EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 1637-1658 ISSN 1307-5543 – ejpam.com Published by New York Business Global Mathematical Analysis and Numerical Simulation on Free-Living Leptospira: A Mathematical Modeling Perspective Rudianto Artiono1,∗, Budi Priyo Prawoto1, Dian Savitri1, Dimas Avian Maulana1, Nur ‘Izzati Hamdan2, Nurul Syaza Abdul Latif2, Normi Abdul Hadi2 1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Surabaya, Surabaya, East Java, Indonesia 2 School of Mathematical Sciences, College of Computing, Informatics and Media, Universiti Teknologi MARA, 4050 Shah Alam, Selangor, Malaysia Abstract. A bacterial disease called leptospirosis is very typical in both tropical and subtropical regions. It is a well-known animal-borne illness that is brought on by spiral-shaped bacteria (Leptospira spp.). Both directly and indirectly, the disease can spread to humans through the urine of sick animals or polluted water, soil, or food. Two phases might appear in leptospirosis symptoms. The patient will have mild symptoms during the first phase, which is known as the Septicemic phase. In the meantime, the Immune phase, the second, is more severe. This study aimed to create a mathematical model of leptospirosis disease using free-living bacteria. In the model, interactions occur between people, free-living Leptospira, animal hosts, and animal vectors. The population’s characteristics are used to build the model, and the actual issue is used to identify the disease’s transmission paths. While the endemic equilibrium is investigated numerically through ODE45 solver, the disease-free equilibrium is analyzed theoretically. The paper demonstrates that for the established mathematical model with an epidemic threshold R0, analytical and numerical solutions produced the same outcome. 2020 Mathematics Subject Classifications: 92B05, 34C11, 47H10, 00A71. Key Words and Phrases: Mathematical Model, Leptospirosis, Free-Living Bacteria, ODE45 Solver, Environment. 1. Introduction The Leptospira genus of spiral-shaped spirochetes is the cause of leptospirosis, a dan- gerous zoonotic illness that affects people worldwide. Leptospirosis is a serious zoonoses ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5178 Email addresses: rudiantoartiono@unesa.ac.id, budiprawoto@unesa.ac.id, diansavitri@unesa.ac.id,dimasmaulana@unesa.ac.id, izzatihamdan@uitm.edu.my, syazalatif@uitm.edu.my, normi683@uitm.edu.my https://www.ejpam.com 1637 © 2024 EJPAM All rights reserved. Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1638 disease that affect people all over the world and is caused by spiral-shaped spirochetes belonging to the Leptospira genus. This disease spreads widely not just in tropical regions but also in subtropical regions, including Europe[9, 23], Africa [6], America [26, 31], Asia [1, 7, 29] and Australia [16, 28]. The disease is also well-known as one of the world’s most typical zoonoses, with a prevalence of 1.03 million cases and 58.900 cases every year [18]. First recognized as an occupational disease, leptospirosis infected sewer workers in 1883. In 1886, German physician Adolf Weil discovered four patients with clinical man- ifestations of this disease, which included symptoms of severe jaundice, fever, enlarged liver, and kidney failure [27]. Subsequently, in 1987, Goldsmith named the disease Weil’s disease. This disease also has other names in several countries, such as Swamp fever, Swineherd’s disease, Mud fever, Redwater of Calves, Autumn fever (Akiyami), Rice-field fever, Canicola fever, Cane-cutter’s fever, Hemorrhagic Jaundice, Stuttgart disease [4, 12]. Furthermore, in 1915, Inada succeeded in isolating the Leptospira icterohemorrhagiae bac- terium as a cause of Weil’s disease [12]. Up to now, there are 20 genus Leptospira species based on DNA hybridization studies consisting of three major, namely, pathogenic, inter- mediate, and saprophytic (non-pathogenic) leptospires [4, 14]. Pathogenic bacteria infect humans and animals. Rats are well-known as vector animals for this disease. Even though bacteria can infect it, rats still usually live. Almost all species of rats are a good reservoir for the spread of leptospirosis disease [2, 21]. The bacteria breed and grow well in the rat’s kidney. These bacteria spread through rats’ urine and then contaminate the environment. Humans and animals in contact with contaminated water, soil and mud have easily been infected by leptospirosis disease [5]. In human cases, these bacteria enter the body through mucosae such as the mouth, nose, and eye. It also enters the human body through broken skin. People who work as farmers, ranchers, gardeners, and butchers would be more susceptible to the disease [3, 15]. Some animals like cats, dogs, pigs, horses, and cows are also easily infected by the bacteria. These kinds of animals are called host animals, and the bacteria can kill them. There are two main phases of infection in the human population. The first