# I. Introduction he model uses the techniques of epidemiological models, the idea is to abstract away the particular details of an infection and express individuals as progressing through a set of states at different rates. Child mortality and morbidity is a factor that can be associated with the well-being of a population. It is also taken as one of the development indicators of health and socioeconomic status in any country (Alderman and Behrman, 2004). In order to reduce child mortality and morbidity which is one of the important Millennium goals, there is need to develop an effective and efficient model that can be used to assess the attributes that are responsible for the prevalence of the diseases in pediatrics patients that are having Lung Respiratory Tract Infections (LRTIs). In this epidemiological model, individuals transition from a Passive Immunity Infant to Susceptible state to Latent period to an Infectious one to a Recovered state at a certain rate, and become Susceptible again at a different rate. This model is called the MSEIRS model, because individuals move between them M (Passive Immunity infant), E (Latent period) S (Susceptible) and I (Infectious states) R (Recovered). The Passive Immunity for Infant -Susceptible -Latent -Infected -Recovered -Susceptible (MSEIRS) model was introduced by Kermack andMcKendrick, in 1927 (Leah Edelstein-Keshet 2005). In the model, the population is divided into three distinct groups of: the Passive Immunity for Infant (M), Latent period (E), Susceptibles (S), Infecteds (I) and Recovereds (R) where M, E, S, I and R represent the number of children in each of the groups respectively and the total population ?? = ?? + ?? + ?? + ?? + ??. The Susceptibles are those who are not infected and not immune, the Infecteds are those who are infected and can transmit the disease, and the Recovered are those who are immune to re-infection. The characteristic feature of LRTI is that immunity after infection is temporary, such that the recovered children can become susceptible again if all the risk factors are still present. # II. Mathematical Model Formulation Passive Immunity is an immunity obtained from external source: immunity from disease acquired by the transfer of antibodies from one person to another, e.g. through injections or between a mother and a fetus through the placenta looking at the case of infection spread on the population of children, there is an arrival of new susceptible population. In this type of situation, births and deaths rate must be included in the model. The above assumptions lead to the following differential equations for LRTI. An additional feature of LRTI is employed. By this Newborn babies whose mothers are immune are taken into consideration .As a result, these children are protected by the antibodies present in their mothers. Thus, group M of children who are completely protected by these antibodies are considered. The ratio of these newborn babies M is equal to the ratio of the general population that is immunized after recovering from infection. Protection reduces and these children M become susceptible at a rate . Under the above assumptions, the following are the results. ???? ???? = ?? ? (+??)??, ??(0)???? ???? = ?? ? ?????? ? ???? + ????, ??(0) ???? ???? = ?????? ? (?? + ??)??, ??(0) ???? ???? = ???? ? (?? + ??)??, ??(0) ???? ???? = ???? ? (?? + ??)??, ??(0) # III. Model Analysis a) Two Classes of Epidemiology Models To introduce the terminologies, notation, and standard results for epidemiology models, two different types of models are formulated and analyzed. They are Epidemic models and Endemic models. Epidemic model is used to describe rapid outbreaks that occur in less than a year due to the availability of some risk factors, while endemic models are used for studying diseases of longer periods, during which there is a renewal of susceptibles by births or recovery from temporary immunity. The two classic SIR models provide an intuitive basis for understanding more complex epidemiology modeling results. ?M = µS ???? = ? ?? But ?? = µ ?? + µ Thus, ? µ * ?? ? + µ = ? ? + µ Finally for, virus free equilibrium, the solution set is as follows: ??? = ?? ? + µ , ?? = ? ? + µ , ?? = 0, ?? = 0, ?? = 0 ? b) Local stability for Virus-free Equilibrium We linearize the system of equations given, using the Jacobian matrix approach to obtain: Evaluating the Jacobian matrix at the virus -free equilibrium E give We defined the characteristic polynomial equation for the J (E) solve for the eigen valves, to get: After a while, the eigen values ? 