THE DYNAMICS OF PSITTACOSIS IN HUMAN AND POULTRY POPULATIONS: A MATHEMATICAL MODELLING PERSPECTIVE

Psittacosis is a disease in human beings that is commonly associated with pet birds such as cockatiels and parrots, and among poultry such as ducks and turkeys. This paper proposed and developed a deterministic epidemiological model that explains the transmission dynamics of Psittacosis infection in humans and poultry. The Psittacosis deterministic model was analyzed to determine positivity of the solution set, the invariant feasible region, the basic reproduction number, the disease free equilibrium points, the endemic equilibrium points and the corresponding stability of each of the equilibrium points. The basic reproduction number is calculated using the next generation matrix and it was found to be entirely dependent on the poultry population parameters. The study established that whenever R0 1, Psittacosis keeps persisting in the population. Sensitivity analysis was conducted to determine the contribution of each parameter to the basic reproduction number. The more sensitive parameters were found to be responsible for the further propagation of Psittacosis while the less sensitive parameters rarely contributed to the spread. Stability analysis of both the disease free equilibrium and the Endemic equilibrium were conducted. Lastly, numerical simulation was conducted to justify quantitative analysis of the dynamics of the transmission of Psittacosis.


INTRODUCTION
Psittacosis is a zoonotic infectious disease caused by gram-negative obligate intracellular bacteria Chlamydia psittaci. It is also known as ornithosis or parrot fever since birds are the major epidemiological reserviors. The disease has been documented in 467 species from 30 different orders of birds however, birds from the order of Galliformes such as turkeys, chicken and pheasants and those from the order of Psittaciformes such as budgerigars, lories, cockatoos, parakeets, and parots are ofen identified as the major epidemilogical reserviors [1].
Human infections are occasioned by contact with infected pet birds by inhalling the bacteria from contaminated dust from bird feathers, bird secretions, and dried-out droppings. Humanto human infections are suggested and thought to be rare but there is limitted documentation to back the suggestions [2]. Psittacosis has an incubation period of one to four weeks during which an individual develops influenza-like illness. Some of the common syptoms include headache, dry cough, muscle aches, myalgia, rigors, fever, and chills. In some cases, it causes systemic illness which leads to atypical pneumonia which can be fatal [3]. Mostly, antibiotics with intracellular actions are administered as a form of treatment.
The infective dose of the bacteria is unknown since the infected birds shed the agent intermittently or continously for weeks or months [3]. The disease infects humans across all ages and genders but is more prevalent in mid-age groups with a peak at the ages of 35 to 55. Adults in constant contact with birds have higher susceptibility such as zoos employees working with avians, poultry farm or processing plant workers, vetinary technicians, and aviary and pet shop employees [3]. Some outbreaks that have been well documented include the local cluster outbreak in the Netherlands in November 2007. The outbreaks was traced to a bird show that was held in the rural town of Weurt (village of Beuningen) [4]. In this particular case 25 positive cases were recorded leading to cancellation of other bird shows as a precautionary measure. The outbreak is summarized by the graph below Between January 12 th and 9 th April 2013 there was another outbreak in Southern Sweden which was attributed to free-living birds reserviors. During this period there was a total of 25 cases reported which was a spike from the previous years mean of 3.3 cases per annum [2]. The outbreak is illustrated below.
There was an outbreak reported in the United States between August and September 2018 in Virginia. All the hospitalized persons tested positive for C. psittaci leading to the suspension of operations at the chicken slaughter house [5]. The suspension was followed by deep cleaning of the facility and inspection to ensure safety of workers. Another, outbreak was reported at in Georgia on September 12 in chicken slaughter house leading to swift measures to contain the situation. Three of the hospitalized patients of symptoms of psittacosis tested positive for C.
New South Wales reported an outbreak in November 2014 at a veterinary school and a local equine stud. All these cases originated from exposure to an equine fetal memabrane of Mare A, which subsequently tested positive for Chlamydia psittaci. A cohort study of those exposed reveled that five were infected with psittacosis arising from the exposure [6]. In particular, this exposure was unique since there were no birds involved. As such, it was a clear inciator of colonization of Chlamydia psittaci in mammals.
Mathematical models usually explains the dynamics of the transmission of diseases and can predict the spread or die out of the infections in the system with time [7,8,9,10,11].