phase, Septicemic, occurs in 3-7 days with some symptoms like fever, rash, vomiting, muscle pain, headache, chills, and red eyes [8, 19, 24]. Some symptoms look like flu disease, such that people with these symptoms usually think that the influenza virus has infected them. It is also one reason why this disease is sometimes un-diagnosed or misdiagnosed. However, in order to know whether someone has an infection with leptospirosis or not, the patient needs to do a blood test. The second phase, Immune, occurs in 0-30 days. Some symptoms that probably appear in this phase are jaundice, acute rash, bleeding, kidney failure, heart failure, liver failure, or meningitis. In this phase, some leptospirosis patients probably will die [11, 25]. There is also interphase and defervescence, lasting 2-3 days. In this phase, someone with leptospirosis will feel better. Unfortunately, bacteria are still in their body, so after this phase, they will go to the Immune phase [20]. Some research in mathematical epidemiology has been developed to analyze the dynam- ics and spread of leptospirosis in a homogeneous/heterogeneous population [10, 17, 22]. Unfortunately, none of these research had considered assessing the effect of free-living leptospira in the surroundings. In the last decade, research conducted to develop mathe- Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1639 matical models of leptospirosis disease only considers the interaction between humans and animal vectors, animal hosts with animal vectors or humans with free-living leptospira alone, so this interaction model cannot describe the overall dynamics of this disease. The research gap that occurs is that a mathematical model needs to be developed that looks at all the factors that cause the spread of leptospirosis. Mathematical models of diseases that are strongly influenced by environmental factors and spread through the intermediary of several animals, both vectors and hosts and also can cause serious illness in humans, need to be constructed and analyzed so that it can be known its spread in the human population which in turn can be done effective prevention of the spread of this disease. In this paper, a framework model of leptospirosis disease with free-living bacteria in the environment is rigorously studied, which has been developed from similar research about schistosomiasis, which has been done by Garira et al. l [13]. 2. Mathematical Model The construction of a leptospirosis model in a closed population was considered a system with no seasonal effect. With regard to the characteristics of the disease, the human population at any time t was divided into susceptible human individuals SHu(t), exposed human individuals who relate to the septicemic phase EHu(t), infected human individuals who relate to the immune phase IHu(t) and recovered human individuals RHu(t). The animal population was divided into a host animal population and a vector population. Similarly with the human, the host animal population at any time t was divided into susceptible host animal individuals SHo(t), infected host animal individual IHo(t), and recovered host animal individual RHo(t), while the vector animal population at any time t was divided into susceptible vector animal individuals SHo(t) and infected vector animal individual IHo(t). Free-living leptospira at any time t is L(t). There are three possible transmission routes for susceptible humans to become infected. They have direct interaction with free-living leptospira in the surroundings at the rate of βHu1 , they have contact with infected host animals at the rate βHu2 , or they have contact with infected vector animals at the rate βHu3. It was assumed that there was only one transmission route for susceptible animals to become infected. Host and vector animals have contact with free-living leptospira at the rate βHo and βV , respectively. There are also two possibilities for the exposed human population to become infected at the rate γHu or recover at the rate σHu. The number of bacteria in the environment only comes from the urine of the infected population at the rate κHu,κHo, and κV . We made some assumptions to construct the model, as follows: a) each population is assumed to be closed; b) every individual in each population is blending homogeneously; c) every infected individual sheds leptospira bacteria in the environment; d) there is no recovered vector animal population; e) there is no direct transmission between host animal and vector animal; f) recovered individuals can be re-infected; and g) there is death-caused by the infection in human and host animal, whereas in vector animal population there is no death-caused by the infection. By incorporating a few assumptions that can provide support when creating a mathe- Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1640 matical model, the spread of the illness can be illustrated in a component diagram similar to the one shown in Figure 1. With regard to the assumptions and the