Dynamic analysis and optimal control of leptospirosis based on Caputo fractional derivative

1.   Introduction

Throughout human history, the emergence of infectious diseases has not only affected the health and quality of life of many people but also claimed many lives. At the same time, it also significantly impacts the economy, healthcare system, and social stability and poses a severe threat to public health. Therefore, preventing and effectively controlling infectious diseases can reduce the economic losses caused by diseases and have great significance for protecting personal health, improving public health, and maintaining social stability. Leptospirosis is a zoonotic disease. According to the World Health Organization (WHO), the incidence of endemic human leptospirosis is estimated at 5 per 100,000 per year, and the incidence of epidemic human leptospirosis is 14 per 100,000 per year ^[1]. However, recent estimates suggest that there are 1.03 million cases of leptospirosis each year and that most leptospirosis cases and deaths occur in tropical regions. Men aged 20–29 years have the highest incidence, while older men aged 50–59 years have the highest estimated mortality ^[2]. According to studies, the symptoms of infected people after infection are varied; there are many sporadic clinical manifestations, from asymptomatic or flu, fever, to various organ failure, and death ^[3,4,5,6]. Because leptospirosis is an acute febrile disease, usually characterized initially by fever which can be confused with diseases such as dengue fever, malaria, and typhoid through carelessness, coupled with the rapid progression of the disease and poor sanitation, leptospirosis has spread in many developing countries ^[4,5]. The reproduction of this pathogenic bacteria occurs mainly in the renal tubules of mammals ^[3,5,7]. The bacteria is excreted into the environment through urine, and humans become infected through contact with an infected animal's urine (or other bodily fluids other than saliva) or contact by water, soil, or food contaminated by an infected animal ^[7]. As a result, leptospirosis is more likely to infect people with frequent contact with animals, contaminated water, or soil. Examples include slaughterhouse workers, veterinarians, hunters, sewage treatment workers, farmers, construction workers, aquaculturists, etc. ^[7,8,9,10]. Outbreaks of leptospirosis are often associated with natural disasters, with flooding proving to be one of the most critical drivers of infection on islands and in Asia, along with and potential risks to rodent exposure and hygiene ^[11]. Therefore, studying the spread and control of leptospirosis is of great practical significance to safeguard people's lives and health and maintain social stability and order.

Mathematical models are often built to analyze the characteristics of disease transmission. For example, Guo and Li established a differential equation system to simulate the transmission process of the novel coronavirus. They established a zoning model to explore the competitive transmission characteristics of the Omimi and the Delta strains, which provided scientific suggestions for controlling the spread of the epidemic ^[12,13]. However, in recent years, fractional calculus has been used more and more in biological and medical modeling than traditional integer models. For example, Zhang et al. discussed the stability of integer-order and fractional-order systems in depth and intuitively demonstrated many advantages of fractional-order systems through time response ^[14]. Tuan proposed a mathematical model of Caputo's fractional derivative transmission of COVID-19, solving the system and numerical simulation of worldwide transmission based on obtained reproduction numbers indicating that the epidemic is continuing ^[15]. Using Matlab, Almeida proposed a model based on Caputo fractional derivatives to simulate a chickenpox outbreak among schoolchildren in Shenzhen, China ^[16]. Therefore, fractional models are more accurate in capturing the dynamic characteristics of biological systems. Introducing fractional differential equations in studying infectious disease models provides a novel and accurate mathematical tool for studying infectious disease transmission.

