Microsoft Word - 792-Article Text-3045-1-18-20200630.doc Adv Syst Sci Appl 2020; 02; 20-31 Published online at https://ijassa.ipu.ru. Global Stability Analysis of Typhoid Fever Model Olumuyiwa James Peter1*, Adebisi Ajimot Folasade2, Michael Oyelami Ajisope3, Fidelis Odedishemi Ajibade4, Adesoye Idowu Abioye5, Festus Abiodun Oguntolu6 1,5 Department of Mathematics, University of Ilorin, Ilorin, Nigeria 2 Department of Mathematics, Osun State University, Oshogbo, Osun State, Nigeria 3Department of Mathematics, Federal University Oye-Ekiti, Oye-Ekiti, Nigeria 4Department of Civil Engineering, Federal University of Technology, Akure, Nigeria 6Department of Mathematics, Federal University of Technology, Minna, Niger State, Nigeria E-mail: peterjames4real@gmail.com Abstract: We analyze with four compartments a deterministic nonlinear mathematical model of typhoid fever transmission dynamics. Using the Lipchitz condition, we verified the existence and uniqueness of the model solutions to establish the validity of the model and derive the equilibria states of the model that is, disease-free equilibrium (DFE) and endemic equilibrium (EE). The computed basic reproductive number R0 was used to establish that the disease-free equilibrium is globally asymptotically stable when its numerical value is less than one, the disease will be under control. In addition, the Lyapunov function was applied to investigate the stability property for the (DFE). The model was numerically simulated to validate the results of the analysis. Keywords: typhoid fever, equilibria, stability, nonlinear mathematical model 1. INTRODUCTION Typhoid fever is an infection caused by Salmonella typhi bacteria. Typhoid is usually triggered by the ingestion of food or water contaminated with feces or urine of infected individuals and is, therefore, a typical illness in regions with poor sanitation. (Brooks [1]; Roumagnac et al., [2]). In developing economies, typhoid fever outbreaks occur from time to time in overwhelmed areas and refugee camps with high population density. The disease causes high morbidity in children below ten years of age, with at least seventeen million new cases globally and nearly 600 000 deaths annually. The disease is not rampant in North America: In the US, an estimated four hundred cases are reported every year; seventy percent of the cases are traced to those who return from endemic regions. The mortality of typhoid fever is ten percent, but with adequate treatment, it can be limited to one percent. (Lin et al., [3]; Sinha et al., [4]; Hyman, [5]). Intestinal fever treatment is anchored on the blood culture condition of the patients. If the species is sensitive, the oral antibiotic is used. Typhoid fever is becoming an increasingly common illness worldwide, making antibiotic treatment more expensive and harder. Enteric fever signs vary and are similar to the signs of different microbial infections. Symptoms of typhoid fever are as follows: variable level of high fever in 75 cases, body aches and muscle pain, chills, shriveled loss of appetite, abdominal pain in 20 to 40 cases, nose bleeds, headache, dizziness, rashes on the skin, bowel constipation or looseness, weakness and fatigue, sore throat and cough (Lifshitz [6]). Some mathematical models have been formulated on the transmission of typhoid fever. (Adetunde [7]; Lauria et al., [8]; * Corresponding author: peterjames4real@gmail.com GLOBAL STABILITY ANALYSIS OF TYPHOID FEVER MODEL Copyright ©2020 ASSA. Adv. in Systems Science and Appl. (2020) 21 Kalajdzievska [9]; Mushayabasa [10]; Cvjetanovic et al.,[11]; Moffact [12]; Pitzer et al., [13]; Date et al.,[14]; Muhammad, et al.,[15]; Watson & Edmunds [16]; Nthiiri [17]; Moatlhod & Gosaamang [18]; Mushayabasa [19]; Peter & Ibrahim [20]; Peter et al.,[21]; Peter and Ibrahim [22]). The aim of this study is to extend and complement previous works by formulating a model that captures the following controls; vaccination and education. 