compartment Figure 1: The route of transmission from Free-Living Leptospira into the human population, animal-host population, and animal-vector population. diagram, the route of transmission of leptospira bacteria at time t is disempowered by the following nonlinear ordinary differential system. Human population dSHu dt = ΛHu − (λHu + µHu)SHu + θHuRHu, (1) dEHu dt = λHuSHu − (γHu + µHu + σHu)EHu, (2) dIHu dt = γHuEHu − (µHu + δHu + αHu)IHu, (3) dRHu dt = αHuIHu + γHuEHu − (µHu + θHu)RHu, (4) Animal-Host Population dSHo dt = ΛHo − (λHo + µHo)SHo + θHoRHo, (5) dIHo dt = λHoSHo − (µHo + δHo + αHo)IHo, (6) Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1641 dRHo dt = αHoIHo − (µHo + θHo)RHo, (7) Animal-Vector Population dSV dt = ΛV − (λV + µV )SV , (8) dIV dt = λV SV − µV IV , (9) Free-Living Leptospira in the environment dL dt = κHu(EHu + IHu) + κHoIHo + κV IV − µLL, (10) where, λHu = ( βHu1L L0+∈L + βHu2IHo NHu + βHu3IV NHu ), λHo = ( βHoL L0+∈L), λV = ( βV L L0+∈L), All parameters for equations are assumed to be non-negative for all times t¿0 with the initial condition given by SHu(t0) ≥ 0, EHu(t0) ≥ 0, IHu(t0) ≥ 0, RHu(t0) ≥ 0, SHo(t0) ≥ 0, IHo(t0) ≥ 0, RHo(t0) ≥ 0, SV (t0) ≥ 0, IV (t0) ≥ 0, L(t0) ≥ 0 Here, t ≥0 represents the time in days and t0 represents the beginning of the leptospirosis illness spread where each symbol can be written as follows in table 1. Table 1: Summary of parameter’s description Parameter Description ΛHu the rate at which the human population is being recruited ΛHo the rate at which animal host populations are being recruited ΛV the rate at which animal vector populations are being recruited µHu the rate of death in the human population µHo the rate of death of animal host population µV the rate of death of animal vector population µL the rate of death of free-living leptospira in the environment θHu the transmission rate of humans from the recovered population to the susceptible population θHo the transmission rate of the animal host from the recovered population to the susceptible population γHu the transmission rate from the septicemic phase to the immune phase σHu the transmission rate of humans from the exposed population to the recovered population Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1642 Table 2: Summary of parameter’s description Parameter Description δHu mortality rate caused by sickness in the human population δHo mortality rate caused by sickness of animal host population αHu the recovery rate of the human population αHo the recovery rate of animal host population βHu1 the infection rate of the human population by free-living lep- tospires in the environment βHu2 the infection rate of the human population by infected animal host population βHu3 the infection rate of the human population by infected animal vector population βV infection rate of animal host population by free-living leptospira in the environment L0 infection rate of animal vector population by free-living leptospira in the environment ε saturation constant of leptospires bacteria κHu a reduction in the growth rate of leptospires bacteria, given the rise in cases κHo the excretion rate of leptospira from the human population into the environment κV the excretion rate of leptospira from animal-host populations into the environment 3. Result and Discussion 3.1. Feasibility Region of the Equilibria of the Model Mathematically and epidemiologically, the system is well-posed by constructed it from the real phenomenon of disease spread and the system of differential equations, meaning it is biologically meaningful when all model parameters and state variables for the model system are assumed to be non-negative and consistent with human and animal populations for all time t ≥ 0. Additionally, it can be confirmed that for the model system, all bounded and non-negative solutions with non-negative beginning conditions continue to exist. Leptospirosis transmission dynamics model in the equations (1) - (10) shall thus be examined in a suitable, feasible area, which will be determined as follows. Letting NHu = SHu +EHu + IHu +RHu, and adding equations (1) - (4) in the system, it will give us. dNHu dt = ΛHu − µHuNHu − δHu ≤ ΛHu − µHuNHu, This implies that lim t→∞ supNHo ≤ ΛHo µHo , Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1643 Analogous for another two populations Letting NHo = SHo + IHo + RHo, and adding equations (5) - (7) in the system will give us. dNV dt ≤ ΛV − µV NV , This implies that lim t→∞ supNV ≤ ΛV µV , From equation (10), we have dL dt ≤ κHu(EHu + IHu) +κHoIHo +κV IV −µLL, This implies that lim t→∞ supNV ≤ κHu(EHu + IHu) + κHoIHo + κV IV µL , Therefore, every possible systemic solution is positive and will eventually reach the invari- ant attractive area. Ω = {(SHu, EHu, IHu, RHu, SHo, IHo, RHo, SV , IV , L) : NHu(t) ≤ ΛHu µHu , NHo(t) ≤ ΛHo µHo , NV (t) ≤ ΛV µV , L(t) ≤ κHu(EHu+IHu)+κHoIHo+κV IV µL }, Thus, whenever ΛHu > µHu, Ω is attractive and positively invariant, examining the