Regarding the mathematical model of leptospirosis, Mukdasai uses the advantages of random number sets supervised neural networks to simulate a fractional leptospirosis model numerically ^[17]. Ullah and Khan developed a fractional model of leptospirosis with the Atangana-Baleanu (AB) derivative, and the results show that fractional order plays a vital role in better understanding the dynamics of the disease ^[18]. Okosun et al. has studied control measures, vaccination, and treatment to control the spread of leptospirosis and advocates that policymakers should work on vaccination and treatment regimes that will better help control the epidemic ^[19]. Baca-Carrasco et al. developed a susceptible and infected model and proposed several intervention techniques to control the disease, suggesting that rodents and other animals that serve as bacterial hosts and are bacteria-free in the environment can infect humans ^[20]. Khan et al. analyzed leptospirosis with saturation incidence ^[21]. Alalhareth et al. used fractional derivatives to analyze the transmission dynamics of leptospirosis with environmental effects and bifurcations ^[22]. Ngoma et al. investigated a fractional model of the transmission dynamics of leptospirosis in the environmental sub-compartment ^[23]. Qu et al. analyzed the dynamics of leptospirosis in the context of piece smart classical global and classical fractional operators ^[24].

In the above studies on leptospirosis, different scholars considered different influencing factors and established different mathematical models to analyze the transmission dynamics of leptospirosis. Few scholars have proposed control measures to optimize disease control. Therefore, based on previous studies, this paper mainly made the following contributions: (1) The Caputo fractional differential (CFD) equation model with good disease fitting effect was introduced to study the transmission characteristics of leptospirosis. Considering the order mismatch between the two sides of the model, fractional order parameters are introduced on the right side. (2) To make the fractional-order model established by us meaningful, we analyze its dynamic characteristics including non-negative, bounded, local, and global stability of local equilibrium point (EEP) and disease-free equilibrium point (DFEP). (3) To control the spread of leptospirosis, we proposed three control functions, developed three different control strategies, and adopted the forward-backward scanning algorithm to optimize the control functions. Finally, numerical simulation is carried out using the fde12 solver. The influence of different fractions on disease transmission and the effect of different control strategies on disease control were analyzed.

The structure of this paper is as follows: The second section gives some theorems of fractional calculus. The third part constructs a fractional model of leptospirosis according to the existence mode and transmission characteristics of leptospirosis. The fourth part proves the fractional order model's non-negative, boundedness, and stability. Section 5 establishes the optimal control model of leptospirosis according to the three control functions we proposed, and the forward and backward scan algorithm is introduced to optimize the control function. The sixth part is the numerical simulation results using the fde12 solver. The seventh part is the conclusion.

2.   Database description

This section introduces some theorems and properties of fractional calculus ^[25].

Definition 2.1. ^[26] When $\alpha > 0$, the left Riemann-Liouville fractional integral is

or

$\Gamma(\cdot)$ is the gamma function.

Definition 2.2. When $\alpha \in (n-1, n]$, the left CFD is

where $n$ is a positive integer.

Definition 2.3. The Laplace transform of $g(t)$ is $G(s)$, and when $\alpha \in (n-1, n]$ ($n$ is a positive integer), the Laplace transform of the left CFD of $g(t)$ is

Definition 2.4. Two-argument Mittag-Leffler function is

Satisfy the following properties ^[27]:

Definition 2.5. The Laplace transform of $t^{r-1}E_{l, r}(\pm \lambda t ^l)$ is

Definition 2.6. $($Routh–Hurwitz stability criterion$)$ The closed-loop characteristic equation of the known system is

Suppose all coefficients in the above equation are real numbers and $a_{n} > 0$. In that case, the necessary condition for system stability is that all coefficients of the system characteristic equation are positive numbers.

3.   Fractional order model

Considering the existence mode and transmission characteristics of leptospirosis, inspired by article ^[18], we choose the following mathematical model as the basis model and then introduce the CFD to study the transmission of leptospirosis. The model consists of a human host and a susceptible vector. The human host is divided into three compartments: susceptible population $S_{h}(t)$, infected population $I_{h}(t)$, and recovering population $R_{h}(t)$. The medium consists of two compartments of susceptible animal $S_{v}(t)$ and infected animal $I_{v}(t)$. Thus, the following differential equation form is obtained:

Here, the overall $N_ {h}(t)$ is by susceptibility, infection, and recovery of three room area, namely, $N_ {h}(t) = S_ {h}(t) + I_ {h} (t) + R_ {h} (t)$. At the same time, medium total $N_ {v}(t)$ is composed of infection and restoring the two districts, that is, $N_{v}(t) = S_{v} (t)+I_{v}(t)$.

