Team:UCAS-China/Dynamic Inf

the Dynamic of Hp Infection

Understanding how Hp infects humans and causes the disease is the first step of figuring out a suitable therapy. Inspired by the work of Blaser and Kirschner, we establish a one-cell model that illustrates the Hp-Human interaction. The model explains the incubation of Hp and answers the question of how an asymptomatic infection turns into a suffering patient.

part 1. Dynamic of Hp-Human interaction

We modified Blaser and Kirschner’s model to make it capable to explain the ongoing process of our project. As a notation, we take subscript \(M\) to represent bacteria living in mucus and subscript \(A\) to represent bacteria adherent to mucosal epithelial cells. The variables involved in the model are listed below.

(Note: unit "b" means unit bacteria, "N" means unit nutrition, etc.)

Variable Meaning Unit
$$H_M$$ Population of HP living in the mucus b
$$H_A$$ Population of HP adherent to mucosal epithelial cells b
$$N$$ The amount of nutrients N
$$E$$ Intensity of inflammation E
$$I$$ Meaning I
$$H_M$$ Intensity of immune response I
$$N_0(t)$$ environmental nutrients N

Table 1. Variables Involved in the Model

The parameters involved are listed below.

(Note: unit "d" means simulation day, it has similar orders of magnitude with real world day but they are not the same.)

Parameter Meaning Value Unit
$$g_M$$ Nutrient allocation ratio of mucus layer 0.9 1
$$g_A$$ Nutrient distribution ratio of the attachment layer 0.1 1
$$r_H$$ Maximum growth rate of HP 10 d-1
$$N_H$$ Nutrient dependent constant of HP activity 0.5 N
$$\tau$$ Reducing parameters of HP natural clearance 0.85 1
$$\mu_M$$ Bacterial clearance rate of mucous layer 0.4 d-1
$$\mu_A$$ Bacterial clearance rate of the attachment layer 0.2 d-1
$$\alpha_H$$ Attachment rate of HP 0.4 d-1
$$A$$ Maximum attachment amount 300 b
$$\beta_H$$ Dissociation rate of HP 1 d-1
$$k_N$$ Maximum release rate of nutrition 30 N/d
$$k_E$$ Induction rate of inflammation 0.01 E/(bd)
$$k_I$$ Induction rate of immune factors 0.1 I/(Ed)
$$I_N$$ Attenuation index of immunity towards nutrition release 0.5 I
$$E_N$$ Facilitation index of immunity towards nutrition release 3.4 E
$$\xi$$ Monod coefficient 3 1
$$a_H$$ Extra nutrition for breeding of HP 0.01 N/b
$$b_H$$ Nutrition for growth of HP 0.005 N/(bd)
$$D_z$$ Diffusion coefficient 1 d-1
$$I_0$$ Maximum intensity of immune response constant 2 I
$$\eta_E$$ Maximum degradation rate of inflammation factors 2 d-1
$$\eta_I$$ Maximum degradation rate of immune factors 2 d-1
$$E_I$$ Control index of E to I 2 E
$$I_E$$ Control index of I to E 2 I
Table 2. Parameters Involved in the Model


Assuming that bacteria migrate in mucus vertically, and the adhesion rate is related to the maximum adhesion amount. We get basic equations:

$$ \begin{aligned} \dfrac{dH_M}{dt}&=increase(H_M)-reduce(H_M)-conv(H_A,H_M)\\ \dfrac{dH_A}{dt}&=increase(H_A)-reduce(H_A)+conv(H_A,H_M) \end{aligned} $$

where the growth rate can be described by Monod equations.

$$ increase(H)=r_H\dfrac{N}{N+N_H}gH $$

The reduce rate is linear, so

$$ reduce(H)=\tau\mu H $$

Suppose migration of \(M\) to \(A\) is related to the maximum adhesion amount while migration rate of \(A$\) to \(M\) is linear, we get

$$ conv(H_A,H_M)=\alpha_H\left(1-\dfrac{H_A}{A}\right)H_M-\beta_HH_A $$

Considering the interaction between HP and human body, we analyze the changes in human body.

$$ \begin{aligned} \dfrac{dN}{dt}&=inflammation(E,I)-breed(H_A,H_M)-grow(H_A+H_M)+diffuse(N_0(t)-N)\\ \dfrac{dE}{dt}&=HP\_induction(H_A+H_M)-decay(I\rightarrow E)\\ \dfrac{dI}{dt}&=E\_induction(E,I)-decay(E\rightarrow I) \end{aligned} $$

We can use a combination of Monod equation and Hill equation to describe the absorption of nutrition. Note that the inflammation-induced nutrition growth is the result of a series of complicated reactions take places in different groups of cells, it should have a high Hill number.

$$ inflammation(E,I)=k_N\dfrac{I_N}{I_N+I}\dfrac{E^\xi}{E^\xi+E_N^\xi} $$

Nutrition consumption is proportional to the breed and growth:

$$ \begin{aligned} breed(H_A,H_M)&=a_H(increase(H_M)+increase(H_A))\\&=a_Hr_H\dfrac{N}{N+N_H}(g_MH_M+g_AH_A)\\ growth(x)&=b_Hx \end{aligned} $$