system’s solutions in Ω suffices. Results for the system’s existence, uniqueness, and contin- uation hold in this area, and all solutions beginning in Ω stay there for all t ≥ 0. Therefore, the model is well-posed both mathematically and epidemiologically. Examining the dy- namics of the flow produced by the model in Ω suffices. Unless otherwise indicated, we will assume in all that follows that ΛHu > µHu. 3.2. Determination of disease-free equilibrium We subsequently ascertained the solution of the system of nonlinear ordinary differen- tial equations based on the created mathematical model (1) - (10). In order to determine this solution, the right-hand side of each equation was taken to be equal to zero. This allowed the system to admit two equilibrium states: the endemic state and the disease-free equilibrium. dSHu dt = dEHu dt = dIHu dt = dRHu dt = dSHo dt = dIHo dt = dRHo dt = dSV dt = dIV dt = dL dt = 0, Infection in the populations of humans, animals serving as hosts, and animal vectors occurs when there are no free-living leptospira in the environment or in a state of disease-free equilibrium. As such, the model system’s disease-free equilibrium is given by E0 = (ΛHu µHu , 0, 0, 0, ΛHo µHo , 0, 0, ΛV µV , 0, 0) 3.3. Calculation of the Reproduction Number The basic reproduction number, or R0, is the most intriguing quantity in many epi- demic models that highlights the key players in the disease transmission cycle. Its defini- tion is the mean quantity of secondary infections caused by a single infectious host that is Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1644 introduced into a population that is completely susceptible. A basic reproduction number is one of the most crucial tools for evaluating disease outbreaks. For the majority of illness outbreaks, if R0 < 1, then with time, the outbreak will end, whereas if R0 > 1, then The epidemic will continue at endemic proportions. The basic reproduction number has been obtained using the operator approach of the next-generation matrix. The form can be used to write a model system. dX dt = f(X,Y, Z), dY dt = g(X,Y, Z), dZ dt = h(X,Y, Z), where X = (SHu, RHu, SHo, RHo, SV ), Y = (EHu, IHu), Z = (IHo, IV , L), Component X represents the quantity of vulnerable, while component Y represents the quantity of those with the infection who do not spread the illness. The component of Z represents the number of people that are able to spread the illness. Define g̃(X∗, Z) by g̃(X∗, Z) = g̃1(X ∗, Z), g̃2(X ∗, Z) with g̃1(X ∗, Z) = (βHu1L Lo+ϵL + βHu2LHo NHu + βHu3IV NHu ) ΛHu µHu 1 γHu+µHu+σHu , g̃2(X ∗, Z) = γHu µHu+δHu+αHu (βHu1L Lo+ϵL + βHu1LHo Lo+ϵL + βHu2IHo NHu + βHu3IV NHu ) ΛHu µHu 1 γHu+µHu+σHu , Let A = Dzh(X∗, g̃(X∗,0), 0) and further presume that A can be expressed as follows: A = M - D where M ≥ 0 and D > 0, a diagonal matrix. Next, A turns into dIHo dt = ( βHoL L0+ϵL).ΛHo µHo - (µHo + δHo + αHo)IHo, dIV dt = ( βV L L0+ϵL).Λµ - µV IV , dL dt = κHu((βHu1L L0+ϵL + βHu2IHo NHu + βHu3IV NHu ).ΛHu µHu . 1 (γHu+µHu+σHu) + γHu (µHu+δHu+αHu) (βHu1L L0+ϵL + βHu2IHo NHu + βHu3IV NHu ).ΛHu µHu . 1 (γHu+µHu+σHu) ) + κHoIHo + κV IV − µLL, A = a11 a12 a13 a21 a22 a23 a31 a32 a33  Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1645 where a11 = −(µHo + δHo + αHo), a12 = 0, a13 = ΛHo µHo βHoL0 (L0+ϵL)2 , a21 = 0, a22 = −µV , a23 = ΛV µv βV L0 (L0+ϵL)2 , a31 = κHuβHu2 (γHu+µHu+σHu) + γHuβHu2 (µHu+δHu+αHu) . 1 (γHu+µHu+σHu) + κHo, a32 = κHuβHu3 (γHu+µHu+σHu) + γHuβHu3 (µHu+δHu+αHu) . 1 (γHu+µHu+σHu) + κV , a33 = κHu(( βHu1L0 (L0+ϵL)2 ).ΛHu µHu . 1 (γHu+µHu+σHu) + γHu (µHu+δHu+αHu) ( βHu1L0 L0+ϵL)2 ).ΛHu µHu . 1 (γHu+µHu+σHu) )− µL, Since A = M - D, we deduce matrices M and D to be M = m11 m12 m13 m21 m22 m23 m31 m32 m33  where m11 = 0, m12 = 0, m13 = ΛHo µHo βV L0 (L0+ϵL)2 , m21 = 0, m22 = 0, m23 = ΛV µV βV L0 (L0+ϵL)2 , m31 = κHuβHu2 (γHu+µHu+σHu) + γHuβHu2 (µHu+δHu+αHu) . 1 (γHu+µHu+σHu) + κHo, m32 = κHuβHu3 (γHu+µHu+σHu) + γHuβHu3 (µHu+δHu+αHu) . 1 (γHu+µHu+σHu) + κV , m33 = κHu(( βHu1L0 (L0+ϵL)2 ).ΛHu µHu . 1 (γHu+µHu+σHu) + γHu (µHu+δHu+αHu) ( βHu1L0 (L0+ϵL)2 ).ΛHu µHu . 1 (γHu+µHu+σHu) ), D = d11 d12 d13 d21 d22 d23 d31 d32 d33  d11 = µHo + δHo + αHo, d11 = 0, d12 = 0, d13 = 0, d21 = 0, d22 = µv, d23 = 0, Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1646 d31 = 0, d32 = 0, d33 = µL, The basic reproduction number is the spectral radius (dominant eigenvalue) of the matrix MD−1, that is R0 = ρ(MD−1) = √ r a . p c + s b . q c , where a = 1 µHo+δHo+αHo , b = 1 µV , c = 1 µL , p = ΛHo µHo βHoL0 (L0+ϵL)2 , q = ΛV µV βV L0 (L0+ϵL)2 , r = κHuβHu2 (γHu+µHu+σHu) + γHuβHu2 (µHu+δHu+αHu) . 1 (γHu+µHu+σHu) + κHo, s = κHuβHu3 (γHu+µHu+σHu) + γHuβHu3 (µHu+δHu+αHu) . 