In addition, Table 1 is the meaning of the parameters in the model.

The initial constraints:

By definition, the CFD solves the fractional initial value problem in R-L fractional calculus. The definition of the Caputo-type derivative takes the same form as the integer order differential equation, including the restriction of the integer order derivative value to the value of the unknown function at the endpoint $t = a$, and so on. Therefore, the description of the Caputo differential equation is more suitable for nonzero initial value problems. So, we introduce the Caputo fractional differential equation to reconstruct the model. At the same time, to solve the problem of inconsistent orders on both sides of the equal sign, the parameters are modified to fractional order, and the equation obtained is as follows:

Figure 1 shows the transmission process of leptospirosis.

4.   Model analysis

In this section, to ensure that our model (3.3) makes sense, first we prove the nonnegative property of the model and the boundedness. Then, the local stability of DFEP and EEP is proved, respectively. Finally, the global stability of DFEP and EEP is proved by constructing different Lyapunov functions.

4.1.   Positivity and boundedness

We need to look at the generalized mean value theorem to prove nonnegativity.

Lemma 4.1. ^[28] Let $f(x)\in C[c, d]$ and $D_c^{\alpha}f(x) \in C[c, d],$ for $0 < \alpha \leq 1$, then we have

with $c \leq \xi \leq x$, $\forall x \in (c, d]$.

Remark 4.1. Obviously, according to the generalized mean value theorem above Lemma 4.1, when $\forall x \in (c, d]$, $\alpha \in (0, 1]$, $f(x) \in C[c, d]$, $D_c^{\alpha}f(x) \in C[c, d]$, if $D_c^{\alpha}f(x) \geq 0$, then $f(x)$ is nondecreasing; If $D_c^{\alpha}f(x) \le 0$, then $f(x)$ is nonincreasing.

Theorem 4.1. The region

is a positivity invariant set for the model (3.3).

Proof. In ^[29], the problem of fractional differential equations and their initial values is investigated in detail. Then, the nonnegativities of the solution for model (3.3) are

According to Remark 4.1, the nonnegative property of model (3.3) is proved.

Theorem 4.2. Boundedness of region

Proof. Let $N = S_{h}+ I_{h}+R_{h}$. ${N}(t)$ transforms $\tilde{N}$ after the Laplace transform and can be obtained from model (3.3)

That is,

We take the Laplace transform of both sides of this inequality

Then,

From the Eqs (2.1) and (2.2), inverse Laplace transformation can obtain the above inequality.

The bounded range of $N(t)$ is $0\leq N \leq 1$. Besides, for $S_{v}$,

Then,

Similarly, we can get

4.2.   Stability

To start, set the right side of the model (3.3) to 0.

Solving the above equation gives two EPs, which are the DFEP $X^0 = (\frac{b^{\alpha }_{1} }{\mu ^{\alpha }_{h} }, 0, 0, \frac{b^{\alpha }_{2} }{\gamma^{\alpha } _{v} }, 0)$ and the EEP $X^{\ast} = (S_{h}^{\ast}, I_{h}^{\ast}, R_{h}^{\ast}, S_{v}^{\ast}, I_{v}^{\ast})$, where

Here, $\kappa_1 = b^{\alpha} + \rho^{\alpha}, \kappa_2 = b^{\alpha} +\upsilon^{\alpha}, \kappa_3 = \eta^{\alpha} +b^{\alpha}-\theta^{\alpha}$.