Model description and formulation.
1.1.1. Model Formation. In this model we consider population of turkeys and human. Each population is subdivided in to four compartments; susceptible, exposed, infected and recovered.
Susceptible humans are recruited at a rate Λ h either by birth or immigration, and their number increase from individuals that come from sub-classes of psittacosis recovered by losing their temporary immunity with rate of σ and decrease by individuals that move to exposed compartments at a rate of β h and natural death rate with rate µ h . Some exposed human population move to infected compartment at the rate of δ h and the remaining exposed human population who get the drug compartment join the recovered compartment at a rate of τ. The infected human compartment decrease both by natural and psittacosis induced death rate µ h and ω h respectively. Susceptible turkeys are recruited at a rate Λ p either by birth or immigration, and their number increases from individuals that come from sub-classes of psittacosis recovered by losing their temporary immunity with rate of γ and their number decrease by individuals that move to exposed compartments with rate of β p and natural death rate with rate µ p . The exposed population of turkeys which get drug go to the recovery by rate of α and the remaining which not get the drug with time go to the infected class. The infected population of turkeys reduces by natural death (µ p ), diseased induce rate (ω p ) and the removing parameter(ω 2 ). Total human population is given by; N(t) = S(t) + E(t) + I(t) + R(t) The dynamics of psittacosis transmission in human and turkeys population is represented in the schematic diagram as shown in Fig: 1 Based on the assumptions and interrelation between the variables and parameters in figure 1, the following system of ordinary differential equation generated. Considering the human population at any time t; The feasible solution of human population of model system in equation (1); Moreover, considering turkey population, denoted by N p ; The feasible solution of the human population of model system in equation (1) the region Feasible solution of human population of model system in equation (1); and R p of the system of the equation (1) are positive for all t > 0 [12,13,14] Proof: By applying the same approach for;

Diseases Free Equilibrium point.
Disease free equilibrium point of the system in equation (1) in the absence of rabbies infections is determined.
1.2.4. Basic reproductive number (R 0 ). This is a threshold value that governs the dynamics of rabbies. By employing the "Next Generation Matrix" [15,10,16]. Considering; This can be written as; Where V −1 is given by; Determining the product of F and V −1 ; Let K = FV −1 , the eigenvalues of FV −1 can be obtained, Hence, eigenvalues, The dorminant eigenvalue is the spectral radius (basic reproductive number).
1.2.5. Endemic Equilibrium Point. Endemic equilibrium points are steady state situations where the disease persists in the population. To determine the endemic equilibrium point we put the right side of equation (1) equal to zero.
The endemic equilibrium point of the model is written below; where; Proof: To prove local stability of disease free equilibrium, we obtained the Jacobean's matrix of the system (1) at the disease free equilibrium (DFE). Then the Jacobean matrix become Where Therefore, from the Routh-Hurwitz criterion of order two, it implies that the conditions, R 0 < 1 therefore DFE is locally asymptotically stable. The prove is completed.

Global Stability of Disease Free
Equilibrium. According to [18,19,20] to get the global stability of disease free equilibrium point of system (1) we write our system as follows: For the global stability of DFE we need to prove the following.
(1) B should be a matrix with real negative Eigen values.
(2) B 2 Should be a Metzler matrix Using system (1) together with the representation in (43) the two equations can be written as follows: Matrices B, B 1 and B 2 are order 4 × 4 matrix Using non -transmitting elements of the Jacobean matrix of system (1) and representation in (43) we get The Eigen value of matrix B is negative and the off diagonal elements of matrix B 2 are non -negative which is Metzler matrix. This proves that the DFE point of system (1) globally asymptotically stable in the region R 8 and R 0 < 1.  (1) is globally asymptotically stable whenever R 0 > 1 Proof To determine the local stability of endemic equilibrium point from the differential equation (1) first we determine the Jacobean matrix at E * .