1 (γHu+µHu+σHu) + κV , 3.4. Local Stability of the Disease Free Equilibrium Based on the theorem proposed by van den Dreissche and Watmough [30], The disease cannot spread throughout the population if the basic reproduction number R0 is smaller than one, indicating that the disease-free equilibrium is locally asymptotically stable. This is condensed into the subsequent theorem. Theorem The point E0, When R0 < 1, the model system is locally asymptotically stable; otherwise, it is unstable. Proof Since local stability of the disease-free equilibrium is a result of the van den Driess- che and Warmough theorem, there is no need for the proof. 3.5. Global Asymptotic Stability of Disease-Free Equilibrium The theorem proposed by van den Dreissche and Watmough [30] states that when R0 < 1 and R0 > 1, the disease-free equilibrium is locally asymptotically stable and unstable, respectively. In order to ensure the global asymptotic stability of the disease- free equilibrium, we must first enumerate two conditions. Fill out the form with the system’s information. dX dt = F (X,Z), dZ dt = G(X,Z), G(X, 0) = 0, Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1647 with X = (SHu, RHu, SHo, RHo, SV ) comprises of the uninfected components Z = (EHu, IHu, IHo, IV , L) comprises of infected and infectious components E0 = (X∗, 0) = (ΛHu µHu , 0, 0, 0, ΛHo µHo , 0, 0, ΛV µV , 0, 0) denotes the disease-free equilibrium To ensure asymptotic global stability, the condition H1 and H1 must be hold H1 : For dX dt = F (X,Z), X∗ is globally asymptotically stable (g.a.s) H2 : G(X,Z) = AZ−G(X,Z), G(X,Z) ≥ 0 for R(X,Z) ∈ R+ 10 where A = DZG(X∗, 0) is an M-matrix and R+ 10 is the area in which the biological model makes sense. Here, we have F (X, 0) =  ΛHu − µHu.SHu 0 ΛHu − µHu.SHu 0 ΛV − µV .SV 0  and A =  −(γHu + µHu + σHu) 0 βHu2 NHu .ΛHu µHu βHu3 NHu .ΛHu µHu βHu1 L0 .ΛHu µHu γHu −(µHu + δHu + αHu) 0 0 0 0 0 −(µHo + δHo + αHo) 0 βHo L0 .ΛHo µHo 0 0 0 −µV βV / L0 .ΛV µV κHu κHu κHo κV −µL  Z =  EHu IHu IHo IV L  Next, we have to find G̃(X,Z) from G(X,Z) = AZ − G̃(X,Z) G̃(X,Z) =  ( 1 L0 .ΛHu µHu − 1 L0+ϵLSHu)βHu1L 0 ( 1 L0 .ΛHo µHo − 1 L0+ϵLSHo)βHoL ( 1 L0 .ΛV µV − 1 L0+ϵLSV )βV L 0  Since S0 Hu = ΛHu µHu then 1 L0 ≥ 1 L0+ϵL , S 0 Hu = ΛHo µHu then 1 L0≥ 1 L0+ϵL , and S0 V = ΛV µV then 1 L0 ≥ 1 L0+ϵL therefore, it is clear that G̃(X,Z) ≥ 0 for all (X,Z) ∈ R+ 10. It is also clear that matrix A is an M -matrix since the off-diagonal elements of A are non-negative. This result can be summarized that the disease-free equilibrium E0 = (ΛHu µHu , 0, 0, 0, ΛHo µHo , 0, 0, ΛV µV , 0, 0) is globally asymptotically stable of the system if and the condition H1 and H2 are fulfilled. Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1648 3.6. The Endemic Equilibrium State This section contains our findings regarding the presence of a constant solution or endemic equilibrium for the model system. We’ll use a threshold parameter, which we’ve already designated as R0, to do this. There is one endemic equilibrium in the pro- portionately specified model solution given by E1 = (SHu = S∗ Hu, EHu = E∗ Hu, IHu = I∗Hu, RHu = R∗ Hu, SHo = S∗ Ho, IHo = I∗Ho, RHo = R∗ Ho, SV = S∗ V , IV = I∗V , L = L∗); with (S∗ Hu, E ∗ Hu, I ∗ Hu, R ∗ Hu, S ∗ Ho, I ∗ Ho, R ∗ Ho, S ∗ V , I ∗ V , L ∗) all non-negative, the threshold parameter determining their existence and characteristics R0 > 1 or more specific determined by the existence of free-living Leptospira in the environment. Suppose we have L∗ from the system, and then we have the endemic equilibrium presented. S∗ Hu = ΛHu + θHuR ∗ Hu (λ∗ Hu + µHu) , E∗ Hu = λ∗ HuS ∗ Hu (γHu + µHu + σHu) , I∗Hu = γHuE ∗ Hu (µHu + δHu + αHu) , R∗ Hu = αHuI ∗ Hu + σHuE ∗ Hu (µHu + θHu) , S∗ Ho = ΛHo + θHoR ∗ Ho λ8 Ho + µHo , I∗Ho = λ∗ Ho (µHo + δHo + αHo) S∗ Ho, R∗ Ho = αHoI ∗ Ho (µHo + θHo) , S∗ V = ΛV λ∗ V + µV , I∗V = λ∗ V µV S∗ V , where λ∗ Hu = (βHu1L ∗ L0+ϵL∗ + βHu2IHo NHu + βHu3IV NHu ), λ∗ Ho = ( βV L L0+ϵL∗ ), λ∗ V = ( βV L L0+ϵL∗ ), The analysis for this equilibrium could not be written in this article due to the com- plexity of the results, and it will be explored in the numerical simulation in the next Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1649 section. 3.7. Numerical Simulation This section displays the system’s result as a graphics produced by ODE45 Solver. The parameter value is given in Table 2. Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1650 Table 3: Summary of parameters used in the system Parameter Value Units Reference/Rational ΛHu 10451 Human day−1 247.949.975 65x365 ΛHo 1000 Animal- Host day−1 Assumed ΛHo 5000 Animal- Vector day−1 Assumed µHu 0.000042 day−1 1 65x365 (Lifespan of human with average lifespan about 65 years) µHo 0.00027 day−1 1 10x365 (Lifespan of animal host about ten years) µV 0.0013 day−1 1 2x365 (Lifespan of animal vector about two years) µL 0.07 day−1 1 14 (Lifespan of leptospire bacteria about 14 days)l θHu 0.01 day−1 1 100 (re-infection could be happen after three months) θHo 0.01 day−1 1 100 (re-infection could be happen after three months) γHu 0.05 day−1 1 20 (Septicemic phase to immune phase oc- curs in 20 days) σHu 0.005 day−1 1 200 (Exposed to recovered population oc- curs in 6 months) δHu 0.0003 day−1 30 100,000 (it is about 30 individual human deaths over 100,000 population) δHo 