MATHEMATICAL ANALYSIS OF TYPHOID FEVER TRANSMISSION DYNAMICS WITH SEASONALITY AND FEAR

With deteriorating infrastructure in many countries on the African continent and declining funding for infrastructure development, diseases such as typhoid fever are a problem. In many of these countries, the disease occurs periodically. This disease, which is caused by Salmonella Typhi bacteria, is associated with poor hygiene, poor waste disposal systems and seasonal rains. Recently in Zimbabwe, the infection has been found to exist due to dilapidated infrastructure. The existence of the infection remains a huge public health problem. In this paper, we study the typhoid fever transmission dynamics with fear in periodic environments. We formulate a nonlinear system of differential equations in which the infection rate is time dependent. The model’s steady states are determined and the stability analysis carried out. Numerical simulations are carried out to determine the impact of the vital parameters on the course of the disease. Sensitivity analysis is also done to determine parameters that influence disease progression the most. The role of fear and seasonality are discussed and the public health implications of the results are articulated.


INTRODUCTION
Typhoid Fever is a life-threatening bacterial infection caused by Salmonella Typhi [37]. The transmission mode of typhoid is identical to that of cholera-that is, direct transmission (humanhuman) and indirect transmission (environment-human). This disease adversely affects the Recticuloendothelical system, the gall bladder and the intestinal lymphoid [23]. Known estimates of the incubation period for the typhoid fever disease range from ten to fourteen days [23]. The case fatality rate of typhoid fever was 10−20% before the advent of treatment, whilst, with prompt treatment, the case fatality rate was reduced to less than 1% [13]. It was observed that the number of deaths caused by typhoid fever in the year 1990 was 181 000 [1]; in the year 2000, it was 217 000 [8]; and in the year 2013, it was 161 000 [1]. In the Democratic Republic of Congo, more than 42 000 people contracted the typhoid fever disease during the years 2004 and 2005 [43].
For a number of incurable communicable diseases, modellers typically make the assumption of homogeneous mixing of susceptible and infected individuals in a given population [11].
This assumption is reasonable for most non-lethal infections, such as influenza and chickenpox.
However, for lethal infections such as COVID-19, typhoid, cholera and HIV/AIDS this assumption is debatable. In the case of HIV/AIDS, examples of partner preferences can be seen when individuals reduce the number of sexual partners they have, and through stigmatization of those who are already infected [33,39]. During a cholera, typhoid or COVID-19 outbreak, fear drives individuals to self-isolate and improve personal hygiene in order to reduce the contact rate of these diseases [12,31,41]. Indeed, a reduced contact rate, due to fear, has a huge bearing on the dynamics of a lethal infection. The role of fear has been considered in mathematical models with interacting species, see [9,11,42]. Recently, the role of fear has been modelled for Ebola virus disease [17]. Very few mathematical models have considered fear as an essential component in human response to infection.
Mathematical models with seasonality have been considered by a number of authors. In [7], a malaria model with seasonality is considered in which a system of differential equations is analysed. A model for malaria was also considered in [38] in with climatic factors where considered to influence the biting rate. Other infections in which seasonality was considered include brucellosis [26], clostridium difficile [20], schistosomiasis [19], respiratory syncytial virus [21], buruli ulcer [3] and cholera [28]. In these models, seasonality is modelled by incorporating a trigonometric function in the force of infection. It is also common to come across mathematical models for diarrhoeal diseases such as cholera, typhoid and many others that assume a constant rate of infection. The primary reason of making such an assumption is that it makes for relatively easy analysis, in that, it produces models that contain systems of autonomous ordinary differential equations [30]. To this end, there are numerous tools for analysing systems of autonomous ordinary differential equations in the literature [15,36]. The major drawback of assuming a constant infection rate for a seasonal disease is that the accuracy of the model might be compromised, thus compromising model predictions. Empirically, it has been shown that typhoid fever is highly seasonal and it peaks during the rainy seasons [32], hence a seasonal mathematical model is befitting to study such a disease.
To the best of our knowledge, the dynamics of typhoid fever's seasonality coupled with a behavioural change in the population due to the fear of infection has not been investigated. A comparison of a seasonal mathematical model of typhoid fever and one where the assumption of seasonality is relaxed is carried out. We seek to understand the effects of seasonality on the basic reproduction number, the number of steady states, the stability of these steady states and to carry out the stability analysis of the steady states.
The paper is arranged as follows; in Section 2, we formulate and establish the basic properties of the model. The model is analysed for stability in Section 3. In Section 4, we carry out some numerical simulations. Parameter estimation and numerical results are also presented in this section. The paper is concluded in Section 5. individuals who recovered from typhoid R(t). Thus, N(t) = S(t)+I(t)+R(t). The model has an additional compartment B(t) which represents the Salmonella Typhi concentration in the environment. The dynamics of the model developed in this paper follows from the model developed by Mushanyu et al. [22].
We assume that susceptible individuals acquire typhoid fever either through person-to-person transmission or by ingesting Salmonella Typhi from contaminated aquatic reservoirs at the rates respectively. The parameter β t 1 denotes the person-to-person typhoid transmission rate of susceptibles and is defined as the product of the probability of typhoid transmission per contact and the effective contact rate for typhoid transmission to occur. The parameter β t 2 denotes the environment-to-humans per capita contact rate for susceptibles and the Salmonella Typhi in the contaminated environment and κ t denotes the half saturation constant relative to the Salmonella Typhi. Here, we assume that individuals under treatment are infectious but cannot infect susceptible individuals since they will be confined to a certain place and separated from the general population where they will be released upon successful treatment or due to mortality (natural or disease related).
Infected individuals in class I experience disease related death at a rate given by δ . Individuals in the infectious state I excrete Salmonella Typhi bacteria into the environment at rate α. Individuals in the recovered class R are temporarily immune to typhoid infection, and immunity wanes at a rate given by ρ, leading to the individuals being susceptible again.
The Salmonella Typhi bacteria population is generated at a rate g b B and its growth is enhanced by individuals in the infectious state I. We assume that the Salmonella Typhi bacteria in the environment becomes non-infectious at a rate µ b . The constant recruitment into the susceptible population is represented by Λ, while the natural death rate for the general population is represented by µ h . We assume that individuals in each compartment are indistinguishable and there is homogeneous mixing. The schematic diagram for the model to be analysed in this work is given below.
Given the schematic diagram in Fig 1 and the given model assumptions, we formulate the typhoid fever model as follows where q = µ h + δ + ε, with initial conditions with initial conditions Here κ t +B .