Next, the basic reproduction number (BRM) is calculated through the next generation matrix (NGM) ^[30]. Let compartment $\varphi = (I_{h}, R_{h}, S_{v}, I_{v}, S_{h})^{\top}$, then the model (3.3) is rewritten as

Because the basic regeneration number is ${R}_0 = \rho(\mathbb F \mathbb V^{-1})$, the matrices $\mathbb{F}$ and $\mathbb{V}$ are as follows ^[31]:

By calculating $R_0 = \mathbb{F}\mathbb{V}^{-1}$, we get the BRN ${R}_0$:

The BRN ${R}_0$ means that if ${R}_0 < 1$, the disease will become extinct. On the contrary, if ${R}_0\; > \; 1$, the disease will persist. Next, we will explain the stability of $X^0 = (\frac{b^{\alpha }_{1} }{\mu ^{\alpha }_{h} }, 0, 0, \frac{b^{\alpha }_{2} }{\gamma^{\alpha } _{v} }, 0)$ and $X^{\ast} = (S_{h}^{\ast}, I_{h}^{\ast}, R_{h}^{\ast}, S_{v}^{\ast}, I_{v}^{\ast})$.

Theorem 4.3. If ${R}_0 < 1$, then the disease-free equilibrium point $X^0$ of model (3.3) is locally asymptotically stable.

Proof. To begin, solve the Jacobian matrix

Bring in the DFEP, and get by simplification $X^0$

We know that to judge model (3.3) to be locally asymptotically stable at $X^0$, all eigenvalues of its matrix must satisfy the following equation ^[32].

The characteristic equation obtained from $|(\mathcal{M}(X^0)-\lambda I)| = 0$ is as follows:

Obviously, $\lambda = -k_{2}$, $\lambda = -\mu ^{\alpha } _{h}$, $\lambda = -\gamma ^{\alpha }_{v}$ all satisfy Eq (4.1). Therefore, we only need to discuss the following formula:

When ${R}_0 \leq1$, it satisfies

Then, $\rho_1\rho_2\geq \rho_3$, therefore, according to the Routh-Hurwitz criterion, the DFEP $X^0$ of model (3.3) is locally asymptotically stable.

Theorem 4.4. If ${R}_0 \geq1$, then the EEP $X^{\ast}$ of model (3.3) is locally asymptotically stable.

Proof. The $\mathcal{M}$ at the EEP $X^{\ast}$ is

For easy calculation, the matrix is expressed as

where

Its characteristic equation is expressed as

where

According to the Routh-Hurwitz criterion

According to the Routh-Hurtwiz criterion, if $D_{5} > 0$ and ${R}_0 \geq1$, $G_i > 0\; (i = 1, 2, 3, 4, 5)$, then the eigenvalue of the eigen-equation is negative. The EEP $X^{\ast}$ of model (3.3) is locally asymptotically stable.

Next, we will introduce a lemma so that we can better construct Lyapunov to prove the global stability of the model (3.3) ^[33,34,35].

Lemma 4.2. When $t\geq0$, $\alpha\in(0, 1)$, $f(t)\in {R}^+$ is continuous and differentiable.

Satisfy the following relation

and

The inequality takes an equal sign if, and only if, $\alpha = 1$.

Theorem 4.5. If ${R}_0 < 1$, the model (3.3) at the DFEP $X^0$ is global asymptotically stable.

Proof. Define $F_1(t)$ as a Lyapunov function:

$t\geq 0$, $F_1(t)$ is continuous and positive definite, and when $S_{h }(t) = I_{h }(t) = R_{h }(t) = S_{v }(t) = I_{v }(t) = 0$, it satisfies $F_1(t) = 0$.

Now, we have

Bring into $X^0 = (\frac{b^{\alpha }_{1} }{\mu ^{\alpha }_{h} }, 0, 0, \frac{b^{\alpha }_{2} }{\gamma^{\alpha } _{v} }, 0)$, then

Thus,

So if ${R}_0 < 1$, then ${R}_1 < 1$ and ${R}_2 < 1$, thus $_{0}^{C}\!D_t^{\alpha}F_1(t)\leq 0$. Besides, $_{0}^{C}\!D_t^{\alpha}F_1(t) = 0$ if, and only if, $I_h(t) = I_v(t) = 0$. Therefore, based on Lassalle's invariance theorem, all solutions tend to the maximum invariant set ^[36,37]. When ${R}_0 < 1$, the DFEP $X^0$ of model (3.3) is globally asymptotically stable.