0.0005 day−1 50 100,000 (it is about 50 individual animal host deaths over 100,000 population) αHu 0.1 day−1 1 10 (Human recovery in 10 days) κHu 0.007 day−1 Estimated κHo 0.007 day−1 Estimated κV 0.005 day−1 Estimated αHo 0.3 day−1 1 3 (Animal Host recovery in 3 days) βHu1 0.001 day−1 Fitted βHu2 0.009 day−1 Fitted βHu3 0.005 day−1 Fitted βHo 0.0005 day−1 Fitted βV 0.0005 day−1 Fitted L0 10000 - Estimated ϵ 0.001 - Estimated Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1651 Figure 2 illustrates the solution profile for all populations when R0 < 1 (disease-free equilibrium). This condition means that there are no free-living bacteria in the environ- ment for an extended period, so then no infected population will be found in the human and animal populations. The numerical solution shows that only the susceptible human, animal host and animal vector population will exist. Figure 2: The simulation for each population in which disease-free equilibrium exists (The condition when R0 < 1). Figure 3 illustrates the dynamics of the human population, animal-host population, animal- vector population and free-living leptospires in the environment with condition R0 > 1. This result demonstrates a correlation between the existence of free-living bacteria in the environment and the existence of leptospirosis disease in human, animal-host and animal vectors. When the bacteria exist, the disease will exist in all populations. This implies that the improvement in the free-living leptospires in the environment can reduce the risk of the spread of disease in all populations. As the route of transmission depends on the infection rate parameter (β), which is the infection occurs not only in the human population but also in the animal host and animal vector population then, the next figures will illustrate the numerical solution for several values of infection rate (βHu1, βHu2, βHu3, βHo, βV ). Fig- ure 4 shows the graph of the numerical solution model for the infected human population, recovered human population, animal host population, and animal vector population with parameters βHu1 = 0.1, βHu1 = 0.01, and βHu1 = 0.0001. In the infected human popu- lation, there is no significant difference between βHu1 = 0.1 and βHu1 = 0.01, whereas, Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1652 Figure 3: The simulation for each population in which endemic equilibrium exists (The condition when R0 > 1) for βHu1 = 0.0001, the spread of disease is quite slow in the beginning, then increases to the endemic equilibrium. This phenomenon also occurs in the animal host and vector population. It slowly increases in the beginning for βHu1 = 0.0001. The graph of the recovered human population indicates the same thing as the other graph in Figure 4. Figure 5 illustrates the graph of the numerical solution model for the infected human population, recovered human population, animal host population, and animal vector pop- ulation with parameters βHu2 = 0.9, βHu2 = 0.09, and βHu2 = 0.009. All population in figure 5 shows the same phenomenon that any difference value of βHu2 does not have a significant result. Infected-human, animal-host and animal-vector population increase almost in the same interval time. Figure 6 illustrates the numerical solution model for the infected human population, recovered human population, animal host population, and animal vector population with parameters βHu3 = 0.5, βHu3 = 0.05, and βHu3 = 0.005. All populations show the same thing for the first interval time, and all of them increase to the endemic equilibrium. However, in the first interval time, all of the three parameters show a significant difference for each population dynamic. Figure 7 illustrates the numerical solution of the model for the infected human population, recovered human population, animal-host population, and animal-vector population with parameters βHo = 0.05, βHo = 0.005, and βHo = 0.0005. For the infected-human pop- ulation, recovered-human population, and animal vector population, the parameter does not have a great effect on the dynamic, whereas, for the infected-animal host population, Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1653 Figure 4: The simulation numeric for infected recovered human population and infected animal host and animal vector with several values of βHu1 (βHu1 = 0.1, βHu1 = 0.01, βHu1 = 0.001) Figure 5: The simulation numeric for infected recovered human population and infected animal host and animal vector with several values of βHu2(βHu2 = 0.9, βHu2 = 0.09, βHu2 = 0.009) this parameter has a significant effect on the dynamic. Figure 8 shows the graph of the numerical solution model for the infected human population, recovered human popula- tion, animal host population, and animal vector population with parameters βV = 0.05, βV = 0.005, and βV = 0.0005. Almost all of the population have the same dynamic behav- ior with several values of parameter βV . However, the infected animal vector population has significant results compared to the other population, precisely when βV=0.0005. 