2. MATERIALS AND METHODS The model comprises four compartments: Susceptible class; represents the proportion of those who are prone to typhoid. Infected class; represents the population of individuals who have been infected with typhoid fever and capable of transmitting the infection to susceptible populations through interaction. Carrier-class; represents the population of individuals who have been infected with typhoid fever but without signs of infectiousness in them. Recovered compartment; represent the number of individuals infected with typhoid fever but are now cured of typhoid as a result of treatment. Recruitment into susceptible populations is by birth or immigration at the rate . It is assumed that a certain proportion in the susceptible class moved to the carrier infectious class at rate while the complement moved to the infectious compartment. We also assume that the rate of disease transmission of carrier individuals will be higher than the disease transmission rate of infected individuals this is because they are more likely to be unaware of their infection. Carrier's disease symptom is noticeable at the rate . Contagious individuals may receive treatment and recuperate at the rate . Susceptible individuals receive vaccination to protect themselves from contracting the disease at the rate . is an educational parameter that serves as the limitation for carriers and indicative persons from transmitting typhoid fever. This parameter lies within the range . When it implies that educational campaign not in position so that the vulnerable population are oblivious of typhoid fever and when it denotes that all vulnerable persons are well informed of the causes of typhoid fever, that is, they explicitly understand what brings about the diseases, how the disease is being transmitted and how to safeguard themselves from being infected by the disease. Figure 1. Pictorial Description of the Model q r r-1 b g a Y f-1 10 ££ f 0=f 1=f O. J. PETER, A. F. ADEBISI, M. O. AJISOPE, F.O. AJIBADE, A. I. ABIOYE Copyright ©2020 ASSA Adv. in Systems Science and Appl. (2020) 22 (1) Where (2) Table 1. Variables and parameters interpretation Variables Description vulnerable populations carrier contagious populations contagious populations recovered populations Parameters Interpretation recruitment rate in vulnerable class mortality rate of vulnerable class mortality rate for carrier infected class mortality rate for infected class mortality rate for recovered class the rate at which individual carriers develops symptoms parameter for education rate of vaccination the probability of newly infected persons becoming asympt omatic or carrier rate of transmission of infection for carrier individuals rate of transmission of infection for infectious individuals force of disease infection rate at which individual in the infectious class recovered ï ï ï ï þ ïï ï ï ý ü -+ +--+-- ---- ---- RIS dt dR IIS dt dI IIS dt dIc SSS dt dS c cc 4 3 2 1 = )()(1)(1)(1= )(1)(1= )(1= µdy dµfaflr faµfrl yflµq IIc gbl += ï ï ï ï þ ïï ï ï ý ü -+ +--++-- ----+ --+-- RIS dt dR IIIIS dt dI IIIIS dt dIc SIISS dt dS cc ccc c 4 3 2 1 = )()(1)())(1(1= )(1))(1(= ))(1(= µdy dµfagbfr faµfgbr yfgbµq ( )S t ( )Ic t )(tI )(tR q 1µ 2µ 3µ 4µ a f y r b g l d GLOBAL STABILITY ANALYSIS OF TYPHOID FEVER MODEL Copyright ©2020 ASSA. Adv. in Systems Science and Appl. (2020) 23 3. SOLUTION OF THE MODEL 3.1. Existence and Uniqueness of the Model The validity of any mathematical model is a function of the solution of the model, provided the solution is unique. We verify the singularity of the solution of the model in equation (2) via the popular Lipchitz condition. Given the system of equation (2) be as follows (3) Theorem 3.1 Assuming the region is denoted by (4) And suppose that R’ meets the Lipchitz condition On each occasion, the pairs and , where is a positive constant. Hence, there is a constant such that there exists a unique continuous vector solution of the system in the interval . It is vital to mention that the requirement is fulfilled by the condition that is continuous and bounded in . Considering the model equation (2), we are of the interest in the region We look for the solution that is bounded in the region and whose partial derivatives satisfy where and are positive constants. Theorem 3.2 Let J represent the region then equation (2) has a unique solution if it is established that are continuous and bounded in R for , For , ï ï ï þ ïï ï ý ü -+ +--++-- ----+ --+-- RISJ IIIISJ IIIISJ SIISSJ cc ccc c 44 33 22 11 = )()(1)())(1(1= )(1))(1(= ))(1(= µdy dµfagbfr faµfgbr yfgbµq  ),........,(=),........,,(=1,, 20102100 noon yyyyyyyyyyacc £-£- 2121 ),(),( yykycfycf -£- ( )1, yc ( ) Jyc Î2, 0³d )(cy d£- 0cc ..1,2,=,, ji x f j i ¶ ¶  .0 ££a 0££ af a d 1,2,3,4=,, ji x j j i ¶ ¶ 1J ¥--+-- ¶ ¶ <))(1(= 1 1 yfgbµ II S j c ¥-- ¶ ¶ <)(1=1 fbS I j c ¥-- ¶ ¶ <)(1=1 fgS I j ¥ ¶ ¶ <0=1 R j 2J ¥-+ ¶ ¶ <))(1(=2 fgbr II S j c ¥---- ¶ ¶ <)(1)(1= 2 2 faµfbrS I j c O. J. PETER, A. F. ADEBISI, M. O. AJISOPE, F.O. AJIBADE, A. I. ABIOYE Copyright ©2020 ASSA Adv. in Systems Science and Appl. (2020) 24 , For , , For , , , The partial derivatives exist, continuous and bounded, so the model has a unique solution that completes the proof of theorem 3.1. 