Theorem 4.6. The model (3.3) at the EEP $X^{\ast} = (S_h^{\ast}, I_h^{\ast}, R_h^{\ast}, S_v^{\ast}, I_v^{\ast})$ is global asymptotically stable if ${R}_0 > 1$.

Proof. Define $F_2(t)$ as a Lyapunov function.

where $Q = {R}_0-1$, and if ${R}_0 > 1$, $Q > 0$. When $t\geq 0$, $F_2(t)$ is continuous and positive definite. $F_2(t) = 0$ if, and only if, $S_h(t) = S_h^{\ast}(t), \; I_h(t) = I_h^{\ast}(t), \; R_h(t) = R_h^{\ast}(t), \; S_v(t) = S_v^{\ast}(t), \; I_v(t) = I_v^{\ast}(t)$. Now, we have

According to Lemma 4.2, we get

Bring into the endemic conditions, and we get

When ${R}_0 > 1$, $_{0}^{C}\!D_t^{\alpha}F_2(t)\leq 0$ and $_{0}^{C}\!D_t^{\alpha}F_2(t) = 0$, if, and only if, $S_h(t) = S_h^{\ast}(t), \; I_h(t) = I_h^{\ast}(t), \; R_h(t) = R_h^{\ast}(t), \; I_v(t) = I_v^{\ast}(t)$. So, based on Lassalle's invariance theorem, all solutions tend to the maximum invariant set. Therefore, when ${R}_0 > 1$, the EEP $X^{\ast}$ is globally asymptotically stable.

In this section, we proved nonnegativity and boundedness. The local stability of the equilibrium point is proved according to the Rouse-Hurwitz criterion. Finally, the Lyapunov function is constructed to prove the global stability of the equilibrium point. It can be seen that the model (3.3) has certain theoretical significance for our study of leptospirosis.

5.   Optimal control problem

In this section, first, because the pathogenic bacteria exists in animal urine, soil, and contaminated water, three control variables $u_1$, $u_2$, and $u_3$ were developed according to the presence status of the bacteria to prevent people from being infected with the bacteria. Among them, $u_1$ represents personal protection; for example, people who often walk in sewage, people who work in slaughterhouses, etc., can wear protective clothing at work, regularly disinfect, and pay attention to hygiene. $u_2$ indicates timely symptomatic treatment and hospitalization for observation. $u_3$ refers to cleaning the environment, landfill treatment of infected animals, and timely cleaning and sterilization of the polluted environment. Second, the control function is optimized by the forward-backward scanning algorithm. According to our control variables $u_1, u_2$, and $u_3$, we get the following equations:

subject to the initial conditions

Our control goal is to reduce the number of $I_{h}$ and $I_{v}$ with leptospirosis and at a low cost, that is, the minimum objective function is

$\mathfrak{m}_1, \; \mathfrak{m}_2$ are weights of infected people and infected animals. The weights of the costs of the three measures are $\mathfrak{n}_1, \; \mathfrak{n}_2$, and $\mathfrak{n}_3$. The implementation cost of measures A, B, and C are $\frac{1}{2}\mathfrak{n}_1u_1^2, \; \frac{1}{2}\mathfrak{n}_2u_2^2$, and $\frac{1}{2}\mathfrak{n}_3u_3^2$, respectively.

For the above problem, the Lagrange function $\mathcal{L}$ and the Hamiltonian function $\mathcal{H}$ are as follows:

and

The adjoint variable of $S_{h}, \; I_{h}, \; R_{h}, \; S_{v}, \; I_{v}$ are $\nu_1$, $\nu_2$, $\nu_3$, $\nu_4$, $\nu_5$, respectively. Next, we will use Agrawal's method to solve fractional optimal control problems (FOCP) ^[38].