1 , i=1, 2, 3, 4, 5 are given as ??(??, ??, ??, ??, ??) ? ? ? ? ? ??? ? µ 0 0 0 0 ?? ???I ? µ 0 ????? ? 0 ???? ?µ ? ? ???? 0 0 0 0 ?µ ? ? 0 0 0 0 ? ?µ ? ?? ? ? ? ? = 0 ?? 1 = -µ ?? 2 = -µ -? ?? 3 = -?-µ ?? 4 = 1 2(? + µ) [?2µ 2 ???? ? 2???? ? ???? ? ???? ? ???? + ??] ?? 5 = ?1 2(? + µ) [2µ 2 ???? ? 2???? ? ???? ? ???? ? ???? + ??]Where ?? = ??? 2 ?? 2 ? 2?? 2 ???? + ?? 2 ?? 2 + 2???? 2 ?? ? 2???? 2 ?? ? 4???????? + 4???????? + 2?????? 2 + ?? 2 ?? 2 ? 2???? 2 ?? + 4?? 2 ???? + ?? 2 ?? 2 From the results above, ?? 1 , < 0, ?? 2 <0, ?? 3 < 0, and ?? 5 < 0 provided ?? 2 ?? 2 ? 2?? 2 ???? + ?? 2 ?? 2 + 2???? 2 ?? ? 4???????? + 4???????? + 2?????? 2 + ?? 2 ?? 2 ? 2???? 2 ?? + 4?? 2 ???? + ?? 2 ?? 2 ? 0 That is, ?? 2 ?? 2 ? ?? 2 ?? 2 + 2???? 2 ?? + 4???????? + 2?????? 2 + ?? 2 ?? 2 + 4?? 2 ???? + ?? 2 ?? 2 ? 2?? 2 ???? + 4???????? + 2???? 2 ?? So also, ?? 4 must be less than zero, i.e, ?? 4 < 0 Hence, ?? 4 = 1 2(?? + µ) (?2?? 2 ? ???? ? 2???? ? ???? ? ???? ? ???? + ??) < 0, Which implies that, ?? 2(?? + µ) < 1 2(?? + µ) (2?? 2 ? ???? ? 2???? ? ???? ? ???? ? ????) That is ?? < 2?? 2 ? ???? ? 2???? ? ???? ? ???? ? ???? Finally, the result is Ro = ?? 2µ 2 + µ? + 2µ?? + µ? + ??? + ??? < 1 A has been defined earlier above. Where R 0 is the basic reproduction number, It is imperative to note that the Basic Reproductive Number, denoted asR 0 , is an important threshold in modelling of infections diseases since it tells us if a population is at risk from a disease or not. Thus, whenever R 0 < 1the new cases (i.e. incidence) of the disease will be on the decrease and the disease will eventually be eliminated. Based on foregoing, the Basic Reproduction number (R 0 ) for our model is less than unity i.e. # Ro = ?? 2µ 2 + µ? + 2µ?? + µ? + ??? + ??? < 1 Then, I(t) decreases monotonically to zero as t??. Therefore, the virus -free equilibrium is locally stable. Local stability for endemic equilibrium we have: Solving for I yields: ?? = ?(?? + ??)(?? 3 + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?????? + ?????? + ?????? + ?????? ? ??????) ??(?? 3 + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? + ??????) Consequently, ?? = ( ? µ + ? )?? The above yields, ???(?? + ??)(?? 3 + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?????? + ?????? + ?????? ? ?????? ??(?? 3 + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? Also, ?? = ? ?? + ? ?? ? ?? ???(?? + ??)(?? 3 + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?????? + ?????? + ?????? ? ?????? ????(?? 3 + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? For mathematical acceptability, (M E, S E, E E, I E, R E ) > 0 Thus, ?? ?? = ???(?? + ??)(?? 3 + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?????? + ?????? + ?????? ? ?????? ??(?? 3 + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? > 0 Let, ?? = ?(?? + ??)(?? 3 + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?????? + ?????? + ?????? + ??????) ?? = ?????? and ?? = (?? 3 + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? + ?????? + ??????)) Hence, ?? ?? = (?? + ??)(?? ? ??) ?? * ?? > 0 = ?(?? + ??)(?? ? ??) > 0 = ?(?? + ??)?? + ??(?? + ??) > 0 = ??(?? + ??) > (?? + ??)?? Dividing through by A, we then have, ?? 0 = ?? ?? > 1 More elaborately, we have: ?? 0 = ?????? ?? 3 + ?? 2 ?? + ?? 2 ?? + ?? 2 ?? + ?????? + ?????? + ?????? + ?????? > 1 If ? 64.5, v = 36, ?= 13, ?= 91, µ =0.041 are all parameter for a period of one year, then we have the following expression: 13 * 64.5 * 91 (0.041) 3 + (0.041) 2 36 +(0.041) 2 13+(0.041) 2 91 + ( 0.041)(36)(13) + (0.041)(36)(91) + (0.041)(91)(13) + (36)(13)(91) > 1 ?? 0 = 2.944076535 > 1 If R 0 > 1 then I (t) increases and reaches its maximum and reduces as ?? 0 ? ? . When the number of children infected increases in this state, it is called the epidemic state. In the long run, the whole population become susceptible if R 0 > 1 # IV. Numerical Solution and Simulation The SEIRS model was solved numerically using Runge -Kutta method. We adopted Matlab ode45 program, which is based on an explicit Runge Kutta (4, 5) formula. It is a one-step solver used in solving a system of first -order ordinary differential equation (ODE). So, in computing ??