
Competing interests
The authors declare that they have no competing interests.
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