Non-negative Trajectories and Boundedness.
We show that all the trajectories of the dynamical system are non-negative. The approach outlined in Yang et al. [44] to show that all the solutions are bounded below by zeros is used in this case.
It is clear that Using Lemma 2 of Yang et al. [44], it follows that the trajectories of model (2) are all nonnegative. The time derivative of the human population is given by This separable differential inequality can be integrated to get the following upper bound for This implies that each class containing humans is also bounded above by Λ/µ h . Since I ≤ N ≤ Λ/µ h , the bacterial class produces the following differential inequality Note that the differential inequality (5) is a derivative of the logistic growth model with carrying On the other hand, if condition (4) is false, then B is bounded above by the constant αΛ/µ h for some t ≥ 0. For the rest of the time points in the domain of B, condition (4) is true, and hence the upper bound for B is (6). Thus, in both cases, The results on positivity and boundedness of the solutions to the typhoid model (2) can be summarized within the feasible region Ω ⊆ Ω c , where

Disease Free Equilibrium and Time Average Reproduction Number
. The disease free equilibrium for system (2) is The basic reproduction number, [R 0 ], is defined as the spectral radius of the next generation matrix [10] or given a completely susceptible population, an alternative definition of the basic reproduction number is the average number of secondary infections that arise out of an average primary case [18].

The new infections vector [F ], transmission vector [V ], and their respective Jacobians [F] and
since the time average of 1 + θ sin 2πt 365 is one. Thus the spectral radius of the matrix where [R 0 ] is the so-called time-averaged basic reproduction number for the typhoid model (2).