Diffusion rate and E induction both satisfy heat equations:

$$ \begin{aligned} diffuse(x)&=D_zx\\ E\_induction(E,I)&=k_I(I_0-I)E \end{aligned} $$

HP induction and interacting decay both satisfy Monod equations, that is

$$ \begin{aligned} HP\_induction(x)&=k_E\dfrac{N_H}{N+N_H}x\\ decay(x\rightarrow y)&=\eta_y\dfrac{x_y}{x_y+x}y \end{aligned} $$

Eventually we get the following equations.

$$ \begin{align} \dfrac{dH_M}{dt}&=g_Mr_H\dfrac{N}{N+N_H}H_M-\tau \mu_MH_M-\alpha_H\left(1-\dfrac{H_A}{A}\right)H_M+\beta_HH_A\tag{1*}\\ \dfrac{dH_A}{dt}&=g_Ar_H\dfrac{N}{N+N_H}H_A-\tau \mu_AH_A+\alpha_H\left(1-\dfrac{H_A}{A}\right)H_M-\beta_HH_A\tag{2*}\\ \dfrac{dN}{dt}&=k_N\dfrac{I_N}{I_N+I}\dfrac{E^\xi}{E^\xi+E_N^\xi}-a_Hr_H\dfrac{N}{N+N_H}(g_MH_M+g_AH_A)-b_H(H_M+H_A)+D_Z(N_0(t)-N)\tag{3*}\\ \dfrac{dE}{dt}&=k_E\dfrac{N_H}{N+N_H}(H_M+H_A)-\eta_E\dfrac{I_E}{I_E+I}E\tag{4}\\ \dfrac{dI}{dt}&=k_I(I_0-I)E-\eta_I\dfrac{E_I}{E_I+E}I\tag{5}\\ \end{align} $$

part 2. Bistability of the System

We run simulations for Hp-Human system with the dynamics obtained.

The simulation result implies that the Hp-Human system has the bistability—with the very same set of parameters, the system has two stable states. The formation of bistability does not rely on certain parameter values and has no relation with the existence of adhered bacteria or biofilms, it results from the synergy effects in nutrition generation, which in our model is described by parameter \(\xi\). Though more process with synergy effects may form a higher level of multi-stability, the coupling effect of reactions usually would reduce the stability of other “stable” states, thus the system would perform as if it is bistable.

For the parameter values showed in Table 2, the variable values of two stable states are listed in the table below.

We name the state with a larger quantity of Hp as p-state (pathological state) and the other state as i-state (incubating state). Apart from having more Hp, p-state also has a much higher level of inflammation and immune system activation when comparing to i-state, which suggests p-state represents disease. I-state is the usual final state in the infection simulation, since the initial amount of Hp is small. However, i-state is semi-stable thus the system has a natural trend of falling to p-state. These characteristics, together with the fact that 90% of the infected people show no symptom, provide strong evidence that in most cases, Hp live and keep living in i-state and the infected people feel nothing, whereas some occasional event could push the system into the p-state leading to illness.

2.3 state transition

Finding out the factors that trigger the state transformation is undoubtedly important. For the system we examine, the change of parameters’ value can be neglected. The stochastic effect of our system is relatively low, since it is a huge system with millions of bacteria and billions even trillions of interested molecules. The most changeable variable in our system is the environmental nutrition \(N_{0}(t)\).It is closely related to ingestion and digestion. Regular meals cause a periodic change on the environmental nutrition, while a feast brings a pulse. Meanwhile, the average level of the environmental nutrition also varies, since people have different living conditions and eating habits. The simulation show that the environmental nutrition does play a dominant role in state transition. The main findings of the simulation are listed below.

(a) The baseline value of the environmental nutrition regulates the balance between i-state and p-state. With the level of baseline values getting higher, p-state gains more stability till i-state is too unstable to exist. Lower level the opposite.

(b) A pulse could trigger the transition from i-state to p-state. The transition may take 60 days. Whether the transition would take place is determined by the baseline value, and the height and width of the pulse.

(c) The periodic change caused by regular meals would not trigger the state transition. This conclusion is independent of the amplitude of fluctuation.

(a) explains the fact that infected people in developed areas usually have a higher risk of suffering critical symptoms. (b) shows the vulnerability of i-state. Even a single feast could break the balance. It is also interesting that the completion of the transition takes place such a long time after the pulse. (c) clearly suggests that periodic change would be convoluted by the dynamics so daily meals bring little risk.

To give more detail of what kind of pulse would trigger the state transition, we make another “ice-cream analysis” and the result is plotted below#. Same with the ice-cream analysis in Part 1.1, we take a square pulse of width \(d t\) and height \(\Delta N_{0}\) over the base line value \(N_{0}\). Through simulation, we find out the boundary surface between i-state and p-state in \(N_{0}-\Delta N_{0}-d t\) space. The result may help to design a special diet that could minimize the risk of Hp infection.