Theorem 5.1. Set $u_1^{\ast}, \; u_2^{\ast}$, and $u_3^{\ast}$ for the control system of control variables. Then, the control variable can be expressed as

where the adjoint variables $\nu_1, \nu_2, \nu_3, \nu_4$, and $\nu_5$ satisfying

have boundary conditions or transverse conditions

Proof. The adjoint variable Eq (5.5) can be obtained from the $\mathcal{H}$

and have boundary conditions or transverse conditions.

Eq (5.4) and the optimal control of the variable $u_1^{\ast}, \; u_2^{\ast}$, and $u_3^{\ast}$ can be obtained by the following

Completion of theorem proving.

In general, one way to calculate $\nu_1$ is to use Eq (5.4) as the iterative formula ^[39]. However, Eq (5.4) is not strictly an iterative formula. Therefore, we use forward and reverse scanning to optimize the control variables.

6.   Numerical simulation result

In this section, because fde12 has good accuracy and stability in CFD numerical simulation ^[40,41], we use the fde12 solver to simulate models of different fractional orders. At the same time, we propose three disease transmission control strategies and optimize the control function by using the forward-reverse scanning method. Detailed steps of the forward-backward scan algorithm and its convergence are in references ^[42,43]. Finally, the characteristics of these three strategies are analyzed. MATLAB (R2018b) was used to simulate the model.

6.1.   Disease control strategies

Control the spread of leptospirosis and take into account its cost. We developed the following three control strategies:

Strategy $\mathbf{A}$: Combination of strengthening personal protection, timely hospitalization of infected patients, and cleaning up the environment after slaughtering infected animals $(i.e., \; u_1, \; u_2, \; u_3\neq 0)$.

Strategy $\mathbf{B}$: The combination of timely hospitalization of infected patients and cleaning up the environment after slaughtering infected animals $(i.e., \; u_2, \; u_3\neq 0$, and $u_1 = 0)$.

Strategy $\mathbf{C}$: Combination of enhanced personal protection and timely hospitalization of infected patients $(i.e., \; u_1, \; u_2\neq 0$, and $u_3 = 0)$.

Parameter values and references are listed in Table 2.

6.2.   Simulation result analysis

Figure 2 shows how $S_{h}$, $I_{h}$, $R_{h}$, $S_{v}$, and $I_{v}$ change over time when $\alpha$ takes different values. We found that over time, susceptible people move into infected people. Therefore, Figure 2a gradually becomes stable after decreasing, while Figure 2b first increases slightly, then decreases, and then becomes stable. Second, looking at Figure 2d,e, we find that the same is true of animals, with the number of susceptible animals transferring over time to infected animals. Finally, the study found that, without any control, leptospirosis stabilized after 86 days ($I_{h} \le 17$, $I_{v} \le 6$).

Figure 3 shows the numerical analysis results when $\alpha$ takes different values under control Strategy A. We can see that the amount of $I_{h}$ in Figure 3b decreases, suddenly approaches a stable position on day 10, and stabilizes after 20 days. The validity of our control strategy is verified. It is worth mentioning that when $\alpha = 1$, the amount of $I_{h}$ goes up and then down. When $\alpha$ takes fractional order, the amount of $I_{h}$ is directly reduced. It has been proved that fractional order is more beneficial for us in studying the spread control of leptospirosis. In addition, in control Strategy A, our control variable $u_{3}$ is to disinfect and clean the environment after killing some infected animals, so the susceptible animals in Figure 3d and the infected animals in Figure 3e directly decline and then stabilize, reflecting our control variable $u_{3}$.