(?? ?? ), it needs only the solution at the immediately preceding time point, ??(?? ???1 ). In general, ode45 is the best function to apply as a first try for most problems involving systems of first order ODES. Runge kutta of order four is also used in plotting the graphs; it's a powerful and popular method because of its accuracy and stability. Also, its simplicity and stability make it one of the most widely used numerical algorithms for stiff and non-stiff equations, while it converges faster than that of order two or three. These are the parameters used in plotting the graphs: although some of it changes are due to the fact that they are the major factors that are determining the situations of the environment that is, if it is of the virusfree and endemic state. In these model, we assume newborn infants of immune mothers that are protected by maternal antibodies. We then introduce a group of M of children that are born completely protected. According to the graph above, we assume the fraction of newborns that are protected is equal to the fraction of the general population that have temporary immunity after recovering from infection . The protected children become susceptible. # V. Conclusion To conclude, while this model would benefit against real world data, in its present form it has been shown to be useful in three areas: providing a systemslevel view, exposing weaknesses and dependencies and evaluating new technologies. With more data this sort of model could provide valuable insight and prediction for the entire LRTI disease. ![The differential equations represent the model which indicates the rate of change of number of individuals in each compartment with respect to time. Below is the Schematic diagram for the single age class M -Passive Immunity Infant, S-Susceptible, E -Latent period, I -Infectious, R -Recovered (MSEIRS) model for LRTI transmission (Weber et al., (2001).](image-2.png "") 1![Fig. 1: Schematic diagram for the single age class MSEIRS (M = Passive Immunity Infants) Susceptible Latent Immune Recovered Susceptible Model for LRTI transmission](image-3.png "Fig. 1 :") 1ParameterDescriptionUnitSSusceptible populationNumber/unit time?Birth rate of the children i.e. the mortality rateNumber / unit timeIInfected populationNumber/unit timeRInfected population that RecoveredNumber/unit timeMPassively immune infantsNumber/unit timeµBirth rate of the children i.e. the mortality rateNumber/unit time??rate of loss of immunityNumber/unit time?Rate of loss of infectionsNumber/unit timeßTransmission parameter (constant rate)Number/unit timeR 0Basic reproduction numberNumber/unit time?Contact numberNumber/unit time?Rate of loss of protection by maternal antibodiesNumber/unit timeThe unit time is (per year) 41ParametersVb 0b 1??????R 0MSEIRS (Virus free State)36500.14910.150.0411.81364.50.9515728172MSEIRS (Epidemic State)36200.20910.150.0411.813272.944076535 * A Childhood acute lower respiratory tract infection in Northern Nigeria: As risk factors AhmedP A YusufDawodu K 2015 * Health and Economic Growth Australian Health Ministers' Advisory Council HAlderman JRBehrman 2004 * A contribution to the Mathematical Theory of Epidemics Published WOKermack AGMckendrick DOI: 10.1098/ rspa.1927.0118 1927. August 927 * Mathematical models in Biology Proc. Roy. Soc. London A 115 2005 Leah Edelstein-Keshet * Respiratory health and disease in Europe: the new European Lung White Book RLoddenkemper GJGibson YSibille European Respiratory Journal 42 2013 * Clinical Predictors of admission in infants with acute bronchiolitis MMarlais .JEvans EAbrahamson 2011 * Development of An Auto-Immune Dendritic Cell Algorithm for Classification problems .BOlubadeji 2016 Akure, Ondo State department of computer science, Federal University of Technology PhD Thesis * Child Adolescent Health. Forthcoming. Report on the Methodology and Assumptions Used to Estimate Costs of Scaling Up Selected Child Health Interventions to 95% in Order to Reduce Under-Five Mortality 2010 WHO Geneva