3.2. Feasible Region and Equilibrium From equation (2) we have that and thus, along with each solution. Also from (2), we see that Where . Therefore, We have omitted R because R does not appear in other equations. This reveals that the model can be studied in the feasible region; . is positively invariant in relation to (2). Once the dynamics are understood, those of R can then be determined from the equation . The first stage of our analysis is to find the disease-free equilibrium states from the equations by setting the right-hand side to zero i.e., The model always has a disease-free- equilibrium . And the endemic equilibrium satisfies From the equilibrium equations we can show that a unique exist with Where ¥- ¶ ¶ <)(1=2 fgrS I j ¥ ¶ ¶ <0=2 R j 3J ¥+-- ¶ ¶ <))()(1(1=3 II S j c gbfr ¥-+-- ¶ ¶ <)(1))(1(1=3 fagfr S I j c ¥+--- ¶ ¶ <)())(1(1= 3 3 dµgfr S I j ¥ ¶ ¶ <0=3 R j 4J ¥ ¶ ¶ <=4 y S j ¥ ¶ ¶ <0=4 cI j ¥ ¶ ¶ <=4 d I j ¥- ¶ ¶ <= 4 4 µ R j S dt dS )( 1µyq +-£ 1 )(suplim µy q + £ ¥® tS t NcRcIcIcSc dt dN c -£----£ qq 4321 }{ 4321 ,,.min ccccc = c tN t q £ ¥® )(suplim î í ì þ ý ü £++ + £ÂÎ=G + c IISSIIS cc q µy q ,:),,( 1 3 G ),,( IIS c RISR 4 ' µdy --= ),,( IIS c 0=))(1(1 SIISS c yfgbµq --+-- 0=)(1))(1( 2 ccc IIIIS faµfgbr ----+ 0=)()(1)())(1(1 3 IIIIS cc dµfagbfr +--++-- ú û ù ê ë é + ,0,0= 1 0 µy qA ),,( **** IISA c= .0,, *** >IIS c *A )( ))(( = 223 23* gµrgµrbµbdrag aµdµ +-++ ++ kk k S )(1= f-k GLOBAL STABILITY ANALYSIS OF TYPHOID FEVER MODEL Copyright ©2020 ASSA. Adv. in Systems Science and Appl. (2020) 25 For to exist in the feasible region , the necessary and sufficient condition requires , or equivalently, Define (4) Then a threshold parameter determines the actual number of equilibria. In the next section, we shall explain the basic reproduction number. Proposal 3.1 If then is the only equilibrium in ; if so, two equilibria exist, , and a unique endemic equilibrium, 3.3. Basic Reproduction Number The basic reproductive number measures the number of secondary cases that the bacterium is able to introduce into the whole population of fully susceptible individuals in a stable demographic state (Kalajdzievska [9]). The basic number of reproduction is a significant quantity in epidemiology since it sets the threshold for the analysis of an outbreak of disease and the evaluation of its control strategies. The value of the reproductive number, therefore, indicates whether a disease becomes endemic or dies out in a population. If , it indicates that each contagious person will not cause an infection and therefore, the disease will always disappear but when the basic reproduction number is greater than one i.e., each contagious person will cause at least one secondary infection which will subject the entire population to the attack of the disease. It is obtained by taking the largest eigenvalue(spectral radius) of the matrix Where is the rate of new infections in infectious class, is the transfer of persons out of the compartment by another means, is the disease-free equilibrium. Applying the next generation matrix approach, the basic reproduction number for the model is given as (5) In equation (5), the expression in the big square brackets is the per capita mean number of secondary infections. This expression is multiplied by , the number of susceptible individuals in the absence of infection to come about the basic reproduction number 3.4. Disease-Free Equilibrium’s Stability In order to check the local stability in the absence of infection A0, the Jacobian matrix will be evaluated at *A G 1 *0 µy q + £< S .1 )( * 1 ³ + Sµy q [ ] )])([( )1()(1 231 23 1 *0 k kk S R aµdµyµ rgµdµbragq µy q +++ -+++ = + = 10 R 0A *A 10 R 1 0 )()(= - ú ú û ù ê ê ë é ¶ ¶ ú ú û ù ê ê ë é ¶ ¶ j oi j oi x xV x xFR iF iV ox ú û ù ê ë é ++ - + + + +++ = ))(( )1( )())(( 23 2 2231 0 kkk kkR aµdµ rgµ aµ br aµdµ ag yµ q yµ q +1 oR ú û ù ê ë é + ,0,0= 1 0 µy qA O. J. PETER, A. F. ADEBISI, M. O. AJISOPE, F.O. AJIBADE, A. I. ABIOYE Copyright ©2020 ASSA Adv. in Systems Science and Appl. (2020) 26 The following stability results indicate that is a sharp threshold. Proposal 3.2 A0 locally asymptotically stable if and is unstable if . Proof One eigenvalue of is < 0. The other two eigenvalues are matrix. We want to show that whenever then the Routh-Hurwitz conditions hold, that is, and . Going by the assumption that and Therefore, and This shows that and the whenever . 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