4. Conclusions In conclusion, our work has built and investigated the mathematical modeling of lep- tospirosis using free-living microorganisms in the environment. With free-living microbes in the environment, various infectious diseases can theoretically be replicated using the generic modeling approach. This study established the disease reproduction number R0 Rudianto Artiono et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1637-1658 1654 Figure 6: The simulation numeric for infected recovered human population and infected animal host and animal vector with several values of βHu3(βHu3 = 0.5, βHu3 = 0.05, βHu3 = 0.005) Figure 7: The simulation numeric for infected recovered human population and infected animal host and animal vector with several values of βHo(βHo = 0.5, βHo = 0.05, βHo = 0.005) and the connected within-animal host and between-animal vector models of human lep- tospirosis using numerical simulations of the whole model and results from the analysis of the endemic equilibrium expression. The endemic equilibrium point and the disease-free equilibrium point are the two equilibrium points that are calculated. Additionally, this work provided a stability analysis for the disease-free equilibrium, which is stable if all of the parameters are assumed to be positive and the stability conditions are met. This is consistent with the numerical simulation that shows a disease-free equilibrium exists. The next generation matrix, which is assessed at the disease-free equilibrium point, is used to compute the basic reproduction number or R0. The presence of the disease in the human population was identified by the state of each basic reproduction number pair. The presence of the endemic and disease-free equilibriums, which rely on the threshold pa- rameter R0, is confirmed through numerical simulation. The presence of the endemic and REFERENCES 1655 Figure 8: The simulation numeric for infected recovered human population and infected animal host and animal vector with several values of βV (βV = 0.5, βV = 0.05, βV = 0.005) disease-free equilibriums, which rely on the threshold parameter R0, is confirmed through numerical simulation. The numerical results also provide a simulation of the dynamics for the selected populations, that is, the populations of infected humans, recovered humans, infected animal hosts, and infected animal vectors. Conflicts of Interest The Author (s) declare(s) that there is no conflict of interest regarding the publication of this paper. Acknowledgements The authors acknowledge the Dean of Mathematics and Natural Sciences Universitas Negeri Surabaya and the Head of the School of Mathematical Sciences, College of Com- puting, Informatics and Media, UiTM Shah Alam, Malaysia, for financial support to do this research. The authors also thank the reviewers for all their valuable comments and suggestions. References [1] Rudianto Artiono. Parameter estimation using inla for disease mapping of leptospiro- sis in bantul indonesia. In Journal of Physics: Conference Series, volume 1417, page 012026. IOP Publishing, 2019. [2] Vinayagamurthy Balamurugan, Nidaghatta L Gangadhar, Nagalingam Mohandoss, Sushma Rahim Assadi Thirumalesh, Moushumi Dhar, Rajeswari Shome, Para- manandham Krishnamoorthy, Krishnamsetty Prabhudas, and Habibur Rahman. REFERENCES 1656 Characterization of leptospira isolates from animals and humans: phylogenetic anal- ysis identifies the prevalence of intermediate species in india. Springerplus, 2(1):362, 2013. [3] Arnau Casanovas-Massana, Gabriel Ghizzi Pedra, Elsio A Wunder Jr, Peter J Diggle, Mike Begon, and Albert I Ko. Quantification of leptospira interrogans survival in soil and water microcosms. Applied and environmental microbiology, 84(13):e00507–18, 2018. [4] An OIE Collaborating Center. Infections in humans incubation period. 2003. [5] M Collares-Pereira, ML Mathias, M Santos-Reis, MG Ramalhinho, and P Duarte- Rodrigues. Rodents and leptospira transmission risk in terceira island (azores). Eu- ropean journal of epidemiology, 16:1151–1157, 2000. [6] Sophia G De Vries, Benjamin J Visser, Ingeborg M Nagel, Marga GA Goris, Rudy A Hartskeerl, and Martin P Grobusch. Leptospirosis in sub-saharan africa: a systematic review. International Journal of Infectious Diseases, 28:47–64, 2014. [7] Léa Douchet, Cyrille Goarant, Morgan Mangeas, Christophe Menkes, Soawapak Hin- joy, and Vincent Herbreteau. Unraveling the invisible leptospirosis in mainland southeast asia and its fate under climate change. Science of the Total Environment, 832:155018, 2022. [8] Francesco Drago, Giulia Ciccarese, Giulia Merlo, Ilaria Trave, Sanja Javor, Alfredo Rebora, and Aurora Parodi. Oral and cutaneous manifestations of viral and bacterial infections: Not only covid-19 disease. Clinics in dermatology, 39(3):384–404, 2021. [9] Julien Dupouey, Benôıt Faucher, Sophie Edouard, Hervé Richet, Angeli Kodjo, Michel Drancourt, and