Local Stability
Analysis of the Disease Free Equilibrium. We begin by analysing the stability of the solutions of model (2) at the disease free equilibrium x 0 . We apply the Routh-Hurwitz criterion [14] in order to find the nature of the eigenvalues.
Theorem 2. The disease free equilibrium, x 0 , for system (2) is locally asymptotically stable Proof. The Jacobian of system (2) at the disease free equilibrium is By inspection, it is clear that the two eigenvalues, −µ h and −(µ h + ρ), of J(x 0 ) lie in the left open half plane. The properties of the remaining two eigenvalues will be obtained from the following sub-matrixJ The characteristic polynomial associated with matrixJ( The Routh-Hurwitz criterion states that the roots ofJ(x 0 ) lie in the left open half plane if and only if tr(J(x 0 )) < 0 and det(J(x 0 )) > 0. Indeed, we study the global stability of the system by casting the system into the following form where X = (S, R) and Y = (I, B). The disease free equilibrium is then written in the form The conditions that must be met in order for the system to be globally asymptotically stable are: and Ω is the region of biological significance. We establish the global stability of system (2) following Theorem (3).
Theorem 4. The disease free equilibrium of the typhoid model (2) is globally asymptotically Proof. Applying condition (H1) to the system gives The Jacobian of equation (8) is We conclude that the fixed point X * is a globally asymptotically stable equilibrium point of system (8) since the system is linear, and all the eigenvalues of the Jacobian are negative and real. Alternatively, the solution for system (8) is where {e 1 , e 2 } is the standard basis in E 2 . Thus, lim t→∞ X = X * .

Endemic Equilibrium.
The endemic equilibrium for the typhoid model (2) is given by setting the time derivative for each class to zero.
In order to show the existence of the endemic equilibrium for the typhoid model (2), we complete the square on the equation: The two roots for the equation (10) are it follows from taking the square root function on both sides of the inequality that k t |µ Hence subtracting k t (µ b − g b ) from the inequalities and then dividing by 2g b yields B (2) ≤ 0 ≤ B (1) . Clearly, B (2) must be discarded since it is negative, thus B * = B (1) . Since λ 1 , λ 2 ≥ 0, it follows that λ 1 + λ 2 ≥ 0. We conclude that the force of infection for model (1) is non-negative.

The Seasonal Typhoid
The partial derivatives of the force of infection are It is clear that the force of infection increases with the number of infected people and the concentration of bacteria. The bacterial growth rates are bounded as follows: whenever B ≥ k t /2. The first inequality shows that the bacterial growth rate increases with the number of infected individuals; the second inequality shows that in the absence of infected people, there exists a threshold, k t /2, such that if the bacteria exceeds this threshold, then the bacterial growth rate decreases with the concentration of the bacteria.
We show, geometrically, that the surface that represents the force of infection, λ 1 + λ 2 , lies below its associated tangent plane at the origin. This means that the remainder term, R 1 , from the truncated Taylor expansion of λ 1 + λ 2 when the degree equals one is non-positive. The second partial derivatives of the force of infection are

Basic Reproduction Number.
We apply the methods outlined in [35,40] to determine basic reproduction number. We show that the system (1) meets the seven assumptions in the article by Wang and Zhao [40]. Let x = (S, I, R, B) T . The disease free equilibrium is x 0 = (Λ/µ h , 0, 0, 0) T and the new infections vector and transfer vectors are as follows The Jacobians of the input rate of new infections and the transfer rate of infections are periodic k × k matrix function and let p = 1 ω ln(ρ(Φ A (ω))). Then gives a solution x(t) = e pt v(t) for some ω-periodic function v(t). Φ A (ω) denotes the monodromy matrix of system (11).
The next infection operator is defined by [40] as follows where Y (t, s) is the evolution operator for the system dy/dt = −V (t)y and φ (t) is the initial distribution function of the infected. The spectral radius of L gives the basic reproduction number R 0 = ρ(L).
In most nonlinear systems, the integral in equation (12) is intractable, thus numerical methods are used to compute R 0 . Since the basic reproduction number is the maximum eigenvalue of the operator eigenvalue problem (Lφ )t = λ φ (t), the authors of [29] constructed an eigenvaluepreserving transformation of the above operator eigenvalue problem into a matrix eigenvalue problem. The spectral radius of the eigenvalue problem is an accurate estimator for the true basic reproduction number.
whenever, (B1) The functions F (t, x) ≥ 0, V + (t, x) ≥ 0, and V − (t, x) ≥ 0 are continuous, continuously differentiable with respect to x on R × R 4 + , and each have a period of 365 with respect to t.
We claim that, for model (1), f conditionally satisfies (B1) to (B6). By inspection, we can see that for each i = 1, . . . , 4, the functions F i (t, x), V + i (t, x), and V − i (t, x) are non-negative, continuous on R × R 4 + , continuously differentiable with respect to x, and 365-periodic in t. If x)). The initial value problem, z = Mz, z(s, s) = I 2×2 , can be written component wise to produce the system The solution and the monodromy matrices to the initial value problem above are, respectively, The spectral radius of the monodromy matrix (13) at t = 365 is The initial value problem Y = −V (t)Y, Y(s, s) = I 2×2 , can be written component wise as Thus a solution to the differential equation is The monodromy matrix of differential equation (14)) is Thus, the spectral radius is ρ(Φ −V (365)) = max{exp(−365q), exp(365(g b − µ b ))}. It is clear to see that ρ(Φ −V (365)) < 1 whenever g b < µ b . We have proven that assumptions (A1)-(A7) of Wang [40] hold for model (1) Corollary 5.1. The following statements are valid for model (1) if g b < µ b : (IV) The disease free equilibrium, x 0 , is locally asymptotically stable if R 0 < 1, and unstable Proof. Since model (1) satisfies conditions (A1)-(A7) of Wang [40] whenever g b < µ b , it follows from Theorem 2.2 of [40] that g b < µ b implies that conditions (I) − (IV ) hold.