Figure 4 compares the control effects of Strategies A, B, and C on leptospirosis. First of all, it can be seen from Figure 4b that when we adopt control strategy A (personal protection, timely treatment of $I_{h}$, killing animals to clean the environment), the number of $I_ {h}$ directly declines and then becomes stable. In the case of Strategy B, there is no personal protection. In Strategy C, no animals were killed to clean the environment. The number of infected people rose to a certain extent before falling, reflecting the importance of personal protection and killing animals to clean the environment and control leptospirosis. Then, according to Figure 4b,e, in terms of the number of $I_ {h}$, the disease was better controlled after the implementation of Strategies A, B, and C on the 6th day, the 8th day, and the 10th day, respectively, ($I_{h} \le 5$). Compared with the three strategies, the first is the best for leptospirosis.

Figure 5 shows the change of the control function over time under three different strategies. Figure 5a shows that we need to ensure that infected people are hospitalized in a timely manner within the first 5 days. After 5 days, depending on the severity of the condition, home treatment or hospitalization can be chosen. After slaughtering infected animals, environmental cleaning must continue until day 42. The control measures can then be reduced according to the actual situation. In contrast, $u_{1}$ (personal protection) has always been close to zero because leptospirosis is rarely transmitted from patients to humans. Figure 5b shows that in the absence of personal protection, the measures we need to take are 93 days of hospitalization of the infected person, 98 days of killing animals to clean up the environment, and then reduce the intensity according to the actual situation. Figure 5c shows that an infected person requires immediate hospitalization when the disease is first detected. At the same time, vulnerable people should immediately take high-intensity personal protection until the pathogenic leptospirosis is eliminated. Due to the increased level of personal protection, the amount of $I_{h}$ has decreased. So, $u_{2}$ rapidly drops to close to zero.

7.   Conclusions

This paper introduces the Caputo fractional differential equation to study the transmission dynamics model of leptospirosis. First, its dynamic characteristics are analyzed and proved, giving the model practical significance. It includes nonnegative, boundedness, and stability. The fde12 solver performs numerical simulations of different fractional order models. The simulation effect of different fractional orders on the transmission of leptospirosis is given. Second, from the perspective of fractional optimal control, combined with the existence mode and transmission characteristics of pathogenic bacteria of leptospirosis, three control variables $u_{1}$ (personal protection), $u_{2}$ (timely hospitalization of infected persons), and $u_{3}$ (disinfection of killed animals and environmental cleaning) were constructed to control the spread of leptospirosis. The forward-back scan algorithm is introduced to optimize the three control functions. Finally, according to the three control variables, three different control strategies were constructed to analyze the spread of leptospirosis. The results showed that strategy A could effectively control the spread of leptospirosis quickly. Without any control measures, the disease was controlled after 86 days. The disease can be effectively controlled after 6 days when the A strategy control measures are taken. It is worth noting that the first strategy is best for leptospirosis. The introduction of forward and backward scanning algorithms has certain optimization effects on the control function.

The above research has achieved certain results. However, there are still some shortcomings. Combined with the actual transmission mode and characteristics of leptospirosis, we found a close relationship between the transmission of leptospirosis and the environment, so the next step can be to establish a model among humans, the environment, and animals for analysis. Also, the experiment lacks some actual data, but we can collect some actual data later to improve the model further.

Author contributions

Ling Zhang: Methodology, Software, Formal Analysis, and Writing - Original Draft. Xuewen Tan: Validation and Writing - Review & Editing. Jia Li: Project administration. Fan Yang: Conceptualization, Visualization and Writing - Review & Editing.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We are grateful to the three anonymous reviewers for their valuable comments and suggestions, which significantly enhanced the presentation of this work. This work was supported by the National Natural Science Foundation of China (Nos. 11361104, 12261104), the Youth Talent Program of Xingdian Talent Support Plan (XDYC-QNRC 2022-0514), the Yunnan Provincial Basic Research Program Project (No. 202301AT070016, No. 202401AT070036), the Yunnan Province International Joint Laboratory for Intelligent Integration and Application of Ethnic Multilingualism (202403AP140014).

Conflict of interest

The authors declare there is no conflict of interest.