Stability Analysis of Endemic
From the above we get From the above λ 1 = C3 it is negative and to see the remains value of λ we use Routh Hurwitz criterion. To obtain the precise number of roots with nonnegative real part, proceed as follow arrange the coefficient of polynomial and values subsequently calculated form them as below: m 0 λ n + m 1 λ n−1 + · · · + m n−1 λ + m n = 0 (61) m 0 λ n + m 1 λ n−1 + · · · + m n−1 λ + m n = 0 (80) (82) By Using the Routh Hurwitz criterion it can be seen that all the Eigen values of the characteristic equation have negative real part if and only if: This implies that the endemic equilibrium point (E * ) is local asymptotical stable.

Global Stability Analysis of Endemic Equilibrium. (E * )
Global stability means that the system will come to the equilibrium point from any possible starting point. (1) is globally asymptotically stable.
Proof. Using the Lyapunov approach in [21], global asymptotic stability of E * : By direct calculating the derivative of V along the solution of (1) we have; Where (109) disease [23,24]. We analysed the reproduction number to determine whether or not treatment of infective and mortality can lead to the effective elimination or control the psittacosis disease in the population.
We determine the most sensitive parameter by using the relation; where mi are parameter, and R 0 is the reproductive number. If P mi < 0, then mi have an effect of controlling the disease If P mi > 0, then mi have an effect on expanding the disease The results of the sensitivity analysis as shown in Table 1 indicates that some parameters are more sensitive to the reproduction number than others. The following parameters; µ p , α, ω 2 , ω p help reduce the spread of the infection. However, the following parameters λ p , β p , δ p increases the spread of the infection whenever their values increases.

NUMERICAL SIMULATION
Numerical simulations are required to study the behaviour of a systems whose mathematical model is too complex to provide analytical solution as in most non linear systems [19,25,26]. Table 2 shows the values of the parameters used in the various simulations.   2.2. Effect of contact rate on exposed birds population. In this section, as we see in

Effects of treatment rate on recovered birds population.
In this section, we simulated the effects of treatment on recovered birds population as shown in Fig: 6.

CONCLUSION
In this study we have formulated mathematical model of psittacosis. The model contains birds and human population. We defined the reproduction number in terms of the parameters and computed it by using next generation operator. The results are depending only on the parameter of birds population. It was also established that for the basic reproduction number, R 0 < 1, the disease free equilibrium point is asymptotically stable so that the disease dies out after some period of time and if R 0 > 1, the disease free equilibrium is unstable and the disease persist.
We also established that when R 0 > 1 then the endemic equilibrium is locally asymptotically stable, and unstable if R 0 < 1. The local stability theorems of disease free equilibrium and endemic equilibrium points of the model are proved by using Jacobian matrix and Routh-Hurwitz criterion. Further more global stability analysis of endemic equilibrium point was computed by using invariance principle. Sensitivity analysis of basic parameters and interpretation of the sensitivity index is also computed. Depending on the value of the sensitivity analysis of parameter the natural death rate, diseases induce rate, removing parameter and vaccination rate (the rate of exposed population of birds join to the recovered population) have an effects on controlling the psittacosis disease in the community and natural birth rate, the rate susceptible population infection by infected animal and the rate of exposed birds infected (incubation period) have an effect on expansion the psittacosis disease in the community. Moreover, numerical stimulation is performed in order to check the effect of each parameter in the expansion as well as in the controlling of psittacosis. Depending on numerical stimulation the removing rate and treatment rate are reduced (decrease) the expansion the disease and the other rate like contact rate and infective rate are increase the expansion of the disease.

ACKNOWLEDGMENT
Authors profoundly acknowledged the encouragement and support from members of department.

SOURCE OF FUNDING
There are no sources of funding for this research. Authors are solely responsible for the entire cost of this research.