Global Stability of the Disease Free Equilibrium.
In this subsection, we prove that the disease free equilibrium of model (1) is globally asymptotically stable. We begin with the following Theorem. Limiting our focus to the two infected classes, I and B, the Jacobian of system (1) is Since R 0 < 1 ⇐⇒ ρ(Φ f (365)) < 1 ⇐⇒ b < 0, it follows that R 0 < 1 implies lim t→∞ e bt v(t) = [0, 0] T . From applying the Taylor expansion to first order and using the R 1 ≤ 0, we obtain the following differential inequality d dt It follows from Theorem (6)  Theorem 7. If R 0 < 1, the following statements are valid for model (1): Proof.
a) We want to show that for any υ > 0, there exists γ ∈ R + , such that if t > γ, then R < υ.
We thus have the following result.

NUMERICAL RESULTS
In this section, simulation results are carried out through MATLAB. We hypothetically choose the following initial conditions and the parameter values in Table (1   the method outlined in [29], we numerically computed the basic reproduction number when ξ = 0. Since the seasonal period of the disease in 365 days, the rate of exponential decay in all expressions containing the period will be fast. This then means that iterations exceeding five in such terms will achieve minimal improvement in the accuracy. It is for this reason that we have set M = 10 and n = 100 as defined in [29].  3.3. Modelling the role of fear. We now consider the potential impact of fear on the model with and without seasonality. We use the logarithmic scale for clarity of presentation of results.
The manner in which fear affects the prevalence of typhoid is shown in Figure 5.

DISCUSSION AND CONCLUSION
Seasonality is a common phenomenon in bacterial infections such as cholera and typhoid in which the diseases are more prevalent in summer than in winter. The spread of these diseases is further compounded by poor hygiene and maintenance of sewage disposal infrastructure. The mechanisms that drive seasonality in typhoid fever are mainly driven by rainfall patterns and poor sewage disposals system especially during summer. While some work has recently been done in [32]. The role of fear, which impacts the rate of infection was not considered. Fear of infection has the propensity to reduce the infection rate and in this paper, our interest was on investigating how the seasonality dynamics of typhoid fever are impacted by fear. In this paper, we propose and analyze a model of typhoid fever that follows the work presented in [22] in the presence of fear. The model is motivated by the fact that in the presence of poor health infrastructure, behaviour changes become critical in the reducing the rate of typhoid fever infection in seasonally fluctuating environments. In addition, we are also motivated by scenarios in many countries in Sub-Saharan Africa, such as Zimbabwe, where typhoid fever remains a problem especially in Summer. The stability of the steady states is carried out and we noted that the disease free equilibrium is globally stable when the basic reproduction number is less than unit. The existence of the endemic equilibrium is also discussed.
Numerical simulations are carried out following some hypothetical initial conditions and some chosen parameter values from the literature. Sensitivity analysis is also carried out using the Latin hyper cube sampling technique and the model is sensitive to the addition of susceptible individuals and the natural mortality rate of the human population.
It is important to note that the model presented in this paper has a number of limitations, as is the case with all mathematics models in which various assumptions are used in the construction of the models. The model is not validated by data. In the presence of data (which was not readily available in this case) the model would have been more robust, and in this case remains a theoretical model. This forms the basis of our future work. Despite this short coming, the model remains of great interest in the investigation of the role of human behaviour, such as fear in bacterial infections.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.