Bernard Davoust. Human leptospirosis: An emerging risk in europe? Comparative Immunology, Microbiology and Infectious Diseases, 37(2):77–83, 2014. [10] Habtamu Ayalew Engida, David Mwangi Theuri, Duncan Gathungu, John Gachohi, Haileyesus Tessema Alemneh, et al. A mathematical model analysis for the transmis- sion dynamics of leptospirosis disease in human and rodent populations. Computa- tional and Mathematical Methods in Medicine, 2022, 2022. [11] Ralph D Feigin, Donald C Anderson, and Clark W Heath. Human leptospirosis. Crc critical reviews in clinical laboratory sciences, 5(4):413–467, 1975. [12] Swamp Fever, Mud Fever, Autumn Fever, Rice-Field Fever, and Canicola Fever. Infections in humans incubation period. [13] Winston Garira, Dephney Mathebula, and Rendani Netshikweta. A mathematical modelling framework for linked within-host and between-host dynamics for infections with free-living pathogens in the environment. Mathematical biosciences, 256:58–78, 2014. REFERENCES 1657 [14] DR OTTO R GSELL. Epidemiology of the leptospiroses. In Symposium on the leptospiroses. Med. Sci. Publ. Army Med. Serv. Grad. School, number 1, pages 34–55, 1953. [15] Dhanze Himani, M Kumar Suman, BG Mane, et al. Epidemiology of leptospirosis: an indian perspective. J Foodborne Zoonotic Dis, 1(1):6–13, 2013. [16] Anthea L Katelaris, Keira Glasgow, Kerryn Lawrence, Paul Corben, Anthony Zheng, Suhasini Sumithra, John Turahui, Janet Terry, Debra Van den Berg, Daneeta Hen- nessy, et al. Investigation and response to an outbreak of leptospirosis among rasp- berry workers in australia, 2018. Zoonoses and public health, 67(1):35–43, 2020. [17] Muhammad Altaf Khan, Saeed Islam, and Sher Afzal Khan. Mathematical modeling towards the dynamical interaction of leptospirosis. Applied Mathematics & Informa- tion Sciences, 8(3):1049, 2014. [18] Atanaska Marinova-Petkova, Irene Guendel, Jonathan P Strysko, Lisa LaPlace Ekpo, Renee Galloway, Jonathan Yoder, Amy Kahler, Aileen Artus, Alex R Hoffmaster, William A Bower, et al. First reported human cases of leptospirosis in the united states virgin islands in the aftermath of hurricanes irma and maria, september–november 2017. In Open forum infectious diseases, volume 6, page ofz261. Oxford University Press US, 2019. [19] Kristin Ohlson. The soil will save us: How scientists, farmers, and foodies are healing the soil to save the planet. Rodale Books, 2014. [20] World Health Organization et al. Leptospirosis: Fact sheet. 2009. [21] Mathieu Picardeau. Virulence of the zoonotic agent of leptospirosis: still terra incog- nita? Nature Reviews Microbiology, 15(5):297–307, 2017. [22] B Pimpunchat, GC Wake, C Modchang, W Triampo, and AM Babylon. Mathematical model of leptospirosis: Linearized solutions and stability analysis. 2013. [23] S. Richard and A. Oppliger. Zoonotic occupational diseases in forestry workers - lyme borreliosis, tularemia and leptospirosis in europe. Annals of Agricultural and Environmental Medicine, 22(1), 2015. [24] Kyung-Taek Rim and Cheol-Hong Lim. Biologically hazardous agents at work and efforts to protect workers’ health: a review of recent reports. Safety and health at work, 5(2):43–52, 2014. [25] Ignacio Santecchia, Frédérique Vernel-Pauillac, Orhan Rasid, Jessica Quintin, Maria Gomes-Solecki, Ivo G Boneca, and Catherine Werts. Innate immune memory through tlr2 and nod2 contributes to the control of leptospira interrogans infection. PLoS pathogens, 15(5):e1007811, 2019. REFERENCES 1658 [26] Maria Cristina Schneider, Deise Galan Leonel, Patricia Najera Hamrick, Eduardo Pacheco de Caldas, Reina Teresa Velasquez, Fernando Antonio Mendigana Paez, Jusayma Caridad Gonzalez Arrebato, Andrea Gerger, Martha Maria Pereira, and Sylvain Aldighieri. Leptospirosis in latin america: exploring the first set of regional data. Revista Panamericana de Salud Publica, 41:e81, 2018. [27] E Sembiring. Diagnostic approach in leptospirosis patients. In IOP Conference Series: Earth and Environmental Science, volume 125, page 012089. IOP Publishing, 2018. [28] Simon Smith, Yu-Hsuan Liu, Angus Carter, Brendan J Kennedy, Alexis Dermed- goglou, Suzanne S Poulgrain, Matthew P Paavola, Tarryn L Minto, Michael Luc, and Josh Hanson. Severe leptospirosis in tropical australia: Optimising intensive care unit management to reduce mortality. PLoS neglected tropical diseases, 13(12):e0007929, 2019. [29] Wei Leong Tan, Shahrul Aiman Soelar, MA Mohd Suan, Narwani Hussin, Wee Kooi Cheah, Khebir Verasahib, and Pik Pin Goh. Leptospirosis incidence and mortality in malaysia. Southeast Asian J Trop Med Public Health, 47(3):434–40, 2016. [30] Pauline Van den Driessche and James Watmough. Reproduction numbers and sub- threshold endemic equilibria for compartmental models of disease transmission. Math- ematical biosciences, 180(1-2):29–48, 2002. [31] Anahi S Vieira, Priscila S Pinto, and Walter Lilenbaum. A systematic review of leptospirosis on wild animals in latin america. Tropical animal health and production, 50:229–238, 2018.